Counterintuitions and the well-ordering theorem
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From: Herman Jurjus on 10 Nov 2009 02:22 Bill Taylor wrote: > ... utter incredulity to the suggestion > that it be in some way false, or at least dangerous and dark. Just in case this wasn't clear before: I agree with your assessment of the power set axiom. > Mathematics, now, runs on ZFC the way Freecell runs on Windows. > > So what would math be with a non-Cantorian set theory as its > platform or operating system? Most people might say, hugely > different, but I think they would be wrong. Obviously some > stuff that is intimately entwined with set theory would have > to change, but not greatly. We could still have Cartesian > Products, ordered multiplets, sequences, continuous functions, > and the whole panoply of C19 math - differential equations, > optimization, analysis, tensors, etc etc virtually unchanged, > without PS. No-one has ever tried this (outside constructivism), > mostly because, "Why bother?" - a comment often levelled at > constructivists as well, intriguingly. But it could be done, > without too much trouble, as well. What do you think about the separation axiom schema? Should it be restricted to formulas in which all the quantifiers are bounded (i.e. 'for all x in V' instead of 'for all x' sec)? If you allow the current 'full' version, you'd still allow (a.o.) quantifying over all subsets of N. Does that make sense, when throwing out PS? -- Cheers, Herman Jurjus
From: Bill Taylor on 11 Nov 2009 23:19 Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Bill, I will get back to you onyourmumblingsabout the powerset > axiom, definability, what not, later. Excellent. I look forward to some helpful mutterings from you. > Here I'd just like to note this > thread started out with a (perfectly ingenuous though possibly not > very serious-minded) enquiry Possibly not. And it goes along with your notable article here about what kinds of joy you (and I) get from Usenet debates. > as to the alleged /counter-intuitiveness/ > of the well-ordering theorem, this enquiry prompted by my (possibly > erroneous and arbitrary) notion that people's statements about I was surprised by the initial inquiry, coming from *you*, as it did. It's more the sort of thing that John Jones would ask. Surprising, especially in view of the (I think obvious) fact that large numbers of people DO find it unintuitive, as the well-known and here-repeated anecdote attest. > counter-intuitiveness, evidence, etc. often are almost totally > arbitrary -- that is, it is often impossible to get any explanation > whatever of e.g. what intuitions are contradicted by this or that. Again, I am surprised, as I would have thought the elucidations were common enough, even if not up to rigorous logical standards. It seemed to me, almost, as if your original enquiry were provocative. But then, so what if it was! > I don't recall if it's been mentioned already, but Sol Feferman's paper > _Mathematical Intuition vs. Mathematical Monsters_ is an enjoyable and > relevant read in this context.) Thanks for the ref! It sounds fun, indeed! Will read asap. -- Breathless Bill
From: Bill Taylor on 12 Nov 2009 00:08 Herman Jurjus <hjm...(a)hetnet.nl> wrote: > Just in case this wasn't clear before: > I agree with your assessment of the power set axiom. Excellent! We are now TWO voices alone in the wilderness. :) > What do you think about the separation axiom schema? > Should it be restricted to formulas in which all the quantifiers > are bounded (i.e. 'for all x in V' instead of 'for all x' sec)? I'm not fully sure I follow the question, (e.g. what was that "sec" in the last line?) However, noting that quantifying over sets of naturals is (encyptically) quantifying over reals, there is still a considerable difference, seemingly, between quantifying over reals and quantifying over SETS of reals. Quantifying over reals doesn't seem to hold many terrors, (though I could be wrong); but quantifying over sets of them is a whole nother matter. It is in this latter that LUB makes its appearance. Typically, the LUB of a set of reals occurs at a greater definability level than any of its boundees, and (presumably) of the set of them. In a similar way, Cantor's uncountability theorem has its content subtly altered, though not if its original statement is kept in its proper form - that for any list of reals there is a real not on the list. This now is seen as a rather simple corollary of the fact that reals occur in distinct levels, which are indefinitely extensible. It seems that any statement involving sets of reals, must inevitably be interpreted as thinking of *some* set of reals at *some* level. So no, not all reals. And the consequent of the statement must allow the possibility of going beyond that original level. But nevertheless the theory itself need not, cannot, make any mention of the levels, much as ordinary ZF never mentions the class V, its intended model. > Does that make sense, when throwing out PS? I hope it makes *some* sense. Any further comments would be most welcome. -- Baffled Bill
From: Daryl McCullough on 12 Nov 2009 13:23 Bill Taylor says... > >Herman Jurjus <hjm...(a)hetnet.nl> wrote: > >> Just in case this wasn't clear before: >> I agree with your assessment of the power set axiom. > >Excellent! We are now TWO voices alone in the wilderness. :) People skeptical of the power set usually balk at the very first application that gets you something really new: P(omega), or the reals. So abandoning the power set axiom means, in this case, considering the collection of all reals to be a proper class, rather than a set. But what does that really mean? What does it mean to say that the collection of all reals (which exists in the sense that it is a definable class) does not exist *as* *a* *set*? It seems to me that we only accept the set/class distinction because it is *forced* on us by consistency. In the case of P(omega), it's *not* forced on us by consistency considerations, so why consider it a proper class, rather than a class? I can't understand the motivation for that. -- Daryl McCullough Ithaca, NY
From: Rupert on 12 Nov 2009 18:01
On Nov 13, 5:23 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Bill Taylor says... > > > > >Herman Jurjus <hjm...(a)hetnet.nl> wrote: > > >> Just in case this wasn't clear before: > >> I agree with your assessment of the power set axiom. > > >Excellent! We are now TWO voices alone in the wilderness. :) > > People skeptical of the power set usually balk at the very first > application that gets you something really new: P(omega), or the > reals. So abandoning the power set axiom means, in this case, > considering the collection of all reals to be a proper class, > rather than a set. > > But what does that really mean? What does it mean to say that > the collection of all reals (which exists in the sense that it > is a definable class) does not exist *as* *a* *set*? It seems > to me that we only accept the set/class distinction because > it is *forced* on us by consistency. In the case of P(omega), > it's *not* forced on us by consistency considerations, so > why consider it a proper class, rather than a class? I can't > understand the motivation for that. > > -- > Daryl McCullough > Ithaca, NY Unrestricted comprehension gets you into trouble, right? Most set theorist types want as much comprehension as you can get without inconsistency. But there's also a case for saying we should be conservative. We know unrestricted comprehension gets you into trouble, so maybe we should have as little comprehension as is indispensible for scientifically applicable mathematics. That is the point of view of the predicativist such as Feferman. |