Dear Juan,

You just reformulated  Russell's paradox (also known as Russell's antinomy). 

Dear Ismat,

There are opinions different from yours. I know several solutions for Russell paradox. For example, negating the actual infinity existence can be solved. However, the solution widely accepted consists of negating the existence of self-contained sets. You can see in this web

http://www.math.ups.edu/~bryans/Current/Journal_Spring_1999/JEarly_232_S99.html

one of them based upon the rejection of the property S∈S.

It is just this interpretation what makes my paradox essentially different. We cannot negate the existence of self-referential definitions. Consider that every one-solution equation is a self-referential definition for its solution. For instance, the equation ax+b=0 can be written as

x= (a+1)x+b

which is a self-referential definition for the number x= -b/a. Negating self-referential definition existence is the negation of almost all math.

Nevertheless, my paradox can be also solved negating actual infinity existence. I cannot see other way, and this is why I would like to know whether there is some different solution.

Dear Juan

If you may formulate the problem easier, we may code it and let the computer try to find a solution.

The solution is that a definition that is used to define itself in fact by definition is no definition (the defined is not within the definition), that is not well defined . That is, we must change the erroneous expression " Def: = " The SET K of all non -self -referential definitions "" for the more general term " Ref := " The SET K of all non -self -referential referential."" because you can make a reference to some concept using the concept or not use it (in which case it is a definition ). So in fact, " Ref : = " The SET K of all non -self -referential referential " Ref : ="The SET K of all definitions "". And what I stated at the beginning is not paradoxical since that the definition of definition in fact is the set of all the definitions and therefore no self-reference because it will not use a superset of itself but it use an element, a definition. And then, if a thing is self-reference then is not within the set of definitions, but it is in the superset of the references (such as self-reference); and one thing that is not self-reference is a reference that also belongs to set of the definitions. So "Ref:=" The SET K of all non-self-referential referential " Ref: ="The SET K of all definitions "" is a reference (self-reference and not definition) for definition.

Dear all,

I have said that a solution can be obtained, rejecting the "actual infinity existence". Recall the axiom stating actual infinity.

AX: Every mathematical construction can form an actual and completed totality.

The solution exposed by Rodriguez rejects AX implicitly. I describe a solution based upon this rejection.

Let H1 be a definition set. Now, we can state the following definition:

Def1 = "The set of non-self-referential definitions lying in H1".

Of course, Def1 does not belong to H1, and it is not self-referential; otherwise leads to a contradiction. Now, repeat the process with H2=H1 ⋃ {Def1} obtaining a new definition Def2. If we assume that every infinity is potential, the process never ends. By contrast, assuming AX, the construction process can be completed, obtaining the set H of all non-self-referential definitions, the definition of which is paradoxical.

Accordingly, this solution, like the Rodriguez's one, only can be stated after rejecting axiom AX, that is, the actual infinity existence. My question consists of knowing whether there is a solution compatible with actual infinity concept. I do not think that there is such a solution, and this is my challenge.

Some time ago, some russian contributors in RG refused actual infinity existence.

To know about this topic, see this website:

https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory

But the usual mathematical literature assumes actual infinity.

1. Russel's paradox has nothing to do with infinity. In fact, if you translate the statement into formal logic, you  see that it always evaluates as false, regardless of the interpretation of P(y,x) (e,g: "person y is shaved by person x", which gives the barber paradox).

(exists x)  (all y) ( not P(y,y)    P(y,x) ) .

In formal logic, such statements are called a contradiction. It is like talking about a circle that is square or a green pencil that is red.

2. Juan-Esteban seems to regard a definition as a statement which is satisfied by one and only one object (then the definition defines that object). If there is none, or if there are two or more objects satisfying the statement, or if the potential subjects of the statement are not specified, then technically it is not a definition, but drifting sands of meaningless verbosity.

Marcel,

A paradox either leads to a contradiction or it is not a paradox.

As I have said before, Russell paradox can be solved rejecting the self-contained set concept. Such a rejection implies the existence of an "infinite" class of sets. See how:

Let H1 denote any set family. The statement "the set of all non-self contained members of H1" defines a set S1. If the existence of self-contained sets are rejected, then S1 cannot be a member of H1. Thus, H2 = H1 U {S1} is a proper superset of H1. Now we can state the definition: S2 = "The set of non-self-contained members of H2". Analogously S2 cannot belong to H2, and so on. The process is infinite unless we accept that for some n, Sn is a member of Hn, hence it is a self-contained set.

If we assume that every infinity is potential, this creation process never ends. By contrast, if the process can be completed because we assume actual infinity existence, then the complete process leads to the Russell paradox. Accordingly, such a paradox can be avoided or solved, rejecting actual infinity existence.

You can see a similar solution in

https://www.academia.edu/9817490/Solution_to_the_Russells_Paradox

In the paper linked above, it is introduced the concept of time to extend the set collection in an endless process. Thus, the infinity is potential, because the universe at the present time is extended in the future and the process cannot be completed unless time stops.

Juan-Esteban,

That's a nice way to avoid the so-called Russell paradox at any finite stage of the process. It feels as if you avoid the paradox by rejecting infinity. If you read my post carefully, you will see that the problem is solved in a more fundamental way with or without  considerations of infinite sets. Formal logic tells you that there cannot be x such that

(all y) ( not P(y,y) P(y,x) ),

whatever you mean by P(y,x) and whatever your universe of discourse may be. If you take the "barber definition" (anyone  shaving all those people not shaving themselves), even in an imaginary infinite society of humans, it turns out that there is no such barber. In fact, it is not a definition because it deals with nothing. Let me illustrate this with a more obvious contradiction:

Remarkable_Weather := weather with rain and yet without rain.

Such a "definition" is verbosity with (literally) no subject, hence with no meaning.

You said: A paradox either leads to a contradiction or it is not a paradox. With your quoted statement at hand, my "claim" that there is Remarkable_Weather deserves being called a paradox (it leads to a contradiction, even stronger: it is a contradiction). It's too cheap, don't you think?

I prefer Webster's definition of "paradox":

A tenet or proposition contrary to received opinion; an

assertion or sentiment seemingly contradictory, or opposed to

common sense; that which in appearance or terms is absurd,

but yet may be true in fact.

The only difference between having a Remarkable_Weather and the Russell paradox is, that the latter involves a slightly more hidden logical contradiction.

   A paradox is (my definition): " a true statement that appears false, or a false statement that appears true." It usually has two kinds of solutions: one correct, but apparently incorrect, and one incorrect, but apparently correct.  

In other words:

A conflict between what is and what seems to be".

The topic of this thread I have open  neither is the paradox definition nor the Russell's one. The topic of this thread, like Russell's paradox, satisfes this definition: A statement P is paradoxical provided that P leads to ~P and vice versa.

Nevertheless, there are some paradoxes, as the Bertrand's one, the nature of which consits of leading to some contradiction using a proper method.

to Juan-Esteban,

There is no problem with self-referential definitions as long as they don't lead to a logical contradiction. All dictionaries are made of words defined by other words.

There is a problem with a statement like n := (n∈ℕ)⋀(n = n⁴)⋀(n > 0) . Il should be written n := {(m∈ℕ)⋀(m = m⁴)⋀(m > 0)} as we don't know, a priori, if there is one, many or no object m that have the required properties. We see that this should be a subset of N. ( n  on the right side is a mute variable used to express properties and is conceptually different from the object it is supposed to define. That is why it can be replaced by any representamen. So the statement is not self-referential. )

If we consider the same type of statement, n := (m∈ℕ)⋀(m = m⁴), n would not be uniquely defined unless it is a set. So this syntax cannot define a set in one case and a number in another.

So n (on the left side) is a set because n could not be more then one thing as this would not be a definition.

Example 2 does define the number 3 because it is the only objet that has the stated properties. But for the same reason, it is not self-referential because it expresses the properties that an hypothetical objet should have to be identified to Def.

As for the third one, this is a semantic question because the statement is of type impredicative. http://plato.stanford.edu/entries/self-reference/ (find "heterological").

"The set K of all non-self-referential definitions." is supposed to define a set and not a definition. So the "definition" does not refer to itself. I must say that I did not find any other source with the " or a set containing it." variation and I consider only as self-referential something that refers to itself and not to a container.

A statement must have a unique truth value. See also Gödel.

Can you give a source for your axiom: Every mathematical construction can form an actual and completed totality?

As for actual infinity, what about limits of convergent sums or the set of natural numbers itself? Non-existence of a set containing all sets and itself does not negate these occurrences.

Remember also that F-->T is true !

1. There is indeed a small difference between Russel's paradox and the problem formulated in the Question: this one deals not with one but two types of subjects: definitions and sets. Yet both "paradoxes" have a quite similar structure and rest upon a weary description.

2. Bertrand's paradox consists in defining different probability spaces to interpret a phenomenon. Knowing this, the "paradox" is not really surprising: when  looked at from different points of view, an object may have different apparent size.

There are simpler and equally striking examples to be found in combinatorics: the so-called birthday paradox  gives the probability that among n random humans there are two with the same birthday.The "paradox" lies in the fact that this probability is much larger than most people naively think. The probability space uses the set of all functions { persons } --> {days of a year }. In addition to this, there is also a Bertrand-type of paradox occurring when (like a psychopath) one considers all humans as indifferent. Then the most appropriate probability space involves number partitions with a somewhat different result. For a skew interpretation, if all people get dressed in a burka that makes them unrecognizable,  the birthday probability suddenly changes...

I wish to note that all these "paradoxes" strictly follow Webster's definition of "paradox" and none of them leads to a genuine contradiction. In general, humans are not good at understanding probability, and many people do not realize that probability involves a choice of a probability space.

The first part of the Question is: "Do you like paradoxes?" As all so-called paradoxes are either apparent contradictions or weirdly formulated situations, my answer is "I appreciate the ingenious setup". It is like appreciating a magician's show while knowing it is all tricks and nonsense.

