Hello Juan-Esteban. The proof includes a step choosing just one of these representations for each such number, i.e. either ending in a string of 0's or a string of 9's. As soon as that ambiguity is removed it will work fine.

Hello James,

What about when using a binary number system? Under this system there are only two figures and the choice is restricted to only one number.

Then the ambiguity is for numbers that end in a string of 0's or a string of 1's - again one can make a consistent choice (i.e. we always choose to write these numbers ending with a string of 1's or we always choose them to end with a string of 0's).

Dear James,

In mathematical literature, I have read published texts from 1950 up to 2015, and I cannot find any version similar to what you describe.

I think that you are talking about "your own diagonal method" because the well-known version of Cantor's one is not consistent.

Please, take into account that the topic of this thread is about the consistence of Cantor's method instead of how to modify it to be consistent.

Have you tried 'Naive set theory' by Halmos or 'Sets, logic and categories' by Cameron? I only know this method because it was in all the textbooks I read :) Even my Open University topology course.

Of course, the paragraph below is copied from the text you have cited:

"...an axiomatic set theory is not necessarily consistent: not necessarily free of paradoxes. It follows from Gödel's incompleteness theorems that a sufficiently complicated first order logic system (which includes most common axiomatic set theories) cannot be proved consistent from within the theory itself – even if it actually is consistent. However, the common axiomatic systems are generally believed to be consistent; by their axioms they do exclude some paradoxes, like Russell's paradox. Based on Gödel's theorem, it is just not known – and never can be – if there are no paradoxes at all in these theories or in any first-order set theory."

I cannot see a sentence that contradicts what I have said about the diagonal method. But this method is used in mathematical literature. According to the quote above, should we accept some inconsistency in set theory?

The great English mathematician, Ramsey, wrote the following sentence:

Suppose a contradiction were to be found in the axioms of set theory. Do you seriously believe that a bridge would fall down?

{Frank Ramsey (1903 - 1930)}.

After these quotations, I can state this question: Is set theory a religion or a science?

Although I don't think that this particular instance is an inconsistency, yes, we have got to accept the possibility that set theory (and indeed arithmetic itself) could contain contradictions. Your suggestion then that it is like a religion is a valid point, and you can choose to be an unbeliever (I for one will not excommunicate you or burn you at the stake, but may the universal set lead you back to the fold and shine on you with his infinite light) :D That probably exhausts my knowledge of the subject - I will let my betters who know more set theory help further.

Best wishes,

James

The initial question in this thread is very simple. I summarize.

In the mathematical literature you can find the use of Cantor's diagonal method. This method is based on the following assumption.

If the figure sequences of two real numbers, r = 0.a₁a₂... and t = 0.b₁b₂..., for some index n, satisfy the inequality aₙ ≠ bₙ. then r ≠ t. Even my barber knows that 0.1000...= 0.09999…

Is such an elementary error admissible in the scientific literature?

I prefer proof from my barber, if it is based on logic, than the opinion of a great man.

Please, if you have any argument against the initial question, do not cite some great genius, I prefer, simply, logic.

Now that I'm back at my desk, I can look up some references.

You are right about Halmos - he doesn't discuss this for the real numbers. Instead he shows it for all subsets of the natural numbers, which is equivalent to all infinite strings of 0's and 1's. Which isn't what you asked - my bad.

Out of the books next to me, these are the parts where they refer to making the mapping well-defined by choosing one of the two possible forms of decimal expansion for each ambiguous case.

'Undergraduate topology' by Kasriel. Exercise 1, p 53

'Set theory and logic' by Stoll, Example 4.5, p 93

'Topology: An introduction to the point-set and algebraic areas' by Kahn, leaves it as an exercise on p 7

'Sets, logic and categories' by Cameron, Proof of Theorem 1.18, p 27,

and my Schaums and Open University textbooks.

In short, if a proof is for decimals, it should deal with this issue. But I cannot find any proofs that are specifically for decimals that do not mention the above argument.

You are right. There are some versions of the diagonal method built on the binary number system.

However, you can find some versions constructed under the conditions

dₙ ≠ 0 and dₙ ≠ aₙₙ

Since the binary numbering system has only two figures, these conditions are not always compatible. This is why these versions are built using the decimal number system.

The versions using the binary one only allow us to define a value for the diagonal. If you can prove that there is a number d which does not belong to the image of an

injective map F:ℕ→[0,1], this map can be extended to the domain ℕ ⋃ {-1} defining F(-1) = d.

You can even extend F to ℤ, if, by the diagonal method, you can define an infinite set of members of [0,1] which do not belong to the image of F.

These are instances of how underlying formalism can restrict the results.

