Dear Ed Darnell ,

(*)The rules of mathematics break down at infinity. I agree that there are some puzzles still under discussion with mathematicians as Zeno's paradoxes and similar arguments. But in general, mathematics conquers and defeats the infinity. Mathematical analysis and almost all branches in science are witnesses about the superpower of mathematics. Mathematics solved completely the asymptotic behavior of virtually all natural phenomena. Control theory and the study of equilibrium state stability is a simple example of the qualitative studies where matrices and eigenvalues describe the behavior of solutions entirely as time goes to infinity.

Best regards

Dear Ed Darnell .

I do not agree with your thesis. For one thing, the roots of math go deaper and are richer than just "1". Second, I do not see any thing flawed with our current definitions of cardinality; I find it extremely elegant and convincing. Third, the fact that the set of natural numbers is "infinite" is just a semantic, no mathematician really considers "infinity" to be a technical term. It simply meand "lack of a last element". You seem to associate more meaning to the inocuous word than to its actual content.

Finally, keep in mind that there were many historic attempts to redefine the roots of Math, including the concept of infinity, but none of them managed to perceptibly jolt the math mainstream.

An aside: it is funny that so many people seem to have a problem with the "absence of something", regardless of whether the something is "lack of an end" or "a God". It is the same psychological problem, it seems, as the fairy-tales concept of a "bottomless well" or, more simply, the "end of one's own life (death)". All there concepts, IMO, have something common in them, something that many people find hard to accept.

Stan Sykora Indeed, but Cantor did "redefine the roots of maths" and I find the definition inconsistent (as almost all mathematicians at the time did and many like me still do today). The real problem is the definition of unbounded. Unbounded is not the same as infinite. Finite is unbounded. Unbounded is a potential infinity, required by mathematical consistency to always be incomplete. To treat unbounded as infinite is to make finite and infinite the same, which is inconsistent. Yes, we deal with many "infinite" concepts in mathematics (geometry, continuity, integration, etc.) but we always approach them from the consistent, unbounded, inside. The rules of mathematics break down at infinity.

In every day life you deal with finite, so this is more reasonable. You know that 1/2 is one half, not the same number.

Dear Ed Darnell ,

(*)The rules of mathematics break down at infinity. I agree that there are some puzzles still under discussion with mathematicians as Zeno's paradoxes and similar arguments. But in general, mathematics conquers and defeats the infinity. Mathematical analysis and almost all branches in science are witnesses about the superpower of mathematics. Mathematics solved completely the asymptotic behavior of virtually all natural phenomena. Control theory and the study of equilibrium state stability is a simple example of the qualitative studies where matrices and eigenvalues describe the behavior of solutions entirely as time goes to infinity.

Best regards

Issam Kaddoura completely agree with the statement about analysis, but it is important to recognise what we do with this. We always stay within unbounded finite mathematics. We consider from the inside what would happen in the limit. This is completely valid. It is however illegal to pull infinity inside, as it then becomes trivial to prove 1=0. Mathematics is the foundation of all science which is why it is so important it is consistent. It is consistent because it builds on itself from a consistent foundation. 1 not infinity is the foundation. Most paradoxes are not paradoxes. They are simply the countable-infinity contradiction. You cannot count to infinity, but finite is a potential infinity. Unbounded finite is required by mathematical consistency to always be incomplete.

Dear Ed Darnell ,

Infinity is not a real number. The usual algebra operations over reals can not be applied to ∞ !! For example, ∞ - ∞ is not zero and 0.∞ is not zero and ∞/∞ is not 1 and so on. We can add ∞ to the reals as compactification of the topological space R.

Mathematics is consistent even within the ∞ terminology. All the attempts that show contradictions are mere illusions. The well known trick to show 0 =1 is not accepted under the right use of the convergent concepts.

You are welcome to discuss any example about such fake results.

best regards

Issam Kaddoura the fundemental issue is that you cannot count to infinity in any consistent mathematics. A countable-infinity is inherently inconsistent. Anything built beyond this inconsistency is at best semi-consistent. So yes you can invent all sorts of consistent rules about what you can and can't do but it will never remove the inherent inconsistency.

"One cannot count to infinity in any consistent mathematics." It depends on the meaning of "to count" and "consistent mathematics". Primarly, these are non-mathematical notions: they are from the super- [or: meta-]language. As long as we are not introducing different levels of logical sentences we will certainly arrive at inconsistencies, and the discussion will become endless (this time words from meta-meta-language!)

