Dear Robert,

Analysing the bijection existence between R and other set without forgetting the topological structure of R, what  is analyzed is the continuous bijection existence instead. Likewise, if the ordering of R is used, what  is analyzed is the monotone bijection existence. Accordingly, any bijection existence or non-existence proof should be built without using the limit concept or the relation ) = ...3, 2, 1 has last member, but no first one.

Peter Robert,

You are right. Nevertheless, those proofs based on infinite structure-free sequences are built avoiding the use of the limit concept ; hence my question has no relation with these proofs. I do not negate the existence of this kind of proof. What I have stated is the existence of mathematical literature in which the proofs are built using some structure of R, in order to prove structure-free results.

Robert Lowen

I understand that there are people who do not see any problem in it, but what I need is somebody who is able to show that there is no problem. Personal preferences and viewpoints are not part of logic laws..

"Nevertheless, it is not difficult to find mathematical proofs using the topological structure of R while it is assuming that the result is concerned to the underlying set of R exclusively"

"What I have stated is the existence of mathematical literature in which the proofs are built using some structure of R, in order to prove structure-free results ... I understand that there are people who do not see any problem in it"

Let me bring a different example. I'll prove that there exists a bijection between R2 and R3, in a way which would seem to be problematic to you.

1) Endow both sets with their Euclidean metrics.

2) For any n, Rn with the Euclidean metric is a complete separable metric space.

3) Both sets are uncountable, since they contain a copy of R. For instance, the points of the form (0,x) in R2 and (0,0,x) in R3.

4) There is a theorem saying that all uncountable complete separable metric spaces have the cardinality of the continuum.

5) Since all Rn are bijective to R, they are bijective to each other.

What exactly is the problem with this proof? If I understand you correctly, the problem is that the Euclidean metric is not part of the definition of Rn, so it is not legitimate to invoke in the proof more structure than is directly involved in what we wish to prove.

However, the mere fact that the Euclidean metric can exist forces Rn to have certain properties as a set, properties which are a prerequisite for something like the Euclidean metric to be defined. Therefore, besides being directly involved, some structure can be indirectly involved, and we can probe the lower structure (set) by detecting its indirect consequences on the higher one (metric space). 

Dear Pedro Teran

When I have said that there is some problem in the proof you have described above?

Indeed, you have shown that in this case there is no problem. Can you prove that there is no problem in every case? Counterexamples are only adequate under universal quantification; but in my question I use the existential one instead. Consider that the existence of a triangle of area equal to 10 does not lead to assume that all triangles are of area equal to 10.

Since you are spanish, let me write a spanish expression. Cuando yo diga viva el Madrid, usted no tiene el porqué entender muera el Barça.

Dear Peter T  Breuer,

I would like to pointing out some issues about your answer.

You identify infinite sequences in {0,1} with real numbers. This identification implicitly assumes the standard topology, since a sequence c1 c2... denotes a real number r in [0,1] whenever the binary expansion of r is 0.c1c2... and the meaning of this symbol sequence is

lim (c1/2+c2/22+....+cn/2n)

when n -> infinity. Neglecting the limit concept the corresponding meaning vanishes.

Now, consider the following facts. For each n in N, let In = {1,2,3,,.n}, and let P denote the countable set P = {In | n in N} U N. Using the index set P instead of N, we can build sequences in {0,1} of the form

S = c(I1)  c(I2)....c(N)

Now suppose that S = 1 0 1 0 1 0.....Which is the value of c(N), 0 or 1?

Without knowing the value of any sequence at N, how can you know whether

S1 = 1 0 1 0 1 0.... is equal to S2 = 1 0 1 0 1 0....?

Without knowing whether two sequences are the same, how can you know whether a map is bijective?

All these questions arise from using the countable index set P instead of N. The difference is that P has a last element, while N has not. However, limits can work as last elements of monotone converging sequences, and the adequate index set is P.

Using the Dedekind cut construction, which only depends on the ordering on the rationals, gives a description of the reals without reference to limits, and without the resulting set having any further structure specified.  (At least not until one looks at how structures on the rationals (order and arithmetic operations) induce corresponding structures on the newly constructed set.)