Here's another example from combinatorics: consider the hexadecimal digits 0,1,2,...,9,a,b,c,d,e,f to build numbers in the range 0 ...264 -1. They can be written in hexadecimal as 16-digit numbers (preceding smaller numbers with "0"s to make them 16-digit. Take a random example, e.g.x:= e39ecb49cf294ede and list the amounts of the occurring digits: four e's, one 3, three 9's, two c's, two 4's, one f, one 2, one d. Such multiplicities are arranged in increasing order and the involved digits are forgotten:

1,1,1,1,1,2,2,3,4

This sequence of positive numbers sums up to 16 and is called a partition of 16. It is called the type of the number x. A more handsome notation uses subscripts like this: {15.22,3,4}

The (counting) probability for a 16-digit hexadecimal number to have a given type can be computed with a formula. For instance, the most probable type of a number appears to be {17,23,3} with a probability of 13.6243 percent. With Maple or Mathematica, you can even make up a complete list of 231 possible types and their frequency.

Now you can perform an experiment with a 16 by 16 matrix filled with hexadecimal digits, each one occurring 16 times, distributed in whatever way you want. Such arrangements are known as equi-n-squares. You must randomly draw a 16-digit number by drawing randomly 16 different positions and writing down the digits in order. You can obtain any 16-digit number in this way. If you let the computer do this in huge amounts, you can make up a statistics. My prediction is: you will confirm that the type {17,23,3} occurs most frequently, but its probability seems rather 14.8601 percent. You will also notice that the order of frequency is not quite as the theory predicts.

It is not an error (unless you have a bad random generator) and it is not a contradiction. It is a Bertrand-type paradox with a simple explanation. My favorite formulation of the paradox is this: random numbers "drawn in nature" have a slightly different distribution than numbers "drawn from a can".

Marcel,

The mathematical literature contains a lot of paradoxes that can be solved easily. Most of them contains self-referential definitions. A self-referential definition contains an underlying equation that need not have solution. If it is the case, the paradox is solved, simply, by proving that the underlying equation is unsolvable. It is sufficient to search with google to find and endless class of such a paradoxes. For instance the Barr's one.

If you are interested, you can open a new thread devoted to this topic. I appreciate that the messages in this thread are related to the paradox I have stated.

Marcel, I would be interested to discuss your equi-n-square situation, but as Juan-Esteban asked, you could start a new thread to help him narrow the scope of this one to his question.

Juan-Esteban,

 Def := "The set K of all non-self-referential definitions." (quote)

You assign the name Def to the sentence which intends to define a set K.

When I look at your example 1:

n := (n∈ℕ)⋀(n = n⁴)⋀(n > 0) (sloppy notation, but let's not quarrel about this)

you assign the name of the defined thing to the definition. (I wouldn't do that but, again, I can live with it). Continuing your notation in a consequent way, the opening line of this post will be written as

K := "The set of all non-self-referential definitions." (*)

Aside of pertinent remarks by other contributors (e.g., what makes you think that any mental collecting effort effectively produces a set?) the symbol K now denotes the defined set and serves as an abbreviation of the definition (*). Could it be that the paradox you hope to generate, is based on a simultaneous assignment of two incompatible roles to K (or Def, if you insist)?

Marcel,

The concept of self-referential definition is widely used in mathematical literature as that definition containing occurrences of the defined object. In my paradox, I have extended this definition considering also to be self-referential, provided that contains occurrences of any set containing the defined object. Thus, the defined object need not be explicitly contained in its definition. You cited the first example, but the paradox fit into the extension I have performed of this concept and I have illustrated in the second example:

Example 2) Def := "The member of ℕ which is the smaller odd prime."

The former example does not contain any occurrence of the defined object, but a set containing it. Your argumentation is biased because you do not take into account the concept extension I have stated. If you are analysing my claims, you must handle the definition I have stated, and it does not matter if you like the underlying notations or definitions.

Bes regards.

It is not a matter of (dis)liking someone's notation or terminology; in fact, I am fairly liberal in these matters. But I am far less liberal if it comes to functionality and meaning. Mathematical communication must be clear and univocal. It should not omit essential ingredients nor should it suggest properties that are not involved.

The original example 1) suggests self-reference by repeating the name of the defined. Christian  already noticed that example 1 should be rewritten as

n := {(m∈ℕ)⋀(m = m⁴)⋀(m > 0)},

which is not self-referential, and to which I add the objection that the defined is taken as a name of the definition (with braces {} omitted again), which is not univocal.

Considering example 2 as an extension of this concept of definition sounds bizarre as I  would rather call it a correction to the strange format of example 1. (It points once more at a confusion between the defined and the definition.) In this example 2, the defined subject "3" remains unmentioned, hiding the fact that this definition is, by standard norms, not self-referential. Christian formulated this very clearly. But the notation itself meets the standards.

Juan-Esteban's objection, that examples1 and 2 refer to the set ℕ and therefore are self-referential, amazes me. With this attitude, every definition becomes self-referential: I could define Unit_Circle in the plane as: "the circle with radius 1 and center (0,0)". This may not seem self-referential but "circle" refers to  the set of all circles in the plane (let's call it: C2) and hence a more correct definition of Unit_Circle should be "the object in C2 with radius 1 and center (0,0)" which is self-referential by Juan-Esteban's (apparent) standards. One can do something similar with any definition of any mathematical object.

Finally, if you wish to follow the style of example 2, example 3 should look like this:

Def := "The set of all non-self-referential definitions."

Now Def is the name of the definition and you can agree afterwards to call the defined set K. Again, there is no self-reference and no paradox: Def is not a member of K. (With the extended notion of self-reference, K can be considered the empty set.)

The proposal lacks consistent notation and clear rules about what makes a statement into a definition, what makes a definition self-referring, what is the universe of discourse. Sets of definitions (well-formed formulas? English sentences?...) are not really standard. And what are all these definitions about?

Most "paradoxes" are a kind of mathematical joke, or a display of ingenuity in constructing a mind-cruncher. There are currently no "genuine paradoxes" (getting mathematics down by a plain contradiction).

Marcel, 

I appreciate your opinions, but  they are only opinions. Unfortunately, I do not know how a notation can be inconsistent. Perhaps, by my poor knowledge on logic, I can only understand the term "consistent" when it is applied to statements, but not to the used symbols.  Excuse me, but I am not interested in self-respect issues.

Thank you a lot.

We speak about the set K defined like:

K := { d \in Definitions | d is not self-referential}

This is a definition refering to the set of all definitions, so refering to a set it is suppose to belong itself. So it is definitely a self-referential definition, but differently of your examples, it is self-referential from outside. It is not an argument or a set of arguments it uses, but just by refering to the set of definitions, where it belongs itself, being itself a definition.

I also believe that we would have no paradox without this self-reference.

Consider Berry's paradox

"The smallest positive integer not definable in under sixty letters."

This definition is self-referential in your sense, because it speaks about a number and refers to it and to the set of numbers, but it is also self-referential in my sense, because it speaks about how we can define or not define the number - so refering again to the set of all definitions. It is just this second kind of self-reference which creates the paradox. Your first kind of self-reference often does not produce any paradox, like in ""n \in N and n^4 = 1" which clearly defines 1. This is self-referential only in the first kind.

So I would dare to say that you produced a Berry-type paradox.

Consistent Con*sist"ent, a. [L. consistens, p. pr.: cf. F.

consistant.]

1. Possessing firmness or fixedness; firm; hard; solid.

2. Having agreement with itself or with something else;

having harmony among its parts; possessing unity;

accordant; harmonious; congruous; compatible; uniform; not

contradictory.

The emphasis is mine; each italicized  expression highlights a feature which notation may (fail to) have.

Criticizing and getting criticism are essential parts of mathematical (in fact, any scientific) activity. It is given respectfully and should be considered with respect.

Dear Marcel,

I appreciate your sense of humor. In my opinion, sense of humor is a noticeable sign of intelligence.

Thank you.

J.E.P.

P.S. I will take care that my symbols do not fall down.

Mihai,

Berry's paradox is of the same type as "This sentence is not true" : they only work if you take a second stand to run the self-reference. It leads to mind-crunchers like some of Escher's drawings.

I always considered such "paradoxes" as a mathematical joke (they cannot be expressed in standard predicate logic, which excludes them). The classical Russel paradox can be formalized (starting with "there is x such that ..") and turns out to be a logical contradiction (i.e., a formula that is always false, no matter how you interpret its parts).

The distinction of being self-referential from the outside/inside seems to be a good point. With this distinction, Juan-Esteban's construction is self-referencing. I insisted on using  more careful formulations of his definitions to go for a classical kind of paradox. But I still do not see a Berry-type paradox emerging: K is defined with a definition that is not a member of K if it is considered self-referencing (in which sense?). 

To use definitions as members of sets is unusual. It is the clever thing about Juan-Esteban's proposal. It  invites to paradoxes based on a second stand, like Berry's. 

Dear, Marcel,

I appreciate your recommendation. The accuracy in maths requires clearly defined concepts. In any case, take into account, that this forum is not a scientific journal and its editor does not accept latex language and it is difficult to display good formal expressions. I think that we must be more indulgent in notations than in a journal because of its limited editing means.

Nevertheless, with no doubt, you know that any mathematical paper contains a lot of definitions, to avoid cumbersome expressions and increase accuracy. If you are reading a text in which I define as self-referential definition those in which there are occurrences of any set containing the defined object, when you read in my text the words "self-referential" definition, you must assume my definition; otherwise your argumentations would be of semantic nature, instead of mathematical deductions.

In any case, if you do not accept that I call self-referential those definitions with occurrences of sets containing the defined object, no problem. I agree to term them by other word, for instance, self-captured definitions.

I understand that your recommendation is good for improving the text quality, but notice that some contributors, upvoted my question, perhaps because they pay attention on the underlying logic, instead of language used.

Regarding Berry's paradox, I have said that self-referential definitions contain underlying equations,. For instance, the following self-referential definition for the number 1,

n = (n > 0) ∧(n=n*n)

is self-referential and contains the solvable equation n = n*n.

However, the self-referential definition

n = (n > 0) ∧(n = n+3)

defines nothing, becuase the underlying equation n = n+3 is not solvable.