By the way:

1. why we HAVE to count to infinity?

2. how without the notion of infinity as it is understood should we be able to communicate Pytagorian theorem for non-Pythagorian triangles? Will this disppear?

3. despite doubts of point 2, some means of avoiding the problem for real numbers would be possible, anyway; e.g. a better fitting formulation of the definition of reals via fundamental sequences of rationals with "forbidding" usage of eg. sqrt{2}; I cannot imagine to myself how much the obtained less-semi-consistency will disturb our life and abstract thinking as well.

Regards, Joachim Domsta

Joachim Domsta counting is a process we define along with a numbering system. Binary is the simplest and arguably the best when understanding fundemental concepts. There is no complex meta-language involved. Having defined this we notice we can count forever, making larger and larger numbers without ever finishing. I use the word "unbounded" to describe this. We define infinity to be beyond counting. It is where we would get to if we could count forever - which we clearly can't from our definitions - but it is a very useful concept. Unbounded and infinite are not the same thing but it is easy to confuse them (as Cantor's mathematics does). Finite is unbounded. If you assume unbounded and infinite are the same thing then you are assuming finite and infinite are the same thing.

If you assume unbounded and infinite - I do not, but what then is such a very practical number sqrt{2}?

JoaD

Irrational numbers are a direct consequence of the distinction between unbounded and infinite. No number can be specified to infinite precision with repect to the continuum (which is an infinite concept). In a 1,1, √2 triangle none of the points can be plotted exactly as an exact point would be of zero size. Whether the value finishes in repeating zeros, a repeating sequence, or a more complex unbounded pattern is irrelevant. We can scale our numbering system so exactly the same points are √2,√2,1. This is exactly what a consistent maths tells us.

√2 , √2 , 2

Unbounded sets, infinite sets, and cardinal number admit different definitions. Denumerable sets are infinite sets, but countable set may be finite or infinite. Don't need to merge definitions that may confuse!

The statement ( We define infinity to be beyond counting) is not correct in mathematics.

If a set is equivalent to one of its proper subsets, then it is called infinite.

For example:

f : Z → E ( a mapping from the integer numbers to even numbers )

defined by f(x) = 2x is a bijection; therefore Z is equivalent to E , hence infinite

Z is an infinite and countable set.

Best regards

" No number can be specified to infinite precision with repect to the continuum (which is an infinite concept)" -

is this assumption or a theorem or a metatheorem?

Issam Kaddoura "we define infinity to beyond counting" is the traditional definition of infinity. The problem with the modern definitions you give is that they are based on infinite set theory. It ends up with a confused, often inconsistent mess. It is hard to un-teach something but if you can forget infinite set theory for a minute you will see all unbounded, bijective mappings generate new unbounded sets. These sets must always be incomplete, and indeed mathematics itself must always be incomplete, because it is the unbounded set of all such unbounded mappings. Hilbert's hotel, Godel's incompleteness theorem, the set of all sets which don't include themselves, etc. etc. are all perfectly explainable provided we recognise mathematics is unbounded (and thus incomplete) not infinite. Infinity is a concept which mathematics defines but it always lies at the edge. We need no more than one such concept but can use it in a mutlitude of ways (geometry, calculus, etc. etc.). We must however always be careful to approach infinity from the consistent inside. Infinite set theory abandons this consistency by axiomatically assuming an infinite set.

Joachim Domsta "No number can be specified to infinite precision with repect to the continuum (which is an infinite concept)". It is obvious, but follows directly from not being able to count to infinity. The decimal (or binary) expansion of a number is unbounded, so we can never complete it. 1.000... must always be specified to some level of accuracy. We can avoid this by symbolic names or lazy evalution or convetions (e.g. 1 means 1.000...) but this does mean we have generated a point on the continuum. The continuum has no discrete points. It is an infinite concept, with infinitely many points, all of zero width.

Ed Darnell "we define infinity to beyond counting" is the traditional definition of infinity. Can you show a reference that supports this (traditional definition)?

This discussion tends to be a philosophical one rather mathematics.

We have nothing to do with such (traditional) definitions that leads to contradictions. Consistent mathematical models of definitions and axioms will end by consistent results. Otherwise, one can expect anything.

