[sharpener]

For the sake of HN archival history...

The executive summary of my paper is that provably there are fatal inconsistencies in Cantor's Diagonal Argument (CDA).

They take a few forms:

(1) Application of basic classical analysis tools reveals that the contradiction sought by CDA does not hold if those tools are used.

E.g. Assume we have the table of all unique infinite length binary strings {0,1}* (where * represents the supremum of the natural numbers, e.g. the first ordinal infinity in set theoretic language).

Assume that those strings represent the fractional part of binary represetations of real numbers in the continuum interval [0,1].

Assume they start big endian, so for some string s, the value v(k) of some bit at the k-th position of s is v(k) = s(k)/2^k, for k = 1,2,3....

Let f: \N x \N -> {0,1}, be a function that accesses bits in the binary strings viewed as a matrix M(i,j).

Create the binary number z as z(k) = 1 - f(k,k).

Now deploy basic analysis.

Note that k -> \infty implies v(k) -> 0.

So no matter what the value of z(k) is, k -> \infty implies |z - f(k)| -> 0, for some suitable k.

Let f be the limit of f(k) as k -> \infty.

Then in the limit k -> \infty the number z = f, and (for all practical purposes) the diagonal z coincides with a binary number in the table f within a countable limit.

This is independent of any ordering of the table rows of M(i,j).

So the contradiction of CDA does not hold convincingly when these basic tools of analysis are used.

(2) Corollary: Platonists would need to prove that infinite digit existence somehow invalidates basic analysis in order to cling to CDA. Moreover, they would have to prove that "higher" infinite digit existence does not invalidate useful concepts of numbers and limits at all.

As far as I know, no one has done that yet.

(3) Using the same basic tools of analysis, key arguments about powersets having cardinalities that are "higher" infinities do not go through, and moreover aspects of "higher" cardinal arithmetic, as presently defined, can be shown to be self-inconsistent.

Therefore: A) When the tools of basic classical analysis are so useful, why would any pragmatic user of mathematics throw them out at some key point within CDA (and CDA only) to endorse a notion of "uncountability" that creates manifold inconsistencies further on?

B) Why should any pragmatic mathematician allow the extra unconvincing machinery of higher infinities into maths?

In the absence of decent answers to those questions I am compelled towards the recommendation is that mathematicians should throw CDA out.

Edit: formatting.

[myWindoonn]

You're presupposing the topology of the Cantor space; how do you know that Cantor space {0,1}* corresponds to real numbers? You're also supposing that an infinite matrix is actually some sort of infinite-plus-one matrix in a handwaved fashion. Limits don't work in the discrete setting; you must invent and justify them.

Pick a k. Note that z(k) is not in rows 0 through k of M, by finite diagonalization. Now, suppose that you try taking k to "infinity" again. z(k) keeps up at every step, by primitive recursion, and M is always missing at least one entry. By what justification do you suppose that M somehow outruns v at "infinity"?

I need you to remember how to be a human for just a moment. Reread your words, "the recommendation is that mathematicians should ..." and consider the precise moral justification by which you make this recommendation. Note that your attempt at the passive voice failed to shift the moral burden, because the given proof is unconvincing (and quite unrigorous). Please consider a dram of humility and give Yanofsky's paper a serious read. It's good.

Also, if you want a better idea of such an infinite Boolean matrix, consider reading about Chu spaces: http://chu.stanford.edu/

[civilized]

Your argument has some gaps. Here are the two basic problems:

1. f(k) is not defined in your notation, only f(k,k) is. My guess is that f(k) is supposed to represent the real number defined by the k^th row of the table.

2. Whatever f(k) is supposed to be in your notation, you have not shown that it has a limit as k approaches infinity, let alone that z = f.

In fact, f(k) cannot possibly have a limit f. If it did, then by the definition of limit, all but finitely many numbers in [0,1] would have to be within epsilon of f, for any epsilon > 0. We could then show that all but finitely many numbers in [0,1] have the exact same first trillion digits of their decimal expansion. Obviously this is absurd.

This is what always happens with Cantor diagonal argument critiques. The critique invariably ignores the logically rigorous argument of the CDA itself, sets up its own new version or extension of the diagonalization construction, and then makes whatever false assumptions are needed for the new construction reach a conclusion contrary to the CDA.

It's like walking into a sturdy bridge with a few pebbles in your pocket, arranging the pebbles into a flimsy tower, knocking the tower over and declaring that you've destroyed the bridge.