The underlying equation in Berry's paradox has no solution, hence dfines noting. Analogously, the self-referential definition in the topic of this thread is unsolvable provided that we assume that every infinity is potential. If the defined set can be completed, becomes solvable and paradoxical·. It is a matter of the assumed infinity nature.

Thank your for your recommendations and best regards.

Dear Juan-Esteban,

"if you do not accept that I call self-referential those definitions with occurrences of sets containing the defined object, no problem. I agree to term them by other word, for instance, self-captured definitions."

Concerning self-reference, I only observed that any definition of some object x of type X can be made "self-referential" by adding "x in X" to the definition (if it isn't there already). That makes each definition self-referencing and then your set K will simply be the empty set. Personally, I agree that it is some kind of self-reference. But if you call it that way, it also penetrates into the definition of K. So it is not a matter of taste, but a matter of consequences.

"If the defined set can be completed, becomes solvable and paradoxical"

Suppose for a moment that we restrict ourselves to definitions written in grammatically correct English sentences with at most 100 words. That is a finite collection (with --I think-- at most 40000100 members). Your definition of K surely requires less than 100 words, and hence is still running. What about the paradox (which I do not see, regardless)?

Dear Marcel

The Berry's paradox is a self-referential paradox arising from an expression like "the smallest positive integer not definable in fewer than twelve words" . To simplify the analysis consider this one that is equivalent: "the smallest positive integer X not definable in fewer than a hundred symbols."

As every self-referential definition there is an underlying equation. Let Len: S ⟶  N be the map sending each sentence into its length, that is the number of symbol it contains, and Def the map sending each object into its definition. The underlying equation in Berry's paradox is

Def(X) = min { n | len(Def(n)) > 100}

Indeed this definition means, implicitly,  that

Def(X) ∈ { n | len(Def(n)) > 100};

therefore satisfies the self-referential definition in the topic of this thread. In addition, we can write this definition explicitly as follows:

Def(X) = min { n | len(Def(n)) > 100} ∧ (Def(X) ∈ { n | len(Def(n)) > 100}

This equation has no solution, since is the conjuction of two incompatible conditions.  Is similar to (n > 10) ∧(n < 2).

My paradox is aldo similar under the scope of potential infinity. See how.

Consider any set of definitions H1. Now, we can define the set K1 of all non-self-referential definitions lying in H1. Of course, K1∉H1 and the underlying equation has no solution. Since K1∉H1, we can build the set H2 = H1⋃{K1}. Iterating the process we obtain a set K2 and so on. If we assume that every infinity is potential the process never ends. By contrast, if the iterating process can be completed, we obtain that K∞∈H∞, which is the paradoxical statement, because leads to its negation K∞ ∉ H∞ and vice versa.

When Cantor assumed the actual infinity is because its diagonal proof becomes inaccurate if the underlying infinite construction process cannot be ended. This is why, the russian mathematician Zenkin claims that Cantor's theorem needs the axiom of actual infinity existence to be true. I think that, in mathematical literature, the term infinity needs disambiguation in each occurrence with either of the adjectives actual or potential. Unfortunately this is not the case, and ambiguous sentences should be avoided.

There are mathematical constructions that need not correspond to any object of the Real World. For instance, when we say that the average height in a population P is 1,67, it is possible that no member of P possesses this height. Analogously, when we handle the concept of infinity, perhaps there is no infinite object in the real world, and actual infinity is an artificial construction.

Dear Marcel,

if a definition refers to the "set of all definitions" - it refers to something which has not been defined before. The quetion if the definition belongs to a special subset of this set can be without object, if it was not in the bigger set from the beginning. If someone wants to formalize this paradox, one should define a syntax for a set of all definitions written according to the given syntax. It is doubtful that one can define it so that the definition of the set of "non-self-referential definitions" is itself a definition acording to the syntax. But even if this will be done, it is very likely to achieve a contradictory sentence as in your example about Russel's Paradox. 

Regarding Berry's paradox, when one writes:

"The smallest positive integer not definable in under sixty letters."

the problem is again the fact that one did not previously formalized the concept of "definition" - because "definable" supposes the existence of a definition, and then this definition must satisfy the condition to be shorter than sixty. One moral of the story is that "English" is not appropriate to define numbers and also not appropiate to define notions like "definition" or "self-referential definition". One needs more rigorous formalizations.

Dear Mihai

You are right, the concept of Definition in Barry's' paradox is not formalized, bur in mathematical literature there, mainly, two concepts of definition.

1) Constructive, that is to say, from some objects and a constructing procedure, the defined object is built.

2) Deductive, consisting of a predicate or property, that only can be satisfied by the defined object.

An equation having a solution can be regarded as a deductive definition for its solution. Since equations of the form f(x) = 0 and similar can be transformed into x = f(x)+x, in general, deductive definitions can be equivalent to self-referential ones and vice versa. Sometimes the underlying equation in deductive definitions are masked.

In my preceding post I have formalized the underlying equation in Barry's paradox.

We can state a lot of deductive definitions the underlying equation of which has no solution, therefore can give rise to paradoxical situations.

The words " the set of all...." can be used in definitions if actual infinity is assumed.

See the definitions of potential and actual infinities.

Ax1(Potential): Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already

(H. Poincaré)

Ax2 (Actual): Every mathematical construction can form an actual and completed totality.

(G. Cantor)

Indeed, if you accept Ax2, you can use the words "The set of all ....." in any definition. This is why I have said that my paradox is easy to be solved (rejected) under Ax1, but is a true paradox in the scope of Ax2.

Juan-Esteban,

Your discussion of Berry's paradox seems close to being perfect. I have one remark: Def() is not simply a function as some objects may have more than one definition (your definition of  "1" in example 1 of the Question is certainly not the same as the standard definition). You could perhaps use further specifications: take the shortest possible definition or choose a non-self referring definition if there is one. If there is still ambiguity, you could take the one with the smallest Godel number. The details are probably totally irrelevant. The important thing is that the function can be properly defined if the issue would show up (are you already thinking of a paradox on the definition of Def()?). An issue that remains for the moment is this: your objects are definitions and sets. Sets probably have a definition, but do definitions have a definition?

Some reactions on the paragraph about Cantor and the subsequent paragraph. I consider potential and actual infinity as philosophical terms which, in the case at hand, refer to the way infinity gets involved in a mathematical problem. So I fully agree that Cantor's diagonal argument needs the "axiom of infinity" as formulated in set theory (it is needed quite explicitly, together with the axiom of specification). Any explicit use of infinite sets falls under the label of actual infinity.

One remark on your last few sentences: so-called finitary set theory rejects the axiom of infinity. All of its sets are finite. But any model of it would automatically be infinite. Similarly, Peano arithmetic is about finite numbers but any model must be infinite. This is a particularly clean way of avoiding infinity: it is there, but not as a property of an object of the theory: there is no ambiguity. 

As to the paragraph starting with "My paradox is also similar under the scope of potential infinity": I will take some time out to formulate some objections properly. Let me suffice for now with a technical remark, which should be clarified before I can make my objections more explicit. The set H1 is a set of definitions. Hence the set K1∉H1 regardles of  K1's content. Then H2 is defined as H1⋃{K1}. This set is no longer a set of definitions. The suggested process does not add any definition. You probably mean to say that K1 is not a subset of H1 and continue with the set of definitions H2 := H1⋃K1.

Mihai,

I have always held the opinion that paradoxes like Berry's or like "This sentence is not true" cannot be formalized because mathematical logic separates language and meaning. We largely owe this to Tarski with his concepts of model and interpretation of language into a model. As a consequence, formal language is explicitly meaningless in se. (This is a source of dislike to many people, even mathematicians.) I do not know Tarski's thoughts but I guess that avoiding slippery linguistic constructions with self-reference must  have been a major concern.

Without disrespect, I consider such paradoxes as mathematical jokes or playful exhibits of some kind of ingenuity.

[...] that "English" is not appropriate to define numbers and also not appropiate to define notions like "definition" or "self-referential definition".

Agreed. Except that "definition" can be specified (in metalanguage) as a statement with unique satisfyability.

Dear Marcel,

It is a pleasure to read your intelligent suggestions. Let me state some doubts about Cantor's diagonal method. Sorry, but to handle this topic, brevity is difficult.

Consider that a theorem consists of a some hypothesis H, a thesis T and a proof leading from H to T. Suppose that the proof is only right, provided that the thesis is true. In this case, the proof proves nothing, because negating the tesis, the proof is wrong. In other words, to be right, the proof needs that the thesis to be true. A proper proof must keep right, when negating the tesis.

Now, consider the classical example of Cantor's diagonal.

Tesis: There is no bijection f:ℕ⟶[0,1]

Diagonal method:

Suppose that a bijection is

1⟶ 0,a11 a12 a13....

2⟶ 0,a21 a22 a23...

.........

We can define the number x=0,b1b2b3...with, for all n, bn ≠ ann. If the thesis is false and the bijection exist, for some positive integer k, bk=akk; therefore the definition containing the expression "for all bn≠ann" is wrong. In addition, under this assumption, the definition of x becomes self-referential and the underlying equation can be unsolvable. Thus the construction of the number x requires the tesis to be true. Accordingly, the diagonal method proves nothing. This does not means that the theorem is is wrong, but the proof is wrong.

To define the number x = b1b2b3... with the condition "for all n, bn≠ann", we need the tesis to be true.

P.S.

Sorry, in my computer, neither Firefox nor Chrome can display subscripts to improve readability.

Cantor's proof  of  T from H (H; the set axioms) is based on a quite general principle called argument by contradiction (assume not T and derive a contradiction with H; I hope you accept that principle, otherwise we have no conversation basis). The second last step of the proof is the constatation of a contradiction. The last step (often omitted) is "Therefore, T follows from H".

Quite recently, I almost fell into the same trap as it seems to happen to you now. I made a proof by contradiction of a very nice and very finite statement on rotating some finite subsets of a regular polygon apart. Some construction during the contradiction argument seemed to offer a lot more information. Wrong, the construction works only in the circumstances of that moment, which will soon turn out contradictory, not in "normal" circumstances.