We know that a little bit change of the fifth axiom of parallel lines in Euclidean geometry evolves a hundred types of non- Euclidean geometries.

For example, in the hyperbolic and elliptic geometries,

the sum of the angles is greater than 180 degrees in the first one and less than 180 degree in the second one and all facts are valid based on the initial assumptions. All geometries are consistent.

Ed Darnell : I still do not agree with your thesis. You say

" The real problem is the definition of unbounded. Unbounded is not the same as infinite."

That is the essence of this pseudo-problem, I think. For me "un-bounded" IS exactly the same as "in-finite". The language itself is absolutely clear. I can't see what is it that bothers you, nor why any other way of naming the "lack of an end" might be of an educational value. Mind you, mine are friendly comments; I would just like to understand what is making you unhappy about "in-finity".

Of course, in metric spaces, ""unbounded" may take on meanings that (apparently) have little to do with in-finite (unbounded sets, open sets, ...). But that is a different story, and only apparently different one, as you sure realize.

To Ed Darnell

You have stated:

ED:>> "No number can be specified to infinite precision with repect to the continuum (which is an infinite concept)". It is obvious, but follows directly from not being able to count to infinity.

Issam Kaddoura Stan Sykora Joachim Domsta many thanks for your time and answers. The problem with much of this is that it turns into rhetoric and philosophy. I reject the axiom of infinity. I reject it becuase it is inconsistent with mathematics built from a foundation of 1. A foundation of 1 does not prevent us exploring all sorts of exotic new mathematics. A foundation of 1 does not require euclidian geometry, it can model whatever we want. A foundation of 1 is however inconsistent with a foundation of infinity. What I don't understand is why there is any reasonable justification for an axiom of infinity given this. Infinity is a less clear concept than 1. From our discussions we clearly don't agree on what infinity is. I suspect our concepts of 1 are far more closely aligned. The beauty I see in mathematics is because of this clarity, simplicity and consistency. Amazingly powerful results build from the simplest of foundations.

Stan Sykora the problem is that in-finite is treated as not-finite (i.e. a binary transition) when counting tells us it is not that simple. Finite is a potential infinity. The easiest way to see this is to consider the set {1,2,3,....}. The size of this set is exactly the same as the size of the largest element (which we never get to). Set theory claims this set is infinite and complete, yet has no infinite elements. These claims are inconsistent. To be consistent the set must always be incomplete in the same way finite must be. 1,2,3,... is required by mathematical consistency to be an incomplete, pending infinity. To describe this as infinite is incorrect and leads to the countable-infinity contradiction. We must distinguish between pending and actual infinity. Sets are composed of discrete objects, so must always be countable. They can be unbounded, pending infinities, but they cannot be infinite. An infinite set is an inherently inconsistent concept.

ED staed: " Set theory claims this set is infinite and complete, yet has no infinite elements. "

Is there any book/paper with this claims?

Joachim Domsta

Joachim Domsta good question. I am not sure if any set theory book covers this. The only works I am aware of which highlight or discuss the issue are those showing the problem with infinite set theory. I have itemised what set theorists have told me. When I claim the complete set of integers must contain at least one infinite element, they tell me it does not. When I claim it must therefore be an incomplete, pending-infinity, they tell me it is complete and infinite.

Ed Darnell , you claim "An infinite set is an inherently inconsistent concept" and also "1,2,3,... is required by mathematical consistency to be an incomplete, pending infinity".

The first statement I simply cannot agree with. There are multiple ways to define and recognize what we know as "infinite set". We could label it in any other way, if you wish, I would not mind. But since such objects exist and can be unabiguously recognized, they merit a name, and that is all a mathematician needs as a justification for using the concept. Mathematicians sometimes construct so-called models of their concepts, but that is just an unnecessary extra. You see no way how to "construct" a practical infinite set? So what? I do not either, but it does not make me unhappy at all.

The second statement is just a reinforcement of what I just said! Every infinity is pending, of course. So what? Limits are pending, too, and many other things.

The discussion reminds me a bit of the Summa Theologiae by Thomas Aquinas with its infinite poring over the "potence to act" and the "act" itself, and how the two could only be reconciled by the [divine] will. After pages upon pages of that I just lost track; in fact, even historically, the "problem" resolved itself by pure exhaustion, it seems.

