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From: Han de Bruijn on 27 Jul 2005 05:24
Tony Orlow (aeo6) wrote:
> Han de Bruijn said:
>>Set theory doesn't deserve such a predominant place in mathematics.
>>After the discovery of Russell's paradox et all, everybody should have
>>become most reluctant.
>
> Not to mention the Banach-Tarski spheres. Doesn't that derivation constitute a
> disproof by contradiction? Isn't the result absolutely nonsensical? And yet, it
> is accepted, somehow, as truth that one can chop a ball into a finite number of
> pieces and reassemble them into two solid balls, each the same size as the
> original, despite all evidence and logic to the contrary. It's a clear sign of
> something wrong in the system, when it produces results like that.

Not to mention Goodstein's theorem, which states that some (ugly formed)
sequences always converge. They prove this finitary statement with help
of infinite ordinal numbers. If the last few axioms of ZFC are rejected,
the finitary result still does exist, while the infinitary "proof" just
vanishes into nothingness. How is that possible?

Han de Bruijn

From: Torkel Franzen on 27 Jul 2005 05:31
Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes:

> Not to mention Goodstein's theorem, which states that some (ugly formed)
> sequences always converge. They prove this finitary statement with help
> of infinite ordinal numbers. If the last few axioms of ZFC are rejected,
> the finitary result still does exist, while the infinitary "proof" just
> vanishes into nothingness. How is that possible?

It's unclear what puzzles you here. What do you mean by saying that
the result "still exists"? What are "the last few axioms of ZFC"?

Goodstein's theorem, by the way, is provable in ACA. As usual,
ZFC is enormous overkill.

From: Robert Low on 27 Jul 2005 05:36
Torkel Franzen wrote:
> Goodstein's theorem, by the way, is provable in ACA. As usual,
> ZFC is enormous overkill.

What's ACA?
From: Torkel Franzen on 27 Jul 2005 05:38
Robert Low <mtx014(a)coventry.ac.uk> writes:

> What's ACA?

Second order arithmetic, with numbers and sets of numbers, and
the full induction principle, but with the comprehension schema for
sets restricted to formulas that do not contain any bound set
variables. ("Arithmetical Comprehension Axiom")
From: Robert Low on 27 Jul 2005 05:54
Torkel Franzen wrote:
> Robert Low <mtx014(a)coventry.ac.uk> writes:
>>What's ACA?
> Second order arithmetic, with numbers and sets of numbers, and
> the full induction principle, but with the comprehension schema for
> sets restricted to formulas that do not contain any bound set
> variables. ("Arithmetical Comprehension Axiom")

OK: without trying too hard to think about it, I can persuade myself
that's enough to let one build the sequence of decreasing ordinals
used in the proof of Goodstein's theorem. (Unless I'm mistaken
about what one needs to prove the theorem, of course.)

Thanks for the clarification.
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