Dear David Yetter

Indeed Dedekind construction is built without reference to limits; but it is no a structure-free method.

Dedekind cut is a partition of a totally ordered set into two non-empty parts A and B, such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element. Thus, neglecting the order structure the Dedekiind construction cannot be defined.

In addition,  to show that the real number set defined by the Dedekind contruction is the same as that defined by limits, we need the limit concept and corresponding topology.

[...] a sequence c1 c2... denotes a real number r in [0,1] whenever the binary expansion of r is 0.c1c2... and the meaning of this symbol sequence is

lim (c1/2+c2/22+....+cn/2n)

when n -> infinity. Neglecting the limit concept the corresponding meaning vanishes. (Emphasis is mine)

Agree. However, in mathematical parlance, the term "neglected" can occur in two distinct circumstances. When used in a judgement of a mathematical argument, it usually refers to "having overlooked something", "forgetting to take something into account". Usually, the result is a plain error. The term "neglected" may also occur within a mathematical argument as part of the argument.

For a simple and relatively innocent example: in set theory, natural numbers are defined as sets: 0:={}, 1:={{}}, 2:={{},{{}}}, etc. Such cumbersome constructions are soon "neglected" in favor of a more pleasing symbolic notation. (Notation is often simplified using the phrase "by abuse of language"). However, at any moment, one may recur to the original definition of 4 as {0,1,2,3}.

More generally, any mathematical result involving a "poor" environment A (e.g., set theory) may have a proof that makes an excursion into a "richer" environment B (e,g., topology, algebra, order,...). The only (and very strict) rule is that the formulation of the  result may not contain explicit references to ("neglects") the richer environment.

In particular, one may use the real line or the Cantor set in set theory and rely on any available construction and every available result as long as the final result (e.g. uncountability) does not refer  to the richer environment.  So It is perfectly allowable to use limits and other sophisticated material in a set-theoretic proof. Note that proof checkers and reasoning assistants are very precise in these matters.

Note. There is one situation in formal logic where Juan-Esteban's objection is taken literally: the "theory of A" (some axiom system and appropriate language) is restricted to those results only that are formulated and proved in the language of A by using only the axioms of A. 

Dear Marcel,

I appreciate your message. However, my interest is very simple, and can be illustrated by the following question.

Can you handle the real number concept using no topology?

(Quoted from a reply to me) Can you handle the real number concept using no topology?

(Quoted from a reply to  David Yetter) Dedekind construction is built without reference to limits; but it is no a structure-free method.

What's the reason for being so picky?

 For each "structure" involved in constructing the real line, replace the name by the structure axioms, and then replace the involved axioms by their verification in the given situation. Replace any reference to a required theorem about the structure by a copy of its full proof.  Putting all these things together properly leads to a purely set-theoretical and fully justified construction. Structures offer an elegant way to divide such complicated stories into manageable units. 

In the attached file I give an (un-elegant, straightforward, short) sketch of one standard construction after removing all references to structures. I formulated no properties either (hence also no proofs) to support the constructions. Every construction step is standard and therefore there is nothing I should justify.

Dear Marcel,

Your message is very interesting. I appreciate the ideas and information it contains. We can learn and clarify ideas from your text. However, the main issue I would like know is whether the concept of real number can be defined without the help of limit notion and topology.

Of course, a concrete real number can be defined easily in a topology-free way. For instance ℼ as a ratio, or √2 as a diagonal length. Is it possible a topology-free definition for the generic real number concept?

Juan Esteban,

My pdf-file describes a construction of the full system of real numbers. Of course you have recognized some traditional features in it. At the same time, it may urge you into an additional sharpening of your criteria. Constructing |R on the fly as in

|R := { x : x in X and P(x) }    (with some available set X and some property P)

is not to be expected. Instead, it may involve intermediate constructions and logical formulae. Where lies the watershed between acceptable formulae/constructions and the ones you find in my file?

Nevertheless, your question hits a point. Both Dedekind cuts and Cauchy sequences connect with topology (order topology and metric topology) and both do assist  in constructing the reals. The real field is also characterized as the only complete and totally ordered field. A structure with such references is expected not to be constructed in a simple way.

Very good answer Marcel.

Thank you.