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From: Aatu Koskensilta on 29 Jul 2010 06:11 FredJeffries <fredjeffries(a)gmail.com> writes: > I would say, rather, that a large cardinal axiom RESTRICTS the > mainstream theory in that it declares that a certain cardinal > definitely exists rather than leaving the existence an open question. A large cardinal axiom restricts mathematics in the sense that if we posit the existence of a cardinal of sort A we can't (with good mathematical conscience) also posit there's no cardinal of sort A. But we know e.g. that if a measurable exists there are sets of naturals that are not constructible, that ZFC + "there is a measurable" is consistent, and so on; on the other hand, nothing interesting whatever about naturals, constructible sets, etc. follows from the assumption that there is no measurable. In addition, all the interesting statements we know of that imply the non-existence of a measurable, such as the axiom of constructibility, hold in well-behaved inner models even in presence of a measurable. As things stand today there is thus no apparent restriction. > Adding the commutativity axiom to the group definition RESTRICTS the > notion of group. Every Abelian group is a group, but there are groups > which are not Abelian. This analogy is not very apt: we don't use the axioms of set theory to single out a class of structures to study. > (If I understand correctly) in ZFC, one neither has nor doesn't have > large cardinals. One of your hypothesized ZFC users doesn't (perhaps) > use large cardinals, but she cannot prove that they don't exist. She > also cannot avoid that there are models of ZFC in which there do exist > (some varieties of) large cardinals. Sure she can. It's consistent with ZFC that there are no models of ZFC at all, with or without large cardinals. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 29 Jul 2010 06:22 Transfer Principle <lwalke3(a)lausd.net> writes: > If so, then perhaps this is one way to raise the reputations of the > sci.math finitists. We're to convince them just to leave the existence > of infinite sets open rather than assert their non-existence, just as > the mainstream leaves the existence of large cardinals open rather > than assert their non-existence. The obvious thing is just to do interesting finitist mathematics and leave all the silly harangues against infinity out of it. What formal theory we might, for this or that purpose, formalize such mathematics in is of not much consequence or interest -- and indeed we can't decide on such formal matters until we have a well developed body of mathematical results, principles, techniques, modes of reasoning, and so on, to formalize. The development of mathematics simply does not consist of people putting forth random formal theories, and fundamental questions about foundations are very rarely purely formal. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Ross A. Finlayson on 29 Jul 2010 12:32 On Jul 29, 3:11 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > FredJeffries <fredjeffr...(a)gmail.com> writes: > > I would say, rather, that a large cardinal axiom RESTRICTS the > > mainstream theory in that it declares that a certain cardinal > > definitely exists rather than leaving the existence an open question. > > A large cardinal axiom restricts mathematics in the sense that if we > posit the existence of a cardinal of sort A we can't (with good > mathematical conscience) also posit there's no cardinal of sort A. But > we know e.g. that if a measurable exists there are sets of naturals that > are not constructible, that ZFC + "there is a measurable" is consistent, > and so on; on the other hand, nothing interesting whatever about > naturals, constructible sets, etc. follows from the assumption that > there is no measurable. In addition, all the interesting statements we > know of that imply the non-existence of a measurable, such as the axiom > of constructibility, hold in well-behaved inner models even in presence > of a measurable. As things stand today there is thus no apparent > restriction. > > > Adding the commutativity axiom to the group definition RESTRICTS the > > notion of group. Every Abelian group is a group, but there are groups > > which are not Abelian. > > This analogy is not very apt: we don't use the axioms of set theory to > single out a class of structures to study. > > > (If I understand correctly) in ZFC, one neither has nor doesn't have > > large cardinals. One of your hypothesized ZFC users doesn't (perhaps) > > use large cardinals, but she cannot prove that they don't exist. She > > also cannot avoid that there are models of ZFC in which there do exist > > (some varieties of) large cardinals. > > Sure she can. It's consistent with ZFC that there are no models of ZFC > at all, with or without large cardinals. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus But wouldn't a large cardinal be a model of ZFC? * Here that's where the cardinal as an ordinal would model the cumulative hierarchy of ZFC, but then it would have the other properties of being a model of ZFC. So, having a large cardinal supports the features of a model of ZFC, i.e., a sufficiently larger ordinal than them to keep all the asymptotics flat, beyond all the cardinals of ordinals of ZFC, convenient one point. With large cardinals, it would only be consistent that there are no models of ZFC with ZFC being inconsistent thus that both are true. Here this is with where everything is an ordinal. Capping the ordinals and cardinals of ZFC with a large cardinal, and then correspondingly larger cardinals ad hoc and generally, these are just moments for transitive induction. * Then there's a difference whether there's a large cardinal or numbers all the way up, one or infinitely many. If there were infinitely many cardinals they reduce to ordinals for transfinite induction schemas. It's easier to prove that where there are simply infinitely many ordinals gives that correspondingly extra-structural cardinals maintaining the ordinals' and cardinals' shared total ordering (heh) model the structural ordinals, they model ZFC. Warm regards, Ross Finlayson
From: Aatu Koskensilta on 29 Jul 2010 12:46 "Ross A. Finlayson" <ross.finlayson(a)gmail.com> writes: > But wouldn't a large cardinal be a model of ZFC? * No, no ordinal is a model of ZFC. V_kappa for a large cardinal kappa is a model of ZFC. > With large cardinals, it would only be consistent that there are no > models of ZFC with ZFC being inconsistent thus that both are true. It follows from the existence of a large cardinal that ZFC is consistent but it is consistent with ZFC that ZFC is inconsistent. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: FredJeffries on 29 Jul 2010 13:55
On Jul 28, 8:47 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 28, 11:05 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > On Jul 27, 8:21 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > One analogy that I often bring up is: > > > Large Cardinal : Mainstream :: Mainstream : Finitism > > > that is to say, large cardinal theories extend the mainstream > > > theory by adding inaccessibles, etc., in the same way that the > > > mainstream theory extends the finitist theories by adding > > > infinite sets. > > I would say, rather, that a large cardinal axiom RESTRICTS the > > mainstream theory in that it declares that a certain cardinal > > definitely exists rather than leaving the existence an open question. I apologize that my language was too strong here (instead of dashing things off, I need to wait until I have time to think things through more thoroughly) -- I meant something more on the lines of your "One could argue that..." > > But in that case, one could argue that the Axiom of Choice is > "restrictive," since although it permits all sets to have choice > functions > and wellorders, all vector spaces to have bases, subsets of R to be > nonmeasurable, etc., it restricts in that it prevents sets without > choice > functions from existing, nonwellorderable sets from existing, Kunen's > cardinals from existing, Dedekind's cardinals from existing, and so > on and so forth. Likewise, the Axiom of Infinity _restricts_ the > finitist > theory ZF without Infinity in that it declares that a certain cardinal > (namely omega) definitely exists rather than leaving the existence an > open question. > > Indeed, almost _every_ axiom could be called "restrictive." |