Obections to Cantor's Theory (Wikipedia article)
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From: Han de Bruijn on 27 Jul 2005 05:56 Virgil wrote: > TO would benefit by learning a little about how these axiom systems were > developed before suggesting that there was no inductive process involved. There was an inductive process involved, without doubt. But the question is whether that part has been implemented as with the same _care_ as the deductive part. I think that has been not the case too often. On the positive side, there _are_ a few mathematicians who have done, indeed, some decent homework on the inductive part. Benoit Mandelbrot with "The Fractal Geometry of Nature" is an outstanding example. Han de Bruijn
From: Torkel Franzen on 27 Jul 2005 06:02 Robert Low <mtx014(a)coventry.ac.uk> writes: > 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.) The ordinals smaller than epsilon_zero kan be represented by a primitive recursive set of notations, and the relation "the ordinal [represented by] n is smaller than the ordinal [represented by] m" is also primitive recursive. The proof of Goodstein's theorem proceeds by well-founded induction on this primitive recursive relation. The non-arithmetical part of ACA enters into the proof that the relation is indeed well-founded. Chapter 11 of my "Inexhaustibility" book contains this stuff.
From: Han de Bruijn on 27 Jul 2005 06:35 Torkel Franzen wrote: > 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"? The infinitary ones. It seems that the "first five" are sufficient for defining set theory for _finite_ sets. > Goodstein's theorem, by the way, is provable in ACA. As usual, > ZFC is enormous overkill. I know that EVA is an Extra Vehicular Activity on the moon. :-) But what is ACA ? Han de Bruijn
From: Han de Bruijn on 27 Jul 2005 06:37 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, got that. Han de Bruijn
From: Torkel Franzen on 27 Jul 2005 06:37
Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > The infinitary ones. It seems that the "first five" are sufficient for > defining set theory for _finite_ sets. ZFC without the axiom of infinity is formally equivalent to PA. However, you haven't explained what you mean by saying "the finitary result still exists", or what is puzzling you. > But what is ACA ? See my earlier response. |