My question to you is: what do you want to achive? Do you really think you can make mathematics educationally more accessible by following your thoughts? Sincerely, I do not think it would work. It looks to me as a blind alley in terms of impact. Though this is just a personal opinion, of course.

To Ed Darnell

Dear Ed, please do nt address to me your ideas, since I am definitely not fllowing it due to simple reason: their formulation does not follow the accepted by mathematician rules of presenting theories. Example: the notion complete, unbouded wtc. are not defined within a CONSISTENT system f primitive notions with axioms about their relations.

Best regards, Joachim Domsta

Stan Sykora "Every infinity is pending, of course." - I'm led to believe set theorists believe their sets are "actual infinities" (thus complete) not "pending infinities" (thus incomplete). https://en.wikipedia.org/wiki/Actual_infinity

Ed Darnell ,

https://en.wikipedia.org/wiki/Actual_infinity

includes some history which followed by hundreds of new ideas.

Anyway, to hide behind general statements will never reach an end.

Have you any specific mathematical statement that makes you confused with the use of infinity? If so, show us one.

Look, Ed Darnell , you are refering me to a Wikipedia article that is 99% philosophy and 1% mathematics. Either we talk about philosophy, or we talk about math, or we talk about math teaching. They are three different things with a limited (even if not negligible) overlap. As for infinities being pending, that sounds obvious to me, but once they are rigorously defined and discernable, they automatically become actual for me, as a mathematician. As a philosopher I may have some lingering puzzlement, but that happens all the time, between Philosophy and all Sciences. It is nice to see also Math confounding the Philosophers now and then, is it not.

The Wikipedia article is relatively good, from the historic point of view, though it leads nowhere: if these philosophical issues are still discussed after 2600 years (and probably much more), it is likely that they will not be settled any soon. I find it interesting as a coctail party banter, but as a mathematician I do not see any problem or contradiction. The set of natural numbers is both infinite and sufficiently actual to me (even if its infiniteness may be somewhat potential to a philosopher). It is the philosophers who have a "problem" to sort out, not the mathematicians!

Math just works and does not give a damn.

Btw: you did not answer my question yet.

Yes, all these different conceptions of the infinite, or unbounded, which in limit theory is quite confusing

needs further work in math teaching I believe. Personally

prefer algebraic definitions for such things as derivatives.

Infinitives are particularly pesky.

Do understand that people use the potential infinity more

than an actual one in analysis, ie. as large or as small as necesary are current terms.

The trouble starts say with what is the smallest positive real number?

Issam Kaddoura 1-based "infinity" is clearly different to set theory "infinity". If we construct mathematics from 1 then we never get to infinity. In this constructed view of mathematics infinity is a single concept which lies beyond all the countable bijections we invent. In 1-based mathematics the route to consider infinity is through analysis. Infinite concepts such as geometry are modelled from the inside, cartesian geometry is one such model. I have yet to find any concept which cannot be modelled by 1-mathematics so see no need to replace this. The concept of 1 is far clearer than the concept of infinity. Provided people are clear on the distinction between infinity-based and 1-based mathematics, and understand that the two disagree that is enough. In time I am confident mathematics will return to a foundation of 1. Strictly non research-level mathematics has never left this solid foundation.

Juan Weisz yes, the smallest positive real number is a nice concept to consider. 1-mathematics is clear. There is no distinction between the smallest +ve real and the smallest +ve (non-zero) rational in 1-mathematics. Both can never be formed. Conceptually the number is an infinite string of zeros followed by one 0.000...01, but this is a concept which lies beyond 1-mathematics in the infinite zone. We can however get as close as we ever need in a model and consider what happens in the limit. In the limit the smallest value reaches zero. There are no open and closed intervals in the limit.

Ed Darnell ,

Do you believe Peano's model ( 1- based mathematics!!) is enough to solve all scientific problems? Did you ignore the irrational numbers? No mathematics exist without the completeness property of the field of real numbers. Otherwise, one will stay in the age before Pythagoras where irrationals were a big mystery!!

Also, I mentioned, in a previous answer, the (1-based mathematics) will never study the control of dynamical systems!!

The integration of all components is essential for modern mathematics. We need more advanced tools to tackles many natural phenomena. I didn't understand your problem with (infinity or any of its equivalents). Topology ensures that any open interval (a,b) is equivalent to R=(-inf,+inf) . The (1- based mathematics) will make one stay with KG level knowledge and will never show the elegant progress of modern mathematics.

