From: Torkel Franzen Subject: Re: cardinality of the power set of the integers Date: 10 Aug 2000 17:25:03 +0200 Newsgroups: sci.math Summary: [missing] Jonathan Hoyle writes: > Surely you also draw a similar line, don't you, Torkel? After all, you > don't honestly believe that Euclid's PP must be true (alternatively, > false)? Don't you agree that the "bifurcating" point of view works well > in this case? Kreisel had an argument for why the undecidability of CH in ZF is more like the impossibility of trisecting an angle using only ruler and compass than it is like the undecidability of the parallel postulate in geometry. The argument is that even the second order axioms of geometry leave the parallel postulate undecided, while in the semantical sense, the continuum hypothesis is decided by the axioms of Zermelo set theory. From this point of view, there is a "bifurcation" with respect to e.g. the substitution axioms of ZF (relative to Z), but not with respect to the continuum hypothesis. Of course this argument is not conclusive, but it does underline that there can be no purely formal justification for a distinction between arithmetical statements and such statements as CH as far as the consequences of their being undecidable are concerned. Like you and like very many people who have thought about these things, I have a strong inclination to regard open arithmetical problems ("Are there infinitely many twin primes?" "Is the Collatz conjecture true?") as turning on questions of mathematical fact, which it would be merely silly to attempt to settle by fiat, while in the case of statements like CH it makes good sense to say that there may not be any fact of the matter, but instead the question turns on what stipulation is most suitable for some particular purposes, or what is most "pleasing to the intellect". What the irritating observations of Kreisel and others (which I have been echoing here) should bring home to us is that there is no facile justification for these different inclinations. ============================================================================== From: kramsay@aol.commangled (Keith Ramsay) Subject: Re: cardinality of the power set of the integers Date: 14 Aug 2000 01:32:58 GMT Newsgroups: sci.math In article <8mvbi8$9ur$1@relay1.dsccc.com>, jmccarty@sun1307.ssd.usa.alcatel.com (Mike Mccarty Sr) writes: |Perhaps I could reword it. There is a difference between an unsolved |question and something which we know is undecidable. | |Suppose Goldbach's conjecture has a proof. Then adding its negation as |an axiom results in an inconsistent system. | |This cannot happen with adding either AC or ~AC. | |Until we know that Goldbach's conjecture is not decidable, we cannot |simply resolve it by adding it (or its negation) as an axiom. It's true that one reason why people are more reluctant to add conjectures of number theory than statements like the continuum hypothesis to axiom systems is that they are less confident that they are independent of the base theory they are starting with. But there is also this philosophical attitude that's been discussed here: the belief that even if we knew it to be independent of the base theory, it still could be "definitely false" and hence an inferior choice. Even if we were to discover that the existence of infinitely many primes of the form n^2+1 was independent of ZF if ZF is consistent, we'd be reluctant to say that whether we regard it as true or not is merely a matter of taste. Number theorists do in fact proceed part of the time as though we had a larger set of axioms... freely presuming that the Riemann hypothesis is true and so on. It isn't to the point where people feel okay about not mentioning where it's used, but using it while noting that you're relying on it is done as freely as if it were the axiom of replacement that was being used (which might well be used without noting it). I have a colleague who remarked that it could be a real problem for him if it turned out not to be true, because he had a whole bunch of results which really depend upon it. Keith Ramsay ============================================================================== From: Mike Oliver Subject: Re: cardinality of the power set of the integers Date: Thu, 10 Aug 2000 13:40:45 -0700 Newsgroups: sci.math Summary: [missing] Jonathan Hoyle wrote: > Surely you also draw a similar line, don't you, Torkel? After all, you > don't honestly believe that Euclid's PP must be true (alternatively, > false)? Actually I think there's a good argument for claiming that the parallel postulate is just simply true. Yes, there are models of the other Euclidean axioms in which it fails. Yes, they are useful and interesting. But they weren't what Euclid had in mind. And what Euclid had in mind *motivated* the parallel postulate, not the other way around. But the argument in the case of CH is different--we don't have to summon up Euclid's ghost and ask what he meant. You've already more or less said that you consider the integers to be a fixed, real object (correct me if I've misunderstood). So then we want to know if its powerset can be wellordered so that every initial segment is countable. Start with the notion of "the powerset of the integers". It's true that different models of set theory have different interpretations for this notion. But they aren't all *correct* about what it is, and we can *tell* -- if model M has a set of integers that model N lacks, then model N is wrong, because it's missing one. Of course model N being wrong doesn't mean model M is right. Perhaps *no* model is right; perhaps you can never get *all* the sets of integers together into a completed whole. But in that case, what we're really saying is that P(omega) does not exist--there are models that think it exists, but they're mistaken. In this case, rather than claiming CH "bifurcates", doesn't it make more sense to say it's meaningless? It contains a term that fails to denote. Once it's granted that P(omega) exists, then we want to know if it has a wellordering whose every initial segment is countable. Such a wellordering could be coded as a *set* of reals (and verifying that it has the desired properties involves quantifying only over reals). (I'm using "real" interchangeably with "set of integers".) So now, once again, we might potentially have two models M and N that *both* are correct about P(omega), but which disagree about CH because M contains the set of reals coding the appropriate wellordering, but N doesn't. But if that's the case, then M is right and N is wrong, because M knows about a set of reals that N is blind to. If this is the situation, then CH is true.