Re: Cantorian pseudomathematics


Keith Ramsay wrote:
> david petry wrote:

> |But wouldn't you agree that a tremendous amount of the mathematics of
> |the transfinite will never have a constructive counterpart?

> An expert in constructivism once said to me that there
> seemed to be areas in classical mathematics that had no real
> corresponding area in constructive mathematics, but I don't
> think he meant it in the sense of individual results, since
> he'd recently been explaining that for each theorem X of
> classical mathematics, there was a corresponding theorem
> AC->X of constructive mathematics, where AC is the axiom of
> choice.

I just don't see how AC->X could be considered to be part of
constructive mathematics.


> When the axiom of choice isn't used, one could
> consider LEM->X instead, where LEM is the law of excluded
> middle. (The axiom of chioce implies the law of excluded
> middle.)

Again, if you put the nonconstructive assumptions back in, you are no
longer doing constructive mathematics.


> And also it's not clear that every
> constructive theorem in the sense he and I were considering
> is constructive in the sense you and I are considering.

I would say not.


> Some results like Cantor's theorem are perfectly natural for
> a constructive mathematician like Bishop.

You might say that, but the constructive version of Cantor's theorem is
not the same theorem as the classical version.


> The general theory
> of cardinality as pursued by classical mathematicians seems
> a bit curious from a constructive point of view. The original
> definition of cardinality is equivalent to asking for various
> sets X and Y whether there is an injection from X to Y. When
> one proves constructively that there is one, in many cases
> that means one has constructed a continuous, computable
> injection from X to Y. So it seems to me that constructively,
> this question has something in common with topology.

That's right, sort of. One way to think about it, although it's not
exactly the right way, is to think that an infinite set is
constructively defined by (constructively) defining a countable
neighborhood base for the set.

.

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