During a proof by contradiction, you are living in a mathematical "what if" dream. You should wake up when the contradiction is reached and recover to reality: that is the last step of the proof, "Therefore, T follows from H". Otherwise, you have a mathematical nightmare.

Dear Marcel,

Math is no a matter of faith, and argumentum ad hominem is not a mathematical method. If a proof is only right when the proved tesis is true, proves nothing, and it does not matter whether the proof is by reductio ad absurdum or by other method.

If inside a proof we include a definition that is only consistent when the thesis is true, then, to be right, the proof needs that the thesis is true; hence proves nothing.

Under the assumption of the existence of a bijection f:ℕ⟶[0,1], the definition of a member of [0,1] different from every member of [0,1] is absurd. As absurd as an endless procedure that can be ended (actual infinity).

Nevertheless, I can prove that such a bijection cannot exist, arranging the set ℕ in the natural order ≤.

As an exercise of imagination, I propose you to solve this question as a challenge. Let p1, p2, p3, p4, p5... = 2, 3, 5, 7, ... the positive prime integer sequence, and denote powers by the accent ^.

Let f1 be a injective map from {p1^n | n∈ℕ} = {2^n | n∈ℕ} into the unit interval [0,1].

Obviously, by Cantor's diagonal method you can prove that img(f1) is a proper subset of [0,1]; therefore K1 = [0,1]-img(f1) is nonempty.

Now, let f2 be an injective map from {p2^n | n∈ℕ} = {3^n | n∈ℕ} into K1 = [0,1]-img(f1) . Analogously, K2 = K1-img(f2)=[0,1]-img(f1)-img(f2) is nonempty. Iterating the process, for every postive integer n, we can define an injective map fn.

The piecewise defined map

F1(k) = f1(k) if k belongs to {p1^n | n∈ℕ}

F1(k) = f2(k) if k belongs to {p2^n | n∈ℕ}

........................

F1(k) = fm(k) if k belongs to {pm^n | n∈ℕ}

............................

is also injective. If  H= [0,1] - img(F1) is non empty. No problem, we can define an injective map from {2^n 3^m| n, m ∈ℕ} and repeat the procedure with two primes.

Can you apply the Cantor's diagonal method to the last  piecewise defined map, when all prime-integer combinations are used?

I think that you cannot. See why.

In an enumeration f:ℕ⟶[0,1], defined in a table like

1 ⟶ 0,a11 a12 a13....

2 ⟶ 0,a21 a22 a23....

...................

between two arbitrary members of the domain there are allways a finite number of elements. By contrast, between two members of the domain of F1, F2 etc there can be an infinite integer set, and the concept of diagonal has no sense. If you cannot apply the diagonal method defining the bijection with the endless procedure above, perhaps we must accept that diagonal method is order dependent, like monotone maps; but between structure-free sets maps must be arbitrary.

I had always the opinion that parodoxes are an index of our not well defined basic concepts, so, here we face the same situation. But, what exact definition is ill defined is a big issue...

argumentum ad hominem.

When I teached courses, I always used lively and imaginative terminolgy, activating the students to associate "dry" mathematical or logical concepts and methods with something more concrete. It worked well, and students showed to appreciate it.

Apparently, this style doesn't work when communicating from a distance by writing. I described a subprocess in a global proof by contradiction as a "what if" dream, and the end (with the contradiction) as the moment to "wake up before it becomes a nightmare". Somewhere, somehow, this has been considered as insulting. If so: sorry, not intentional.

I'm going to explain things once more, but in a "dry" way: a proof by contradiction is organized like this, like a list of statements.

List of hypotheses

Some list of successively derived statements

Assumption --something we claim in order to test if it can be true

~~~~~~~(List of statements derived from all the previous)

~~~~~~~~Contradiction

Not Assumption (the assumption cannot be held)

Finishing the job

QED.

Any statement or construction, performed in the list with ~~~~~ is no longer available.

The error I once made (my example) was similar to what I saw you did,: 

~~~~~~~~~(...)