You didn't reply to my request yet.

Can you provide any problem that shows inconsistency with the use of infinity?

Juan Weisz

Algebra and geometry are twins and their combinations as in differential topology ( or differential geometry) open new horizons to investigate the whole universe.

The mathematicians, as well as physicists, convinced that our universe is NOT EUCLIDEAN and the needs more advanced tools to describe and investigate its frontiers.

(*) You raised the elementary problem

< The trouble starts say with what is the smallest positive real number? >

No troubles at all, the introductory real analysis gives a simple elegant answer

to this question. No least positive integer exists:

If m is the least positive integer, then m/2 < m contradiction.

This motivates the definition of (greatest lower bound).

The greatest lower bound of the positive integers glb(R+ ) = 0.

So, we need more advanced tools to describe the behavior of the real numbers.

The elementary real analysis provides excellent answers to all similar problems.

Issam Kaddoura I am also convinced the universe is non-Euclidean and it is 1-mathematics which tells me this. Once we recognise there is no countable route between discrete and continuous we must ask deeper questions. I believe the route between discrete and continuous is uncertainty. Scientific observation is discrete. We always attach numbers to things, even concepts we describe as continuous. The underlying fabric of the universe in is very possibly continuous, but the nature of observation means this can never be observed directly. At some point, dictated by uncertainty, we describe continuous objects as discrete. I suspect it is a finite (but not necessarily constant) speed of time-space which dictates this. If we consider the universe starting from a point then time-space spreads out, possibly at the speed of light, in four (or more/less) dimensions akin to blowing up a 4-dimensional ballon. Every observer occupies "now" and everything they observe is a discrete view of "history". The origin of this geometry is not at the centre, it is at the edge. Look out into space and you are looking back in time towards the origin. Something which appears to be getting larger is actually getting smaller. I am not sure how much of this has already been explored but I suspect it goes a long way to explaining the odd things we see in quantum theory. If it is correct everything in the universe is natrually 'entangled'.

Issam Kaddoura Did you ignore the irrational numbers? Most definitely not. 1-mathematics predicts irrational numbers. I explain clearly in my paper Preprint The Countable-Infinity Contradiction

Ed Darnell

(*)The origin of this geometry is not at the centre, it is at the edge.

Define center and edge ?

Consciousness will start at your location then extended to unbounded space.

Einstein observer is at the origin, you can see that our universe is without edges.

I mentioned before; there are many open problems still under discussion; this doesn't mean that a mathematics contradiction !!

The preprint includes a philosophical point of view,

1. Lack of mathematical definitions.

2. The different concepts are confused and not clear.

3. The question ( How large can finite be? ) is not accepted.

Infinity is not a real number and not included in the field of ordered numbers!

and not obeys the usual rules of addition, division or multiplication!

4.Wrong understanding of numerical approximations.

5. ( There is no distinction between rational and irrational as far as continuity is concerned ) this statement is incorrect.

I can use simple tools of ( number theory), the fundamental theory of arithmetic, to show you that √2 is irrational!

6.( The reals are one of the most confusing definitions in mathematics )

My advice to read more about measure theory.

7.( generate a potential continuity of the (0,1) interval.!!)

No, this proposition is wrong, you will never cover (0,1) by such b's.

8....... reals are not countable. The conclusion is correct (we cannot count to infinity) . Not countable is not equivalent to say (we cannot count to infinity).

Etc.

You hope to cover the interval (0,1) by a set B that includes numbers of form 0.1, 0.01, ... countable strings of zeros and ones.

Do you know Lebesgue measure theory?

Measure (0,1) =1

Measure B = 0

This will stop any thoughts to cover intervals by countable sets.

Conclusion:

The complete ordered field of real numbers is consistent, and no mature mathematician contradicts this fact.

1. Lack of mathematical definitions. - by what definition of mathematics?

2. The different concepts are confused and not clear. - which concepts? How are they unclear?

My whole point however is that many concepts (like finite and infinite) are not as clear as people think they are.

3. The question ( How large can finite be? ) is not accepted. - by who and why?

I am happy to clarify any terms but the clarification will be based on traditional-mathematics and computing terminology. Rejecting the axiom of infinity makes any explanations which assume this invalid.

