Obections to Cantor's Theory (Wikipedia article)
in [Logic]
|
Prev: Derivations
Next: Simple yet Profound Metatheorem
From: Helene.Boucher on 21 Jul 2005 08:03 Jeffrey Ketland wrote: > There are four schemes of interest: > - the usual induction scheme [v]; > - the scheme of total induction [iv]; > - the least number principle [i]; > - induction "up to z". > These are all equivalent (in first-order PA). More details in R. Kaye 1991, > _Models of PA_, pp. 43-6. But since first-order PA includes the induction scheme, this is misleading. After all, in first-order PA (and so assuming the induction scheme), (1) the usual induction scheme is equivalent to the well-ordering principle scheme is equivalent to just (2) the usual induction scheme implies the well-order principle scheme What would be interesting if you could show (1) in first-order PA minus the induction scheme. But apparently you cannot, since in first-order PA minus the induction scheme you cannot show that the well-order principle scheme implies the induction scheme. See http://groups-beta.google.com/group/sci.logic/browse_frm/thread/ec5f1fa874430bd8/7d553d98e3f2d45c?q=Equivalence+Pigeon+Hole+Principle+Induction&rnum=1#7d553d98e3f2d45c or if this address is too big, search in Google sci.logic with "Equivalence Pigeon Hole Principle Induction" and the relevant thread will be the first choice.
From: David Kastrup on 21 Jul 2005 08:07 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > David Kastrup wrote: > >> The rules are: if you don't like some conclusions, you have to >> change the axioms, and then you lose all other conclusions (many of >> them might be easy to reacquire, but that process is not >> automatic). > > But what if your method is not axiomatic ? Then neither is your reasoning, and there are no truths you can derive. Cherrypicking from the axiomatic mathematics is intellectually dishonest. You can't just put together the conclusions you like and call that a complete building. That's like telling an architect that you want to have a penthouse view from a twelve-story house, but to save costs, he should leave off the lower 10 stories. > I mean, in intuitionism, the emphasis is not on formal reasoning and > axions, but "constructiveness". That's like saying that in the Minotaur's maze, the emphasis should be on finding the right direction and not on following some stupid thread. The line between truth and falseness spins into a hairline when you are exploring the far ranges of mathematics. And this thread always leads back to the axioms from which you started. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Han de Bruijn on 21 Jul 2005 08:39 Martin Shobe wrote: > On Thu, 21 Jul 2005 12:21:38 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >> >>an "all-encompassing definition of 'infinite' must be provided". > > Does a "Christmas tree" in certain engineering contexts actually have > be a conifer? No. But it seems that you have deleted an essential add-on: >> For the simple reason that 'infinity' is not a concept that is limited >> to mathematics alone. It spreads out i.e. into physics, and gives rise >> there to singularities that exist but one can never perceive them, due >> to a Cosmic Censorship that prevents us to take a look into the inside >> of a Black Hole. Thanks to Roger Penrose. Does somebody believe this ? Han de Bruijn
From: Jeffrey Ketland on 21 Jul 2005 09:56 Helene.Boucher(a)wanadoo.fr > Jeffrey Ketland wrote: > >> There are four schemes of interest: >> - the usual induction scheme [v]; >> - the scheme of total induction [iv]; >> - the least number principle [i]; >> - induction "up to z". >> These are all equivalent (in first-order PA). More details in R. Kaye >> 1991, >> _Models of PA_, pp. 43-6. > > But since first-order PA includes the induction scheme, this is > misleading. It depends upon how you formulate PA. They are equivalent over the base theory PA-, which has (a) axioms stating that + and x are associative, commutative, and satisfy the distributive law, (b) axioms stating that 0 is an identity for + and a zero for x, and that 1 is an identity for x; (c) axioms saying that < is linear order. (d) axioms saying that + and x respect order; (e) if x < y, then (Ez)(x+z = y) (f) that 0 is the least number and that < is discrete. See Kaye 1991, p. 16 and pp 45-6. Richard Kaye then defines PA as PA- plus the induction scheme (p. 43). Other authors define PA as the six axioms for successor, plus and times plus the induction scheme. > After all, in first-order PA (and so assuming the > induction scheme), > > (1) the usual induction scheme is equivalent to the well-ordering > principle scheme > > is equivalent to just > > (2) the usual induction scheme implies the well-order principle scheme > > What would be interesting if you could show (1) in first-order PA minus > the induction scheme. But apparently you cannot, since in first-order > PA minus the induction scheme you cannot show that the well-order > principle scheme implies the induction scheme. If by "PA minus the induction scheme", you mean the six usual axioms for successor, + and x, I don't know. But they are equivalent when the above theory PA- is the base theory, as I noted above. --- Jeff
From: Helene.Boucher on 21 Jul 2005 10:10
Jeffrey Ketland wrote: > It depends upon how you formulate PA. > > They are equivalent over the base theory PA-, which has > (a) axioms stating that + and x are associative, commutative, and satisfy > the distributive law, > (b) axioms stating that 0 is an identity for + and a zero for x, and that 1 > is an identity for x; > (c) axioms saying that < is linear order. > (d) axioms saying that + and x respect order; > (e) if x < y, then (Ez)(x+z = y) > (f) that 0 is the least number and that < is discrete. > See Kaye 1991, p. 16 and pp 45-6. > Richard Kaye then defines PA as PA- plus the induction scheme (p. 43). > > Other authors define PA as the six axioms for successor, plus and times plus > the induction scheme. > I don't have Kaye's article handy, but it certainly seems a non-standard definition of PA. Indeed I don't think it's warranted to call this particular axiomatization "PA" unless some kind of warning is given ! <snip> > > If by "PA minus the induction scheme", you mean the six usual axioms for > successor, + and x, Yes that's what I (and I think most other people) mean. |