~~~~~~~~~Contradiction (the assumption cannot be held)

~~~~~~~~~Further Conclusions

Not Assumption (being doubted)

Finishing the job using the Further Conclusions

troublesome QED.

The statements in the list preceded with ~~~, especially the Further Conclusions, have no status afterwards. They should not be used freely. Please inspect your argument or  inspect Cantor's proof (a very clean example of a proof by contradiction) with the above description at hand. 

Dear Marcel,

I cannot see any relation between your post and my claims.  What I have said is that, if in any proof of any nature you include a definition that becomes inconsistent when negating the thesis, then this proof proves nothing.

In addition, you have no mention to my challenge. Remember it: Can you apply Cantor's diagonal method to the supposed bijection F described in my preceding message?

I can build a lot of bijections piecewise defined by endless procedures such that diagonal method cannot be applied. Perhaps, is a method only acceptable for some defining procedures. If it is the case, perhaps Cantor's diagonal method only proves the non-existence of bijections defined by some given procedures.

First, I was reacting on your sentence "argumentum ad hominem is not a mathematical method", which I experienced as a complaint about something that offended you.

"if in any proof of any nature you include a definition that becomes inconsistent when negating the thesis, then this proof proves nothing.".

A proof by contradiction has a sub-process starting with some assumption (an additional hypothesis), and ends with the contradiction. The subprocess is closed, its assumption is dismissed (by negating it).  All statements made during the sub-procedure are  --in principle--  to be considered unreliable thereafter.

In the case at hand, the assumption was "not thesis", so when dismissed, we have "thesis" as our conclusion. This process is completely standard and has been applied for ages in any thinkable circumstances. I really cannot invent a fourth way to explain this technique. Please consult a handbook of logic if you still don't take it from me.

In case of Cantor's argument, it turns out that the "diagonal construction" -- though part of a dismissed process-- appears to be interesting for its own sake and you may reconsider it in a new context. It then shows that, given a sequence of sequences, there is always another sequence.

"By contrast, between two members of the domain of F1, F2 etc there can be an infinite integer set, and the concept of diagonal has no sense."

Two members of that domain are both natural numbers, and between them is a finite number of integers. But even if you change the notion of "between" to have infinitely many in between, no problem. You are defining functions f1,f2,...on distinct infinite parts coded with distinct prime powers, then joining them together in a common extension F1 defined on all numbers involving only one prime factor. Then you start consuming two prime factors, etcetera. This ends at an injective function Super_F() of all the integers into [0,1]. Fine. Now you ask:

"Can you apply the Cantor's diagonal method to the last piecewise defined map?". That is Super_F(). That is a function. I need a sequence of integer sequences to do the Cantor trick. Which sequences? What is the purpose of doing all this?

If your problem is actual infinity, then say so. I will then advise you to fall back on finitary set theory. No dispute. If infinity is not the problem, then consider Cantor's trick for 10 sequences of length 10 to prove that there must be another sequence of length 10.

I know what is a proof by reductio ad absurdum. But my claim is more general. If in some proof, of any kind, it is included the definition of an object and negating the thesis to be proved, this definition becomes inconsistent and defines nothing, then the proof proves nothing, because it is only right when the thesis to be proved is true. I think that it is not so difficult to understand, even with the limitations  in this web page.

Regarding the diagonal proof, I try to be more explicit. Between finite sets, bijections can be stated given any order for their domains and codomains.

In other words: If A and B are finite sets of cardinal n, defining an arbitrary linear ordering ⋞ for A, and another one ⊰ for B, there is always a monotone bijection from (A, ⋞) onto (B, ⊰). Unfortunately, for infinite sets this is false. For instance, there is no monotone bijection F from (ℕ, ≤) onto (ℚ ,≤) in spite of being ℚ countable. This is because for every couple n, m in ℕ, the set {x∈ℕ | n ≤ x ≤ m} is finite, while its image {x ∈ ℚ| F(n) ≤ x ≤ F(m)} is infinite. Accordingly, the existence of bijections between infinite sets cannot be stated in any ordering for their domains and codomains.

When we define a bijection by means of a table

1 -> a11 a12 ....

2 -> a21 a22 ...

.....

the first column is arranged in the natural order ≤. In this form the diagonal concept can be stated. However, arranging the first column as follows

2¹ -> a111 a112.....

2² -> a121 a122...

...

..

3¹ -> a211 a212...

3² ->a221 a222 ...

...

...

5¹ -> a311 a312....

...

...

2¹ 3¹ -> .....

....

The diagonal cannot be defined, containing all rows. In addition, between two members, for instance, 2¹ and 3¹ there are an infinite number of rows.

Accordingly, the diagonal method requires an ordering for the domain equivalent to ≤; hence is order dependent.

Excuse me, but I cannot be more explicit, because these ideas are under consideration somewhere. In any case, perhaps the following considerations can be helpful.

Between structure-free sets bijections can be arbitrary. By contrast, between structured sets bijections must preserve structure (morphisms). In some abstraction level is not possible to define or analyze bijections involving arbitrary real numbers, and neglecting its structure, that is to say, as if they were structure-free sets. ¿Why? Because if you forget the topological structure of real numbers, its definition  vanishes. You can define for instance the identity f(x) = x of f(x) = log(x), that are concrete functions. However, in a higher abstraction level, if you forget the topological structure, you loose the definition of real number. Without definition you cannot handle them. Without forgetting the topology, perhaps, what you can prove is the existence or non-existence of homeomorphisms, but not arbitrary bijections.

The observation about order is to the point. Finite orders are much the same, whereas infinite orders can be distnguished with a host of  properties. There are more parts of your post which make sense to me. But I cannot comment on everything.

Let me quote this sentence, which I think causes the Cantor dispute while there should be none:

"If in some proof, of any kind, it is included the definition of an object and negating the thesis to be proved, this definition becomes inconsistent and defines nothing, then the proof proves nothing, because it is only right when the thesis to be proved is true"

First, I think that the word "definition" in this sentence actually refers to a "construction". A definition is something absolute and static. If it occurs in a proof (e.g., for convenience) it can simply be taken out and placed before the theorem. It is never under the influence of an ongoing proof. I guess you have the Cantor diagonal trick in mind.

You won't object if I would also take this construction out of the proof and formulate a neutral version of it, e.g. in a Lemma. I suggest this:

Lemma. Given a set A with more than one element and sequences an := (an,i)i=1 to oo for n=1,2,3,..., there exists a sequence b := (bi)i=1 to oo such that b != an for all n=1,2,3,---

Note that A may be finite or infinite, this is totally indifferent here.

The proof is an easy variant of the diagonal trick: there must be two different elements in the set A, say: x and y. Then define for each n:

bn := x if an,n != x    and     bn := y if  an,n = x.

It is clear that b != an for each n. QED.

Then you enter the Cantor proof and refer to the Lemma where appropriate To reduce the problem even further, one can even avoid the subprocess of reductio ad absurdum in the same way as Euclid avoided the word "infinite" in his proof that there are infinitely many primes. Thus,

Suppose we have a sequence of distinct real numbers

dn := 0,dn1, dn2, dn3 ... for n=1,2,3....,

where dn,i are decimal digits. Each number dn is transformed into a sequence an := (dn,i)i=1 to oo. The set A of the Lemma is taken as the collection of decimals {0,1,..,9} (with x=0 and y=1  as the different elements). By the Lemma, there is a sequence b := (bn)n=1 to oo with b != an for each n. Therefore, the real number 

0,b1, b2, b3,....

is in the unit interval  [0,1] and is not among the numbers dn for n = 1,2 3,....

If you wish, you can now complete this story with: "Hence the interval [0,1] cannot be countable because otherwise ...." Please also convince yourself that this proof is merely different from the original version, and that the "construction" and the" thesis" are not al all corrupted by reductio ad absurdum as I claimed and reclaimed and reclaimed.

You may still claim to be an intuitionist (no reductio ad absurdum) or you may object to the infinity axiom (no reals, exit Cantor theorem). If not, Cantor's result is as clear as water. Exit dispute.

You have said:

>

Please, do not biases my words. In a previous message, I have stated that I consider both constructive and deductive definitions.

To understand my claims properly, I write two examples. To be accurate these examples need previous lemmata and definitions, that I omit because my only purpose is illustrate my claims.

Let H be the set of all sequences in {0, 1} that can be defined by a finite formula o described by a finite English sentence. Let S be the set of all finite formulas or sentences defining members of H. Thus, the following relation holds.

  (1)      #(S) ≥ #(H).

Theorem 1. The set H is countable.

Proof.

We can define a Gödel-numbering function G:S ⟶ ℕ; hence S is countable and by virtue of (1) so is H.

q.e.d.

Theorem 2. The set H is uncountable.

Proof.

We show this theorem by Cantor's diagonal method. Let F be a bijection from ℕ onto H, pointwise defined as follows.

   1 ⤅ a11 a12 a13....

   2 ⤅ a21 a22 a23...

   .............

Now, we can define a sequence q = b1 b2 b3... such that

  (2)     ∀n∈ ℕ: bn ≠ ann

accordingly q ∉ img(F), but since q is defined by the finite formula (2), q ∈ H; therefore img(F)≠ H and F is not bijective.

q.e.d.

As you can see, Theorem 2 is the negation of Theorem 1. Which of them is true?

If the true theorem is Theorem 1, then Cantor's diagonal method can prove false theorems.

Notice that the proof of Theorem 1 is based upon the definition of Gödel numbering function. Negating the thesis, that is, supposing that H is uncountable, Gödel numbering function definition keeps right. By contrast, negating the thesis of Theorem 2, that is, supposing that img(F) = H, there is no sequence in H satisfying (2), therefore the sequence q cannot be a member of H, and the proof of theorem 2 becomes wrong. Thus, to be a proper proof, the thesis must be true, hence diagonal method proves nothing.

Please, do not ask to me "what is a definition" etc...this is only an illustration for my claim, for understanding purposes. To avoid any incovenient, I need a large article.

I mentioned the definition stuff only because in distant discussions like this, one should proceed carefully. I am not bullying. This being said:

Theorem 2. The set H is uncountable.

According to what is proved, the proper statement is:

Corrected Theorem 2. The set of sequences in {0,1} must contain members which cannot be described with a finite formula.

This fact is known since a long time. It is by extension known for the real numbers: there are more reals than there are descriptions of it.

Juan-Esteban,

in introductory  courses for undergraduate students, Cantor's theorem takes no more than a half page to prove. Students have to do with this. I have already spent the equivalent of several pages to it. That should be enough. Face the proof instead of running to the exit.

This is mathematics, not theology or politics.

=============

I corrected a misquote: the interval [0,1] must be the set {0,1}. so this is about binary sequences to be precize

Ot course, if you substitute my example 2 by another, you are not argumenting against my statements.

I think that you do not want understand, perhaps by prejudices. If, as you have stated,  the set H contains non-finite formulas, does not satisfy its own definition and the proof is wrong. By comntrast, if the thesis of Theore 2 is assumed, everything is right. This is what I am stating in my messages.

Nevertheless, as I have said, this is only an illustration. This is not a forum for accurate expositions. Accurate expositions are for scientific journals.

In any case, you should read these articles.

http://www.ccas.ru/alexzen/papers/CANTOR-2003/Zenkin%20BSL-2.pdf

http://projecteuclid.org/download/pdf_1/euclid.rml/1203431978

Remember that mathematics should be presented in an entirely undogmatic way.  There are several set theories, but those that reject the actual inifinity concept, are less dangerous and more testable then others.  In addition, logic is an empirical science, because logic cannot prove itself.

From Alexander Zenkin's introduction of: CANTOR’S DIAGONAL ARGUMENT: A NEW ASPECT (unpublished document retrieved from the previous post; apologies for the capitals). About Cantor's alternative diagonal argument:

[...] Cantor’s conclusion in the form |X| < |P(X)| is deduced from the fact that the difference between infinite sets, P(X) and X, amounts to one element, that is such conclusion contradicts fatally the main property of infinite sets.

I can hardly believe what I am reading here. The man is opposing to actual infinity (that's his democratic right) but he defends this view with mathematical claims like the above. Here's another quote, taken from: LOGIC OF ACTUAL INFINITY AND G. CANTOR'S DIAGONAL PROOF OF THE UNCOUNTABILITY OF THE CONTINUUM, p. 8

The real basis of the set theoretical paradoxes is just the actuality of infinite sets, and not that any predicate P(x) generates a definite set of all x, for which P(x) is true.

He seems to refer here to the Axiom of Specification, which is exactly the one that (by its absence in Cantor's setup) made Russel's paradox possible. I already explained in an earlier post that infinity does not cause the paradox. It is essentially a general type of logic contradiction (regardless of its interpretation).

One other feat of Mr A.Z. is to take Cantor's diagonal argument to give a "proof" that a test of (un)countability depends on the indexation of the set, concluding that Cantor's argument must be nonsense (rather than his).

Actual infinity may be rejected, but it is a different thing to claim that Cantor's proof is wrong or cannot be formalized. The latter is a more serious complaint, but I did my homework properly:

[See the attached link on formalizing 100  famous theorems]

#63: Cantor's Theorem formally proved / checked by the following reasoning assistents / proof checkers:

HOL Light, John Harrison

Mizar, Grzegorz Bancerek

Isabelle, Larry Paulson

Coq, Jorik Mandemaker

Metamath, Norman Megill

ProofPower, Rob Arthan

NuPRL, Jim Caldwell

To be sure: each of these different computer systems independantly produced and/or verified a proof that has been completely worked out to the smallest formal-logical detail. At http://arxiv.org/abs/0809.4144  I found a claim and proof --by someone else-- that infinity is unique: there are no different infinite cardinals. On the website

http://scienceblogs.com/goodmath/2009/01/28/the-continuum-hypothesis-solve/

the  proof was identified as a quite elementary mistake. Can't these people produce a serious argument to support their opimions? 

I feel bad for making fuzz about a math issue after all that happened in Paris.

http://www.cs.ru.nl/~freek/100/

Dear Marcel,

Thank you for the links, but they do not solve the question. I can prove that it is not possible to build any bijection from ℕ onto [0,1], but this need not be consequence of different set sizes. Indeed, if there is a bijection both are equal in size, but the converse implication need not be true. Sometimes properties of finite sets are supposed for non-finite ones without any proof.

Only as a sample, I can state a proof based on a claim of yours. In a previous message you have said.

"The set of sequences in {0,1} must contain members which cannot be described with a finite formula."

Since in my post I consider also those that cannot be finitely defined or determined, to be more general,  we can  write:

"The set H contains members that cannot be finitely defined."   O.K?

It is not difficult to see that Cantor's Theorem leads to your claim; otherwise any Gödel numbering function sends [0,1] into ℕ.

Suppose, that you are right. There are a subset E of H no member of which can be finitely defined. If you can define a bijective map F between ℕ and H, then for some integer n, the image F(n) belongs to E. If you can define the bijection F, since every integer n is finitely definable, then so is F(n), which contradicts your claim that H contains some subset E  no member of which is finitely definable. Consequently the assumption of the posibility of defining such a bijection F is false.

As you can see this proof is build pointwisely, and  does not depend on the involved set sizes. Thus, it is not a proof of different infinity existence.

Regards.

P.S

Let me joking a little. If every member of a set M of men can be  married with one  member of a set W of women, and there are no bachelors, then both sets have the same cardinal. Nevertheless, the converse implication need not be true. Perhaps there are bachelors because they are very ugly, in spite of being both sets of the same cardinal.

Juan-Esteban,

(says) " Indeed, if there is a bijection both are equal in size, but the converse implication need not be true.".So P-->Q does not imply Q-->P. ok.

The converse would be : If two sets are equal in size then there exists a bijection between them. Q-->P

You say this could be false with infinite sets.

Your example shows that there is no bijection between N and H but you don't use the hypothesis that N and H are of equal size. The above converse uses the size concept.

So you don't really illustrate your point.

Dear Christian,

I have not claimed that they are equal in size. What I say is the they "need not be  different in size", but they can be. For instance, between {1,2} and {1,2,3} it is not possible to define a bijection, just because they are different in size.  In other words, the impossibility of defining a bijection need not lead to unequality in size.

It is to avoid such a bad interpretation I have added the joke. The sets of men and women can be of the same size or not depending of the existence of ugly men/women

"The set H contains members that cannot be finitely defined." O.K? 

To be sure: H is now the set of all sequences N->{0,1}.

"If you can define the bijection F, since every integer n is finitely definable, then so is F(n), which contradicts your claim that H contains some subset E no member of which is finitely definable. Consequently the assumption of the posibility of defining such a bijection F is false."

Apparently you assume that this F is finitely definable to be sure that F(n) is finitely definable. That is not automatically the case as you have just seen that there are sequences that have no finite definition, and, after all, a sequence is also a function on numbers. So the conclusion up to now is: there is no finitely definable bijjection F between N and H.

That's a nice step, now the next step: try to see that there is no bijection at all between N and H. I suggest to take a look at Cantor's proof now :)

P: there exists a bijection

Q: sets are the same size

You assume

S:There are a subset E of H no member of which can be finitely defined

You suppose there is a map and find a contradiction with S.

This is not about  Q-->P (When you say the converse need not be true) because you don't use Q. ; you comfirm this by saying you did not claim Q.)

  So your proof does not relate to: "the converse (of P-->Q) need not be true".

Dear Marcel,

You have pointed out: Apparently you assume that F is finitely definable to be sure that F(n) is finitely definable.

  Of course I did, but under the assumption of potential infinity, an endless definition never is completed and defines nothing. By contrast, under actual infinity assumption, it falls into a fuzzy logic. In fuzzy logic there are no contradictions assigning fractionary truth-values.

For instance, consider the sequence

0, 1, 0, 1, 0,1 .....

If you assume that every infinity is potential, no problem. By contrast, if the infinite sequence can be completed, which is the last element 0 or 1? If you assume that it is 0, nobody can prove that it is false. If you assume that it is 1 we are in the same situation. I think that a well-balanced assignation is 0 with a truth-value 1/2 and 1 with the same truth-value.

Once again it is a matter of what kind of infinity we assume.

If we assume actual infinity existence, countable sets would be enumerated with a complete set. For instance, consider the set K= {I_1, I_2, I_3.....ℕ}; where for each n, I_n = {1,2,3...n}.

Of course, K is countable. Please, try the diagonal metod using K instead of ℕ, and say how can define the diagonal element at ℕ. Notice, that I_(n+1)= I_n⋃{n+1}, but for every n, ℕ⋃{1+n} = ℕ.

"[...] under actual infinity assumption, it falls into a fuzzy logic". Huh?

"If the infinite sequence can be completed, [...]". I never complete sequences or infinite sets, only  superman could do this in a fantasy world. Either I define it with a finite expression or I consider a generic one with perhaps a few properties to argue on (example: using a hypothetical bijection N->H leads to a contradiction). That's how mathematics works: it is processing information by the rules of logic. It does no "stamp colllecting" or "lining up infinities".

If you belong to some mathematical school where a sequence of 0s and 1s  needs a value to be assigned to an imagined last element, then I'm not with you. (Even more bizarre: why should this imagined last element be some kind of judgement on the previous elements?) There is not enough common ground for discussion then.

"Please, try the diagonal metod using K instead of ℕ"

Answer: "Let sn : K -> {0,1} denote the n-th sequence; its k-th element is properly denoted sn(I_k). The diagonal sequence is s: K->{0,1} with s(I_n):=sn(I_n)."

I've used standard mathematical notation and adopted your set K and its elements with the names you suggested. As the author of this little piece of mathematics, I am free to choose to number the sequences like "sn", not like "sI_n", but to prevent you from using this escape route: it works equally (it's just a bit uglier).

Juan-Esteban,

You are entitled to reject infinite sets and you do not even have to explain your reasons. But if you do reject it, then say so: Cantor's theorem no longer exists and this discussion is hot air. If you (even temporarily) accept it, we can discuss Cantor's theorem. Point at a mistake if you can, and if you manage, be kind enough to send a note to the people behind HOL Light,  Mizar,  Isabelle,  Coq, Metamath, ProofPower, and NuPRL that their system needs revision.

"If the infinite sequence can be completed, [...]". I never complete sequences or infinite sets, only superman could do this in a fantasy world. (Marcel)

Of course, neither superman... but recall the definition of actual infinity by Cantor:

"Every mathematical construction can form an actual and completed totality"

as you can see, the god Cantor can complete any infinite object.

Consider the sequence S_1 = 0, 1, 0, 1.... if infinity is synonymous of endless, then S_1 has no last element. By contrast, if S_1 is a complete totality, there is a last element in it. If a human cannot know the last element, cannot compare this sequence with another S_2 = 0, 1, 0, 1 ....to know whether both are the same, unless the identity is assumed by definition or axiom.

In a previous message, you have accepted that there are members of [0,1] that cannot be finitely defined. Suppose that x1 and x2 cannot be finitely defined. In this case only superman or the god Cantor can always know whether x1 = x2 or not. To know that a map is injective you must be able to pairwise compare the members of its image. If the image contains endless definitions only the god Cantor can know whether or not a map is injective. Summarizing, Cantor's theorem leads to the existence of members of [0,1] that cannot be finitely defined. Thus, only the god Cantor and his priests can know whether a map from ℕ into [0,1] is injective even without defining it.

Regards.

P.S.

You have not answered how to define the diagonal element at ℕ, your definition is only for finite sets I_n. If you accept actual infinity existence the index set must be a complete construction with a last element. You cannot work with finite sets, and apply the obtained result to infinite ones.

If to be right, you need the standard notation, perhaps your claims are consequence of this notation instead of logic inference.

Juan-Esteban,

the word Endless is not well defined. It is only one vision of infinity, in a sequential way.

The open interval ]0,1[ is a mathematical entity of infinite cardinality. But there is no first or last element. What is endless is each of the infinite Cauchy sequences associated with a given number. For example the one whose n-th term is the n-decimal places decimal approximation of 1/7. There is no last term in this sequence but the limit exists and is the actual infinity instance of the sequence, even if the sequence itself can be seen as a potential infinity instance.

as for that "Every mathematical construction can form an actual and completed totality", we know this is faulty at least in set theory. There is no such thing as a set of all sets as it would have to be an element of itself as well as a subset of itself.

But this does not mean that actual infinity does not exist in other contexts, like finite limit of infinite sums. In a geometrical view, we can look at this infinite construction based on the golden ratio. The sequence of areas is an instance of potential infinity and the limit is the area of the big rectangle which is finite and is the actual infinity instance of the sequence of areas.

And what is the meaning of the word "exists"?

In the context I used this word, I meant numerical values. We could argue that numbers are abstract constructs of the mind and they don't really exist. But this is the case for all mathematical objects. Rejecting conceptual existence would mean also rejecting all your ideas, don't you think?

I cannot answer unless you define the word exist.

I feel you are having fun. You're like a chess player who plays speculative moves to see what your opponent will play. I could play like you and tell you that I will define the word "exist" if you can tell what a definition is. I hear you say let's see if the guy can really cope with the game when we play out of known variations. This is not without interest!

