Regarding Goedel, I apologize and retract my claim.

Regarding Cantor, indeed, there are many ways to compare the "sizes" of two sets. Cardinality is one such way. It can have applications, for instance Cantor's original paper used it to prove that there are infinitely many transcendental numbers. It is also used in real analysis, in Lebesgue integration, and in measure theory. But it is not the only way to do so. There is a very real sense in which {1,2,3} is a smaller set than {4,5}; that does depend on knowledge of the contents of the set though, not merely the number of elements contained. In a more abstract context where measure or natural density aren't applicable, cardinality can be left as the only well-defined definition of "size", but in most contexts I agree, it's just one of many possible definitions and they should each be used with care.

Arguing about the sizes of different infinite sets is tricky, though. The set of even integers has the same cardinality as the set of all integers, even though it has density 0.5. The cardinality of the set of rationals is somehow "infinitely less" than the set of irrationals, even though both sets are dense on one another. But there are some things we can learn from not having

Me, however, I like cardinality for the aesthetic properties. I like the Axiom of Choice and all the nonsense that follows from it. Cardinal infinities behave quite differently from the well-defined infinities of ordinals, hyperreal numbers, and surreal numbers.

In your article "The Size of a Set", you discuss Formalism. Are you aware of the axiom "V=L"? It's quite neat, because by effectively outlawing any non-constructible sets, the Generalized Continuum Hypothesis follows! If you accept cardinality as a purely aesthetic definition in conjunction with Formalist ideology, you get a very beautiful result. Practical, maybe not, but beautiful.

That is all to say that I agree with a lot of what's in your paper on "the Size of a Set". However, the "Program Analogy" section of the paper contains what I think is a flawed argument:

> [A] Cantor-type conclusion concerning the sizes of sets of programs is absurd. The proper conclusion is simply that there is no program (in some programming language) to generate all and only the infinite-sequence programs (in that same programming language). It is not a conclusion about the sizes of sets. Likewise in the original Cantor argument, the proper conclusion is not about the sizes of sets; it is simply that there is no sequence of all infinite sequences.

I think the correct conclusion here is that to determine which programs are "infinite-sequence" is undecidable. You've reached a contradiction, therefore the last assumption must be flawed; in this case the assumption was that you could write a program D "whose execution neither halts nor goes into a non-printing infinite loop." This assumption is about programs and programming languages, and says nothing about sets, so the analogy is flawed. In Cantor's argument, the assumption is that the set [0,1) can be ordered. You're correct in saying that this is merely a statement about the existence of a bijection between the naturals and the reals, but that is the definition of cardinality. Your issue here is that you don't like saying that cardinality means "size", and that's fair. But I think this analogy is a stretch at best, and irrelevant to your overall argument.