Proving root 2 is irrational simply shows it cannot be replaced with a finite numerator divided by a finite denominator. It is equivalent to stating we cannot simply replace the root operator with the division operator. The division operator is only one lazy-evaluation we can define on the integers. We can however show that root 2 is the unbounded sum of finite rationals (as is pi, e etc). If we could count to infinity then root 2 would indeed be a rational with an infinite numerator and infinite denominator. These concepts do not disprove 1-mathematics, they are created by 1-mathematics and support it.

I do not know Lebesgue measure theory so will read it and explain the 1-mathematics interpretation.

"The complete ordered field of real numbers is consistent, and no mature mathematician contradicts this fact." - an extreme statement. I think you will find this is not true.

I suspect most just accept what they are told without giving it too much thought. I found Cantor's proofs reasonable until I looked at them a little more closely. Most rejected his theories in the era when they were considered rather than taught.

By 1-mathematics the ordered field of real numbers is not complete. The concept of an infinite field is the same as the concept of an infinite set. Unbounded and incomplete is fine. Infinite is not possible.

I think we are talking in different languages.

It is not acceptable to use any mathematical concepts without clear definitions.

What is infinity? What is finite? Infinite? Bounded? Unbounded? Countable?

One needs to show clear definitions such as

( We assume the bijection definition is clear)

Definition: A set X is said to be equivalent to a set Y if there is a bijection from X onto Y.

Definition: If a set is equivalent to one of its proper subsets, then it is called infinite.

Definition: A set S in R is called bounded above if there exists an element r in R

such that x ≤ r for all x ∈ S.

Similarly , we define bounded below and bounded sets, Etc.

For example:

(1) f : Z → E ( a mapping from the integer numbers to even numbers )

defined by f(x) = 2x is a bijection; therefore Z is equivalent to E , hence infinite

Z is an infinite and countable set.

(2) The interval (a, b) is bounded but infinite set.

Let c,d ∈ ( a, b) , c < d.

It is not difficult to show that (a, b) is equivalent to (c, d) and hence

(a, b) is infinite.

All such concepts ( included in the reprint) are not presented in the logical form.

It is not accepted at all to discuss the real numbers without the modern concepts of measure theory.

Limits, differentiation, integration, and modern calculus all are based on the concept of the measure theory.

We are not satisfied with Riemann integration theory, so the new theory extended and modified the old concepts into a more general approach.

You can find functions which are not integrable according to Riemann theory, but they are integrable according to the Lebesgue theory.

To transform and replace any interval (a,b) in the real line into a set with measure zero will create a big problem and destroy the whole mathematics.

Issam Kaddoura 100% agree. We are talking different languages. Language is inherently a problem as it is always open to interpretation. I will try to be clear so at least you can try to understand my language. Having read up on Lebesgue I can see where the confusion comes from and why it is easy to over-interpret the distinction between Riemann and Lebesgue approaches to integration. The underlying problem is the concept of the set R, but you will need to understand my language to understand this statement.

So let me try and explain my language, and we must start with axioms. What is an axiom? Definitions are best made by examples. An axiom is a definition which is sufficiently clear and unabiguous it requires no definition. I appreciate this is a little unsatisfactory but a the building process must start somewhere. In computing terms we must "bootstrap".

Infinity is not an axiom. 1 is axiomatic. Infinity is defined by 1. 1 is not defined by infinity.

Counting and countable mean exactly what they say on the tin. We must define a numbering system and a process of counting, but all of these are sufficiently clear it adds no value for me to redefine them. Something is countable if you can count it.

Infinity is a theoretical concept beyond counting. It is the theoretical place we would get to if we could count forever (but this would be a circular definition). By definition infinity is not countable. If you assume it is countable then you will be able to generate "the countable-infinity contradiction". Creating this contradiction simply follows from the definition, so does nothing of value. Cantor's "proofs" generate the countable-infinity contradiction.

If infinity is the destination unbounded is the journey. A process which never finishes is unbounded. In computing terms it can be thought of as a recursive operation with no terminating condition. Counting is an unbounded process.

Finite is an unbounded concept so great care is needed when dealing with the term. If we have an unbounded counting process, finite can be considered as any value it can theoretically reach. This value is potentially infinte which is where all the paradoxes come from. We cannot allow finite and infinte to be the same thing as that would be nonsense. The definition of unbounded and infinte save us from this but the cost is that finite and unbounded and must always be incomplete. they are a never-ending journey, not a destination.