You know that the concept of existence can only be defined according to one's epistemologic stand. Let's say something exists as soon as someone thiks about it. So if you think about a pink elephant, it exists (for you, in your head). Now i'm sure you can think about a number.

Dear Christian,

Do not run away from the word "existence". Who have used it are you, claiming that actual infinity exists. I have only considered either actual or potential infinity existence as an axiom, but not as a description of an existing object.

As an axiom, we can assume anything, unless this assumption leads to an absurd. By contrast, something exists if can be seen, or  we can obtain some information from it. For instance an electron cannot be seen but it gives rise to some effects that can be observed.

¿Have you seen some actually infinite object to claim that actual infinity exists?

You can claim that Peter Pan exists, at least, as an image in our minds. Have you in your mind some infinite image?

Even quantum mechanics negates the possibility of observing infinitely small objects, as a consequence of Heisenberg's uncertainty principle. Analogously, Relativity negates the existence of an infinite universe.

Regards,

P.S.

If I ask you for your definition of existence, is to play in your arena, that is, I give you all advantages. Since you did not define it, I use the concept of existence, that I prefer.

ok, now you show a card! "something exists if can be seen, or we can obtain some information from it". This is a choice of definition as good as any. By "it can be seen", I guess you also mean "heard, touched, felt or tasted", or accessed by any of human senses or their extensions (devices like microscope, sound amplification, etc.). This is clearly a positivist epistemology. The other part "we can obtain some information from it" is less clear for me. I admit conceptual objects like the number 2. Of course we can see 2 things, but these things are not the number two. I can't get information from the number 2. So I don't know if it exists for you but it does for me. I would take a more general stand and believe something exists if I can put it in relations with something else. Then, the sum of all powers of 1/2 is something that exists because I can write it down with standard math symbols. I can put it in relation with the number 2 by the relation of equality using the formula of the sum 1/(1- 1/2). So I am satisfied because an infinite sum can be identified by a finite representation. I don't say we can always do that. But an integral is another valid example for me.

Your definition of existence is very funny!! It is possible to give existence to every myth.

  I can see a relation between Zeus and Ulises. Surely both exist. Likewise, I can see a relation between the creativity of Mozart and Mozart. This is why the creativity of Mozart exists today and continues creating some music. Of course, this procedure is very easy if the attributes of some object can be separated from it.

What I cannot understand is your relation with the actual infinity. Can you help me? I can conceive an endless process. Unfortunately, the end of an endless process is far from my poor imagination.

Of course there is no end to an endless process.If you divide 1 by 3 you will have 0.3333... or 3/10+3/100+3/1000+...+3/(10^n)+.. which is an endless process. There is no last step but we know that the infinite sum is 1/3. We can compute it with (3 times the sum of 1/10^k) -3 which is 3(1/(1- 1/10) -3 or 10S-S=3.

Another way to see this is take a one unit square. Any way you cut it, the sum of the areas of the parts will always be one unit. So cut it in half, cut a half in half and so on (this is obviously a conceptual process). For any finite number of steps, you know the total area will be one unit. The idea is that "any finite number of steps is arbitrary" so once you decide it, you know there is another number which is larger so you can make more steps.

But 1/3=0.333... is the simplest example for me.

As for "existence", There are many definitions according to what you believe. There have been people to say that nothing exists except what happens in your brain. all the rest being illusion. We can even deny past, saying that we are being created at the very moment, with a false memory of thing that don't really exist and never happened. At the opposite, some will say that existence is materiality. Such views are so extreme that it is hard to argue.

For infinity, there are also many points of view. As soon as you admit that between two real numbers there is a third one (let's say their mean) then you have actual infinity everywhere there is a "width".

You have claimed that the number 2 exists. Perhaps you are right; however, with the help of magic actual infinity, I can get that its existence vanishes. See how.

let S_1 = a_11, a_12, a_13..... be a complete enumeration of ℕ such that a_11 = 2.

Now, swapping the locations of a_11 and a_12 we obtain a new complete enumeration S_2 = a_21, a_22, a_23.. such that a_21 ≠ 2 and a_22 = 2. Swapping the locations of a_22 and a_23 we obtain S_3 = a_31, a_32, a_33 ... such that

∀n < 3: a_3n ≠ 2

Iterating the process k times we obtain a complete enumertation S_k = a_k1, a_k2, a_k3... such that

∀n < k: a_kn ≠ 2

Under the scope of potential infinity no miracle occurs when, in the former expression k⟶∞ . However, assuming actual infinity, the process can be completed obtaining a complete enumeration S = a_1, a_2, a_3.... such that

∀n ∈ ℕ: a_n ≠ 2

As you can see the number 2 has vanished; no longer exists, since a complete enumeration of ℕ does not contain it. Indeed, all enumerations S_1, S_2 ... are complete, since we only swap 2 by another integer. Take into account that swapping objects is not a destructive method. A similar miracle occurs when using the diagonal one.

Regards,

Sorry for notations. My browser does not display subscripts, and I write  LaTeX-like notation.

This is how I see it. "assuming actual infinity, the process can be completed". No, it can't. It is a similar situation as 1/3= 3/10+3/100+3/1000+...+3/(10^n)+.. the right side is a process and stays in the realm of potential infinity, it will never stop. The left side is a limit so it is not one of the states of the process (nor the final step as it does not exists). In the 1/3 example, we have a converging sum. This means that each step brings us closer to the limit but never reaches it (remember, for all epsilon, there is a delta...).

In the case of the swapping process, at each step we still have N, so this is a constant sequence. The limit concerns the sequence of S_k, not a state with a final k.

The limit of a constant sequence is the constant itself, so it is N itself.

No problem for the notation, looks like what I use in Geogebra.

Dear Christian,

You have said: "assuming actual infinity, the process can be completed". No, it can't.

Perhaps you need remember the difference between potential and actual infinity. The only difference is the possibility of completing any infinite procedure. If you assign different meaning to the expression "actual infinity", your argumentation does not fit in this topic.

Remember the axioms of infinity.

POTENTIAL:

Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already. (H. Pointcaré)

ACTUAL:

"Every mathematical construction can form an actual and completed totality." (Cantor)

Then it will probably be my last post here as you request.

First thing I want to thank you because you made me look into my conceptions and motivate me into reading Tsamir (1999) and Fleron (2015) for example.

What you state as axioms of infinity are only citations of Poincaré and Cantor and were influenced by religious concerns about nature of God at the time.  These citations are also more than a hundred years old, Other ideas and nuances have come since then. ZFC axioms to eliminate paradoxes and strong axioms of infinity that could lead to decide CH.

I will state again my view before leaving.

you say: "The only difference (between potential and actual infinity) is the possibility of completing any infinite procedure."

An infinite step by step process cannot end. There is no last step. This is a potential infinity context. It is seen from inside, in a procedural way.

Actual infinity is looking at the object from the outside, where that object is the result  of a ALREADY completed infinite process (without referring to how that process could have been completed). The conceptual gap between a process that cannot be completed step by step and the reification of such a completed process as an object is what causes dialectic between the two.

Have a happy day,

Christian

Dear Christian, 

I appreciate your words.

regards

"Please, try the diagonal method using K instead of ℕ".   (N.B.: K= {I_1, I_2, I_3.....ℕ}; where for each n, I_n = {1,2,3...n} ).

 Answer: "Let sn : K -> {0,1} denote the n-th sequence; its k-th element is properly denoted sn(I_k). The diagonal sequence is s: K->{0,1} with s(I_n):=sn(I_n)."

(I am free to consider the "ℕ" at the end as the "zero-th" index, which I wish to call "I_0". If there is no zero-th sequence s0, I shift the indexes one to the left.) Submit this Question and its Answer to a hundred mathematicians (except Juan-Esteban) and they will all confirm that this is a full description of what is asked. Contrary to politics and religion, mathematical correctness is an objective criterion.

I don't care what "the god Cantor" may have said about infinities. Looking closely at today's set axioms, one can see that only "naive sets" are sometimes conceived as "containers". Most sets are just symbolic labels (like the ones attached to maps and files) which implicitly describe a certain type of content. Whenever an element a of type A occurs, we are entitled to write "a ∈ A". Perhaps one should read this as "a is of type A"  to avoid these silly disputes about completing an infinite listing process. There is no such process involved or required. 

The distinction between potential and actual infinity is a philosophical issue. I consider the following to be a deontological error: to introduce an obligatory process of listing elements of a set in mathematics. It is committed by certain philosophers (not rarely mathematicians who should know better). Did you ever count to one trillion? Adopting these people's view, you are not entitled to use that number until you have completed the obligatory process. 

Of course, you are free of assigning the zero-th index to ℕ. Unfortunatelly you are not free of locating it at the last place.  If you cannot apply the last index to ℕ, then the diagonal method is order dependent. If to show the non-existence of a map F, you need that its domain is arranged in a particular order, perhaps it is a consequence of the assumed order, instead of the set size.

It seems to me that you have forgotten that my claim is that the non-existence of a bijection by a particular constrained method, need not be a consequence of different set size. Remember that, from a certain view-point,  my topic is "actual versus potential infinity."

Consider that, in a previous message, I have proved the non-existence of a finitely definable bijection, based on a claim of yours.

The error consists of extrapolating a property of finite sets to infinite ones. For finite sets, when there is no bijection between X and Y,  can be only a consequence of different size, because between finite sets of cardinal n there are always n! bijections. One for every permutation. This does not hold for infinite sets. For instance, there is no bijection from ℕ onto ℕ being the domain arranged in the natural order, while the codomain is given in the converse order.

You can see in https://logic.wikischolars.columbia.edu/file/view/Jech,+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf

a proof for Cantor's theorem in which the bijection F:ℕ ⟶[0,1] is rearranged by the ordinary relation ≤. Perhaps, in this way, what is shown is the non-existence of a map such that its domain and codomain are arranged in the natural order. This circumstance is also given between countable-infinite sets; hence it is not a proof of different set size, but it proves that in this particular arrangement of the map codomain, there is no bijection.

Finally, notice that I use the word "size" instead of cardinal. If by definition two sets are of the same cardinal whenever there is a bijection between them, then the non-existence of a bijection denotes different cardinality, but need not mean different size, unless they are finite. Thus, since, in general, the meaning of the term "cardinal" is defined by the existence of bijections, I use the less order-dependent word "size".

Finally, to simplify our discussion, to reject my claims, you must prove that the non-existence of some bijection between infinite sets can only be a consequence of different set size, and does not depend on the arrangement of its domain and codomain used in the proof method.

I am looking forward for your proof.

Best regards.

JEP

Dear Marcel,

Why you do not apply you the same trick to the diagonal method?

In the diagonal method we enumerate from 1 up to infinity the interval [0,1] and we obtain a number r_1 that it is not in the image. We can assign r_1 as the image of 0. If you can obtain another one r_2 outside the new image, assign it to -1, and so on. The process is endless. If endless means that cannot be ended, neither [0,1] nor the integer set can be exhausted.

Juan-Esteban,

Of course, you are free of assigning the zero-th index to ℕ. Unfortunatelly you are not free of locating it at the last place. 

Indexation is a choice of notation, which is usually the most convenient one. In Cantor's argument, both the sequences and the members of the sequences were indexed with the usual 1,2,3,... That's indeed the most convenient choice.