In any case, do you think that it is possible to define maps whose domain has some members that cannot be defined by finite expressions in any formalism?

Consider that any Goedel-like numbering function assigns a positive integer to each finite symbol sequence.

Thus, the collection of all numbers that can be defined by finite symbol sequences in any language or formalism is countable.

For instance, Π can be finitely defined as "the ratio of the circumference of a circle to its diameter".

As a consequence, if a set is non-countable, some members of it cannot be finitely defined in any language or formalism.

Thus, supposing that there is a bijection from ℕ to a non-countable set X, if there is no finite method to define such a bijection, then we cannot show whether it exists or it does not.

It sounds like an interesting approach, especially in the context of constructivism. Good luck to you :)

Dear Juan,

Thanks for highlighting on ResearchGate a significant philosophical issue concerning Cantor's diagonal argument: 'Is Cantor's diagonal method consistent?'

Actually, there seem to be two underling issues here that should be of foundational concern:

1. The first is merely one of an obfuscating convention concerning the notation '...'; which is simply a way of indicating that the sequence being considered is non-terminating.

From such a perspective, the Cauchy limit of the Cauchy sequence t = {9/100 +9/1000 + ...} is defined as r = 0.1.

In other words, although it may be informally expressed as r = t for practical purposes, the sequence and its limit are distinct in any formal system that defines the real numbers uniquely.

Mathematically, the former is an 'algorithm'; the latter a 'term'.

To better appreciate the practical significance---and importance---of the distinction for the issue highlighted by your post, please see my book [An22], §20.D.b (The mythical completability of metric spaces):

[quote] Thesis 10. If:

(a) a physical process is representable by a Cauchy sequence (as in the above cases §20.C.a., §20.C.b.);

and:

(b) we accept that there can be no infinite processes, i.e., nothing corresponding to infinite sequences, in natural phenomena;

then:

(c) in the absence of an extraneous, evidence-based, proof of ‘closure’ which determines the behaviour of the physical process in the limit as corresponding to a ‘Cauchy’ limit, the physical process must tend to a discontinuity (singularity) which has not been reflected in the Cauchy sequence that seeks to describe the behaviour of the physical process.

The significance of such insistence on evidence-based reasoning for the physical sciences is that we may then be prohibited from claiming legitimacy for a mathematical theory which seeks to represent a physical process based on the assumption that the limiting behaviour of every physical process which can be described by a Cauchy sequence in the theory must necessarily correspond to—and so be constrained by—the behaviour of the Cauchy limit of the corresponding sequence. [unquote]

2. The second concerns an appropriate interpretation of Cantor's diagonal argument.

(a) Interpreted arithmetically, the argument simply states that if we have an algorithm which can construct a denumerable (countable) set of sequences {a(i, j)}, where i, j range over the natural numbers---such that any two members of the sequence differ in at least one term for some finite j---then we can always construct some sequence {b(i, j)} which is not in the set of sequences {a(i, j)}.

Hence we cannot have an algorithm that can construct all possible sequences {a(i, j)}, where i, j range over the natural numbers, such that any two members of the sequence differ in at least one term for some finite j.

If we now define each sequence {a(i, j)} as a real number, then Cantor's Theorem is simply that the set of reals are not algorithmically computable (i.e., the set cannot be computed by any Turing machine T even in infinite time).

(b) However, Cantor's diagonal argument can also be interpreted set-theoretically; though any such interpretation is essentially non-constructive, since any Set Theory that admits of an Axiom of Infinity cannot have a finitary (i.e., Turing-computable) model.

The issue that you raise in your post implicitly reflects the circumstance that uncritically inherited paradigms---and pedagogy---do not highlight that the distinction between arithmetical (essentially evidence-based), and set-theoretical (essentially platonic), interpretations of Cantor's diagonal argument has consequences which, although complementary, are not only distinct, but of practical significance that has material implications when it comes to the issue of which consequences are categorically communicable.

If of interest, this issue is addressed at length in my book [An22]; both in:

§20.C: Mythical ‘set-theoretical’ limits of fractal constructions; and

§16: The significance of evidence-based reasoning for Cantor’s Continuum Hypothesis.

Kind regards,

Bhup

CV: https://www.dropbox.com/s/ohrhhinjvg5zj7s/

[An22] The Significance of Evidence-based Reasoning in Mathematics, Mathematics Education, Philosophy, and the Natural Sciences. Second edition (Preprint).

https://www.dropbox.com/s/gd6ffwf9wssak86/

Cantor `slash 2´ got solved! See my articles on Cantor here.

i and j stay in relation of 2n

Cantor shortened (wrong) to i = j .