Why is R a problem?

R is generally used to mean the set of all numbers on the continuum but by the above definitions an unbounded set is always incomplete. A set can be suitably dense that it models the continuum, with a closure of the continuum, but a set is never inherently closed.

I believe irrational numbers are what cause most people difficulty with seeing this, and are what historically caused inconsistent definitions.

The modern computing concept of lazy-evaluation is a good way to understand a correct definition for R.

Numbers on the continuum may be defined in terms of lazy-evaluations. If our numbering system is binary and our lazy-evluation delivers a binary representation of 1/3 then the process is unbounded. We can stop the process at any point and get a best approximation to the value, but we will never deliver the final result. The final result would be a value to infinite precision but we can never deliver it. This does not matter as we can deal with the lazy evaluation instead, with no loss of accuracy. Root 2 and pi are exactly the same. We can define them as a lazy-evaluations.

What we cannot do is deliver "all" the lazy-evaluations, because to do so would require us to count the infinite variety of lazy-evaluation algorithms we can define. The set R can never be complete. You cannot count the continuum.

I hope this helps but I am not convinced it will. If it does I will explain the issue with Riemann and Lebesgue and what they show.

Ok, I think that now we got somewhere and we all agree :-) The fact is that philosophers of math would like to go beyond the [rigorous] math, to put it in a broader context, possibly, using common language terms in a non-technical way, something that mathematicians abhor. For a mathematician something that can be defined exists, even if it cannot be modelled in the real world; for a philosopher that may be hard to digest (though eventually they have to give in :-) ). There are many such examples.

I think that this is OK, as long as both mathematicians and philosophers understand and accept each other's somewhat different goals and methods.

Ed Darnell

Are you trying to show a new mathematical model for the real numbers?

If so, we expect to see a set of consistent postulates that preserve the basic algebraic operations and the geometric consequences as well.

I don't know any of the giant's mathematicians dared to do. I appreciate your attempt, but I think that the expected mathematical model to make things better is far - fetched.

I am convinced that the current model is optimum.

Issam Kaddoura I don't think so. My definitions are more historic than new, so they are part of well established work refined and corrected over millenia. The only problem I have with mathematics is "infinite set theory" and this is a relative newcomer. There is nothing I teach or program which relies in any way on infinite sets. I don't have a problem with set-theory (or even unbounded set-theory). I find the syntax of the formal arguments clear and elegant. The problem I have is the axiom of infinity. It is not valid in my world of mathematics to have an ambiguous or ill-defined axiom. Infinity is defined by 1. 1 is not defined by infinity. I do not require axioms to exist. I do however require them to be clear and self-consistent. An infinite set is not a self-consistent or clear concept. An unbounded one is but we do not require an axiom to define this. Set theory is fixable and the fix makes no difference to most proofs. It does however invalidate any proof by contradiction which assume infinity is countable.

... just to justify my statement that the axiom of an "infinite set" is not clear or self consistent.

Consider the binary set:

{0.1, 0.01,0.11,0.001,...} - binary counting reversed.

There are two mathematically valid ways to view this set. We can take an algebraic view or an analytic view (when I studied maths over 30 years ago the two subjects were taught seperately).

Algebraically this set contains no infinite elements and is thus always finite and incomplete, but countable.

Analytically this set delivers the [0,1] continuum as it is a dense set. This is very clear to computer scientists since it is systematically itemising every possible bit combination.

The problem is that we must not hold these two views simultaneously, although our pattern-based brains are good at doing this.

The analytic view requires us to consider the limit. We are pushing at the boundary of inifinity and considering what would happen if we could reach it. We push from the inside, so it is mathematically valid, but the limit is infinity, which does not strictly lie within mathematics. At the limit we have effectively made the unbounded concept of "finite" equal "infinite". This is illegal in algebraic logic.

It is mathematically invalid to combine analysis and algebra in the way infinite set theory does. Either this set is finite and countable, or it is infinite and becomes the continuum. Both views are valid, but not at the same time. We are not allowed to let finite be both finite and infinte at the same time.

The rest of set theory is fine. It is an elegant subject, but not "the one true" mathematics. Choose the right tool for the job. If approaches disagree then examine why. A faulty axiom is the most likely cause. Garbage in - garbage out.