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Logic related to Cantor's argument

Given the structure of Cantor's diagonal argument as a proof by contradiction, is it not necessary that his argument method (diagonalization) hold for all forms of the argument in any number base (the most common form seen uses base 2 representations of the Reals) and for any proper subset of R that is equinumerous with R? Question restated: is it not necessary that the proof lead to the same result of contradiction for all possible ways that N could be put in a 1-to-1 correspondence with R?

That is, if there is a proper subset equinumerous with R for which the diagonalization method leads to an inconclusive result, does that not show that the necessary contradiction is avoided...does not go through to proof...and hence that the proof fails to show what is claimed, namely that the Reals are uncountable (uncountable in the sense that there is a higher infinity above that of the Natural numbers).

It would be arguing from the negative (appeal to ignorance) to claim that such a result (an inconclusive result to the diagonalization argument when applied to a proper subset eqinumerous with the Reals) is of no consequence...as in claiming it proves nothing. The burden of proof is not on the one pointing to a case not yet considered...it is on the author of the original claim to show how all possible cases are handled in his/her argument, no? I am not claiming to prove anything other than that Cantor's proof is not conclusive, but rather inconclusive....it does not cover all cases. Also, note I am not claiming that the Reals are countable.

Regarding the Real interval [0,1] in base 2...When I consider the set of equivalence classes of complements {x, 1-x} therein I ask:

Form two subsets of R[0,1] by choosing arbitrary members of each class, sending one such member of each class to set A and the other to set B. Pair set B with 0 and list the members of set A paired with the natural numbers. Do we not get an arrangement of A and N for which the diagonal argument does not lead to contradiction? For when we try to argue that if we take the diagonal of the N vs. A arrangement and attempt to argue that its complement is not accounted for...it cannot be proved that it is not in B.

We are left with this problem: Either

Sets A and B are not equinumerous with R


Cantor's argument does not hold.

Either way the conclusion of uncountability of R is not proved.

Here's a link to a paper by Haim Gaifman that contains reference to the same general idea... https://academic.oup.com/jigpal/article/14/5/709/620306

Is this the idea behind Gaifman's argument?

"Richard's solution to his paradox, stated at the end of his paper, is that the “definition” of g is no definition, since it suffers from vicious circularity: it uses every definition in a sequence in which it already appears as one of the definitions. Richard's solution could have been stated in clearer form than his original phrasing; but he, and Poincaré who in [1906] endorsed his solution, hit on the crucial valid point. If we follow the procedure given in the definition of g, and if g is defined by the mth string of letters that is classified as a definition, we find ourselves in a non-terminating loop: g(m) is defined in terms of g(m) and there is no other way to determine what g(m) is. Peano [1906] classified the paradox as linguistic, not mathematical, a view that is more or less accepted nowadays: ‘definition’, it is argued, is relative to language and the definition of g is carried out in a language that is on a higher level than the language whose definitions were previously enumerated. What is overlooked in such accounts is the fact that the argument from linguistic levels and the argument from circularity are two sides of the same coin. We can either ascend to higher level languages, or we can keep within the same language paying the price of unavoidable gaps: truth-value gaps, or lack of defined values that results in partial functions. If the second alternative is pursued then indeed, for some m, g = fm, and fm satisfies an equation of the form: fm(x) = h(fx(x)), where h(x)≠ x for all x. Putting x = m, we get fm(m) = h(fm(m)); but the contradiction is avoided by the fact that fm(m) is undefined."

Thank you all for engaging my question on this forum. Let me give some background to my question. I have been studying various paradoxical situations in Quantum mechanics. In particular this paper regarding J. Wheeler's idea of pregeometry https://arxiv.org/abs/gr-qc/0411053 and D. Bohm's book on the implicate order http://www.gci.org.uk/Documents/DavidBohm-WholenessAndTheImplicateOrder.pdf , as well as numerous papers addressing the Elitzur-Vaidman bomb experiment and Hardy's paradox. In particular this quote from Bohm's book interested me, "Yet, in principle, it is possible to start with an infinite variety of very different assumptions for the vacuum state, involving the assignment of definite values to a set of completely discontinuous functions of the field variables, functions which ‘fill’ the space densely and yet leave a dense set of ‘holes’. These new states cannot be reached from the original ‘vacuum’ state by any canonical transformation." (I added the emphasis) I asked what would that structure be like? I thought, an infinite Mobius strip of negative curviture: start with two edges and two sides and by identifying complements we get one edge and one side...we have lost access to the other side and every point is a saddle point. Thus some information previously accessible is now inaccessible. A quote from Gaifman's book added to the idea,

"Ramsey [1925] based his proposed reform of Principia Mathematica on the now accepted distinction between the set-theoretic paradoxes, which he called logical, and the semantic paradoxes, which he called epistemic. In this he followed Peano [1906] who distinguished between mathematical and linguistic paradoxes. I suggest that the distinction is not as sharp as it appears. Philosophically, the distinction is motivated by the fact that when we make statements we use language, not computations. Hence any account of the Liar says something about language per se, including the language in which the account is given. Yet the division is far from absolute. There is a linguistic aspect to recursion theory, or any theory that is broadly concerned with algorithms." (I added the emphasis).

I see the most fundamental structure of R as its complementation...no matter how the order of R is mixed by any function acting on R we would always have access to its complementary structure as a foundation. So I asked can we construct within ZFC a choice function that completely separates R into complementary sets where one is completely inaccessible. I thought in this way, by the Axiom of Regularity every non-empty set A contains an element disjoint from A (essentially a least element). So let’s identify one element of each complementary pair of R/{0} with one such element from outside the set...to the least element 0 and the other complement to an element of N. Then we get an arrangement where that real number is identified with an element not related to N. But, of course, what is a least element in one set may not be such in another that contains that same element...so we do not in fact get the real complement element severed from all ties to its complement. I think that is what Mr. Hubeny’s answer shows.

Realizing that being able to isolate the two complementary sets would mean Cantor's proof cannot proceed and results in a different view of R: instead of an array of infinites beyond N we get a single choice function at infinity...essentially a fundamental oscillator that injects disorder into Q (the rationals). I now understand better that such a view of R is outside ZFC, and that within ZFC Cantor's diagonal argument/construction is fundamental to the structure of R. Thank you all. I intend to pursue my thinking, but with a better understanding of where I am going.