From: Peter Percival Subject: Re: Abject Surrender--Death of a Thread Date: Fri, 20 Jul 2001 19:45:30 +0100 Newsgroups: sci.math Summary: Do second-order axioms of set theory decide Continuum Hypothesis? (No) "David C. Ullrich" wrote: > > On Wed, 18 Jul 2001 19:21:11 +0100, Peter Percival > wrote: > > >"David C. Ullrich" wrote: > >> > >> On Mon, 16 Jul 2001 19:55:14 +0100, Peter Percival > >> wrote: > >> > >> >lbudney-usenet@nb.net wrote: > >> >> > >> >> ... > >> >> > >> >> But this thread needs to die. I've refiled "CH is true or false" under > >> >> "non-facts", and if I ever meet that logician again (whose name I don't > >> > > >> >CH is decided in second order ZF, but no one knows which way. > >> > >> I could _very_ well be wrong about this, but it doesn't seem nearly > >> that simple to me. If you see why it _is_ that simple lemme know: > >> > >> In "second-order whatever" we are allowed to quantify over subsets > >> of the universe. So doesn't saying that CH is decided in second-order > >> ZF commit us to a notion of "real set"? The real subsets of N or P(N) > >> being the ones we quantify over... If we believe that there are such > >> things as well-defined real sets with a well-defined element-of > >> relation then any statement is true or false about these real sets. > >> But I don't know what those real sets are - I don't see how saying > >> we're talking about second-order ZF helps explain what they are. > >> > > > >I almost know what a "real" set is. It's an element of the cumulative > >hierarchy, isn't it? I say almost because I don't know what collection > >of urelements to begin with and I don't know how far up the hierarchy to > >go. Mathematicians seem to favour an empty set of urelements, but what > >axioms of infinity should one use? > > > >Are you claiming that second order ZF settles _all_ questions about > >sets, and hence a fortiori CH? > > I wasn't claiming anything at all - just wondering about what > you're claiming here. It _seems_ to me that just saying "second > order" leaves a few things open that it seems like you're saying > it settles. I had misunderstood you. What I claim is that second order Zermelo set theory (let's call it 2Z) settles CH but there are things that it does not settle-see the next paragraph: > > But the notion of a measurable cardinal > >is third order [see Hanf & Scott _Classifying inacessible cardinals_ > >Notices AMS, 8, 1961], so it can't settle all questions about them, can > >it? > > > >CH or its negation is already a second order consequence of Zermelo's > >set theory with the "ordinary" axiom of infinity. > > I honestly don't know what you mean by this. "Is a consequence" sounds > like some proof-theoretic notion, but if that's what you mean then > it seems like if it's "settled" then it must be possible to say > which way it's settled. I mean this: "Either there is a proof in 2Z of CH or there is a proof in 2Z of not-CH." What I _don't_ mean by "there is a proof" is something like "X has just published a proof of CH (or not-CH) from 2Z in JSL." I mean "Either there is a sequence (having the requisite properties to make it a 2Z proof) which ends with (the formal expression of) CH; or there is a sequence (having the requisite properties to make it a 2Z proof) which ends with (the formal expression of) not-CH." I see nothing wrong, from the classical point of view, with a statement "X or Y, but we don't know which." "X or Y" might be a mathematical fact, but the not knowing is a matter of psychology--the two together do not contradict each other. If the use of the word "settle" suggests that it is known which of X or Y holds, then perhaps I shouldn't use it. > Try not to interpret the following question as an assertion: > Exactly what do you mean when you say that CH or its negation > is a second-order consequence of those axioms? See George Kreisel's _Informal Rigour and Completeness Proofs_ in Philosophy of Mathematics, ed Imre Lakatos, North-Holland 1972. Specifically, see page 150. Part of Kreisel's paper is reproduced in The Philosophy of Mathematics, ed J. Hintikka, OUP 1969; happily it's the relevant part. If I have misunderstood what I read there and hence made an incorrect claim when I said that 2Z proved CH or not-CH, my apologies. > David C. Ullrich