If you want to discuss a "generalized" diagonalization process, then you should first present the rules of this new game. I would expect two identical indexations, as Cantor did, in order to make sense of the term "diagonalization". Having one indexing with a last element and another  without, I can't make sense of "diagonal" (which, by broad consent, is something of type (x,x) for x running through the common index set). And what do you expect from a more general diagonalization process if you already have a problem with the original one?

"I have proved the non-existence of a finitely definable bijection, based on a claim of yours. The error consists of extrapolating a property of finite sets to infinite ones."

What error are you talking about? If this has to do with your notion of "size", consult the sentence below with "1." in front.

[...] a proof for Cantor's theorem in which the bijection F:ℕ ⟶[0,1] is rearranged by the ordinary relation ≤. Perhaps, in this way, what is shown is the non-existence of a map such that its domain and codomain are arranged in the natural order.

Does this refer to Theorem 4.1 on page 44, stating that The set of all real numbers is uncountable? Or perhaps to Theorem 4.3(i) on page 45, stating that Any two countable unbounded dense linearly ordered sets are isomorphic? Both proofs involve the natural ordering ≤ and no sets are rearranged. Are you really suggesting that Jech's  proof of these Cantor theorems merely produce that the order type of natural and real numbers are different? That statement has a one-sentence proof !

"Finally, to simplify our discussion, to reject my claims, you must prove that the non-existence of some bijection between infinite sets can only be a consequence of different set size, and does not depend on the arrangement of its domain and codomain used in the proof method." (Emphasis is mine).

1. You have no definition of "size"; you only mention it is not "cardinality".

2. There are no effective rearrangements in sets: no matter in which order you list the members (if there is a listing at all), it's the same set. You should know this. Hence using a partial order to assist in defining a procedure or function by no means "de-arranges" the set.

3. I have so far disabled about each and every criticism you formulated on the subject of Cantor's theorem. For each one I dismissed, you simply produced a variant one. Your creativity on this subject begins to look like an actual infinity. There seems to be no common basis for a discussion. Even worse, I have problems with your interpretations of proofs and mathematical statements you make.

Dear Marcel,

If you are forgetting what I have stated before, this seems to me the tissue of Penelope.

Let us begin again. We can know that between finite sets of the same cardinal we can define bijections, by arranging the domain and codomain in an arbitrary way. Accordingly, two finite sets are  different in size, whenever we can show that there is no bijection, in spite of being the domain and codomain arranged in an arbitrary order. This criterion does not hold for infinite sets. For instance, there is no bijection between ℕ in the natural order onto ℕ in the reverse order. It is easy to see that there no bijection from ℕ onto ℚ, when both are arranged in the natural order. Does this means that ℕ and ℚ are different in size? Obviously, the answer is no. Accordingly, two infinite sets are of the same size if there is at least one bijection between them. However, two infinite sets are not of the same size if there is no bijection between them, whatever the assigned order. Thus, it is not sufficient some proof requiring a particular arrangement.

If you do not assume this claim, the non-existence of a bijection from ℕ onto ℚ in the natural order would lead to assume that they are different in size.

In the diagonal method the order is not arbitrary. The domain ℕ is arranged in an ordering in which between two elements there are always a finite set of integers.

Accordiingly, proving that there is no bijection in this order does not lead to know that it is not possible in other arrangements.

Consider the set ℕ being arranged in the following way. For every couple of integers m and n, we define the relation  ≾  as follows.

If m = p_1^k1 p_2^k2....p_i^ki

and

n = q_1^h1 q_2^h2...q_j^hj

are the prime factorizations of them, then

m  ≾ n if either p_1 p_1...p_i = q_1 q_2...q_j and m < n, or p_1 p_1...p_i < q_1 q_2...q_j.

With this ordering, between two elements, for instance 2 and 3, there is an infinite set, namely, {2,2^2, 2^3....} = {n∈ℕ | 2 ≾ 3}. Arranging ℕ by the relation ≾, the diagonal method cannot be applied. Perhaps, there is no bijection arranging ℕ by ≾, but a proof is needed. Accordingly, if for you it is sufficient to find out the non-existence of a bijection stated in a particular order, you should assume that ℕ and ℚ are different in size and of different cardinal; because in the natural order there is no bijection between them.

To reject my claim you should show that ordering ℕ by the relation ≾ there is no bijection from ℕ onto [0,1]. If you change the relation ≾ by the ordinary one < , you are applying a property of finite sets to infinite ones. As I have show many times, this property of finite sets is neither  satisfied by ℕ nor  ℚ.

P.S.

With repect to the difference between size and cardinality, it is a matter of assuming a unique infinite, the potential one, or the possibility of actual infinty. If actual infinity is rejected, there is only one size of infinite sets, in spite of not every couple of infinite sets is comparable by bijections. Remember the illustration above for this concept, stated by  two sets of men and women of the same size, but there are a bachelor subset,  because its members are ugly.

Dear Marcel,

Diagonal method is based upon the assumption that if there is a map F from ℕ into X, we can arrange it in a table such that the domain is given in the natural order "≤". This assumption is not admissible. I show why.

Let S_1 = 1, 2, 3, 4... the positive integer sequence. Apply the following transposition: Transpose the first odd integer n with the first even one at its right hand. Thus, we obtain

S_2 = 2, 1, 3, 4...

Applying a similar transposition to S_2, and we obtain

S_3 = 2, 4, 1, 3...

Iterating the process we obtain the sequence

S∞ = 2,4,6,8...

and every odd integer has vanished. Accordingly, applying an infinite set of transpositions to an enumeration, an infinite subset can be lost.

Now, consider the injective map F:ℚ⟶ℕ defined by f(n/m) = 2^n. 3^m. If i and j are two arbitrary members of its image, then the set

A(i,j)={x∈ℚ | F^(-1)(i) ≤ x ≤  F^(-1)(j)}

is infinite; therefore so is its image F[A(i,j)]. If we write the map F in a table, such that its image is arranged in its natural order ≤, then between two arbitrary members i and j of its image the set

B(i,j) = {n ∈ img(F) | i ≤ n ≤ j}

is always finite, hence so is its counterimage F^(-1)[B(i,j)]. Thus, as in the preceding example, in this arrangement an infinite subset of its domain ℚ vanishes.

As a consequence, arranging any map, imposing an arbitrary order to its domain or codomain, we can loose an infinite subset of them. Diagonal method is built upon the converse assumption, that is, applying an infinite set of transpositions to the domain of a map, no member of which is eliminated.

"Accordingly, two finite sets are different in size, whenever we can show that there is no bijection, in spite of being the domain and codomain arranged in an arbitrary order. This criterion does not hold for infinite sets. For instance, there is no bijection between ℕ in the natural order onto ℕ in the reverse order".

As I already said before, I have problems with your mathematical statements, and this is just one example. You do not seem to distinguish sets from ordered sets (i.e., sets with an additional relation of order).  The sets ℕ and the rationals Q have the same cardinality, but the usually assigned order types are heavily different. This has nothing to do with the original Cantor theorem and with cardinality, which is about "naked" sets.

Mathematics is a high-precision language. Please respect these distinctions. There cannot be any discussion if you throw distinct types of results on one heap. There is also no discussion if you only accept "potential infinity". Then there are only finite sets, there is no Cantor theorem, and no playground for alternative diagonal methods. Your project is called finitary set theory. Google it, if you haven't done it yet. Wish you good luck with it.

Dear Marcel,

Dear Marcel,

I appreciate your words. But I have also problems about your mathematical concepts. I cannot figure out the diagonal method in an order-free way. The proof contained in http://www.math.ups.edu/~bryans/Current/Journal_Spring_1999/JEarly_232_S99.html

is also stated using the standard order; however you can understand it.

Structure-free number sets is a delusion, at least for infinite ones. For instance, if you forget the topological structure of ℝ, you lose the real number definition. Without definition hardly can be ℝ handled.

I would like to see how you can state the diagonal method, forgetting the order structure of ℕ.

I already explained this earlier, though not quite explicitly, so I'll give a full definition here. It has nothing to do with order: just switch from the notion of "sequence" to the notion of "function".

Any index set I (which can be just any set) can be used for a diagonal procedure: given functions fi: I->{0,1} for each i in I, the diagonal function is f:I->{0,1} defined by f(i):=fi(i) for each i in I. The image set {0,1} is considered for convenience; any set X could be used instead.

Your are right, because to define a diagonal an index is sufficient. however, diagonal method assumes that each map from ℕ into [0, 1], one is defined can be arranged as a enumeration following the natural order of integers. This assumption is not true. For instance, consider the injective map F:ℚ -> ℕ defined by n/m = 2^n.3^m. If you analize the cardinality of subsets of its domain and codomain yo can see that are all infinite. Since the mapis injective the counter images of two diferent member of the codomain are different. Between two different rational numbes are always an infinite subset X of them. Consequently, the image is F[X] ⊂ ℕ is also infinite. Consequently, if you arrange the codomain in the natural order, then between two different elements there is always a finite; hence an infinite subset is lost. Accordingly, the countable index set K={2^1, 2^2, 2^3...}∪{3^1, 3^2, 3^3.....}∪{5^1, 5^2..}∪... cannot be re-arranged in a enumeration 1,2,3... without losing an infite subset. If you use the index set K, you cannot define the diagonal becauseif you consider that every infinite subset can be completed, for instance 2^∞+n = 2^∞ and the diagonal element at infinity is ambigous.

I am beginning to hate this discussion because I am repeating things over and over.

The choice of |N as the indexing set is due to it being the simplest infinite set. The choice of the natural order  of |N is not needed  for the diagonal construction, it is only there because after the diagonal argument, we have to interpret the sequence as a decimal sequence. By then, the "disaster" (in your view) has already taken place; it is just the aftermath of the proof.

If you don't see this: split the result in two parts: one proving that the set of sequences of 0s and 1s is uncountable, and one proving that this set of sequences can be implemented within the reals.

Dear Marcel,

I try to express my main difficulty to accept diagonal method.

As you know, argument by reductio ad absurdum consists of assuming one hypothesis H from which we can obtain a contradiction.

In the following paragraphs you have two examples.

Example 1) To prove that the set P= {p_1, p_2...} of prime integers is infinite, we assume the following hypothesis

Hp = There are only a finite set P = {p_1, p_2...p_n} of prime integers.

Assuming this hypothesis we define the integer

(1)      q = p_1.p_2....p_n+1

which is not divisible by any member of P; which contradicts Hp; hence the set P is infinite. Notice that,  involving every prime integer in the definition (1) of q, it is only possible if Hp is true. Consequently the definition (1) of q is derived from Hp.

Example 2) To prove that there is no bijection from ℕ onto [0,1] we assume the following hypothesys.

Hd = There is a bijection F:ℕ ⟶ [0,1] such that, for every integer n,

F(n) = 0.a_n1 a_n2 a_n3.....

Now we define a number

(2)     r = 0.b_1 b_2 b_3... such that, for every n, b_n ≠ a_nn;

therefore r ∉ [0,1] and F is not bijective.

Notice that, unlike (1), the definition (2) of r is only possible if Hd is false, otherwise for some k, the digit b_k  satisfies the relation b_k = a_kk; therefore the definition (2) is only consistent if Hd is false. In other words: when we assume that the definition (2) is consistent we are negating the hypothesis Hd. Accordingly, the contradiction

r ∉ [0,1]  ⋀ img(F)=[0,1]

is not obtained from the hypothesis Hd, but its negation. Of course, we can obtain that Hd is false, since we have negated it implicitly among the proof steps. If we assume Hd until the end of the proof, the definition (2) defines nothing, because becomes inconsistent. In fact,  example 2 is not by reductio ad absurdum, because in the definition of r, the initial hypothesis Hd is abandoned.

This was my first argument, however after a sequence of messages we have forget what we are talking about.

I can state a proof by diagonal method that negates  a true statement. If you try, surely you can also  find such a bizarre result.