Counterintuitions and the well-ordering theorem
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From: Rupert on 22 Nov 2009 02:33 On Nov 21, 12:27 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > Daryl McCullough wrote: > > Bill Taylor says... > >> stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > > >>> Does it really make sense to doubt the truth of the > >>> claim that there is a number i such that i*i = -1? > >> That's totally different and completely unobjectionable. > >> Complex numbers can be defined trivially as ordered pairs with > >> the appropriate operations, and all the usual results obtained. > >> It has no logical or philosophical problematicity at all - > >> i.e none beyond whatever the reals that make them up, already have. > > > I don't think it is different at all. The fact that you can > > interpret complex numbers as ordered pairs is just a simple > > case of coming up with a *model* of complex numbers, which > > exists if the theory complex numbers is consistent. The same > > is true of the power set. We define the power set by axioms. > > If those axioms are consistent, then there exists a model > > of those axioms. > > Can you prove "if ZF minus powerset is consistent, then so is ZF > including powerset", without presuming the latter? > > Or even "if ZF minus powerset is consistent, then so is ZF minus > powerset plus (there is a set containing all subsets of N)" ? > > -- > Cheers, > Herman Jurjus- Hide quoted text - > > - Show quoted text - The answer to your first question is no. Let ZF-P be ZF minus powerset. ZF proves that ZF-P is consistent, as Keith Ramsay explained. One can prove in ZF that there is a set of all hereditarily countable sets and that this is a model of ZF-P. However ZF, if consistent, cannot prove that ZF is consistent. However, I think your latter statement might be provable. We could take the theory ZF-P and then add to it a recursive comprehension axiom schema which allows us to form recursive subsets of the hereditarily countable sets, and anything that can be constructed from there using ZF-P. I would think that the resulting theory would be bi- interpretable with ZF-P. That is worth looking into. There is a good discussion in Simpson's "Subsystems of Second-Order Arithmetic" of a theory called ATR_0^set, a weak fragment of ZF-P, which is bi-interpretable with a weak fragment of second-order arithmetic called ATR_0.
From: Keith Ramsay on 22 Nov 2009 04:26 On Nov 20, 5:12 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: |Bill Taylor says... |>stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: |>> Does it really make sense to doubt the truth of the |>> claim that there is a number i such that i*i = -1? | |>That's totally different and completely unobjectionable. |>Complex numbers can be defined trivially as ordered pairs with |>the appropriate operations, and all the usual results obtained. |>It has no logical or philosophical problematicity at all - |>i.e none beyond whatever the reals that make them up, already have. | |I don't think it is different at all. The fact that you can |interpret complex numbers as ordered pairs is just a simple |case of coming up with a *model* of complex numbers, which |exists if the theory complex numbers is consistent. The same |is true of the power set. We define the power set by axioms. |If those axioms are consistent, then there exists a model |of those axioms. The fact that it is more complicated than |ordered pairs doesn't seem particularly important to me. It seems to me that the main difference is that the attempted justification for the use of complex number succeeded with relative ease. (It took a good span of time to settle the issue but since the power set was stated we've had a lot more activity on the philosophical and logical fronts than they were able to do when the complex numbers were a mystery.) I think the success of the project is due to more than just the consistency of some set of axioms for complex numbers, though. The way I like to think of it is as the development of an interpretation of a language. That way, not only is it consistent to have a square root of -1 in a field, but we can rest assured that a question about complex numbers, whether there exists a z such that e^z=z for example, really means the same thing as the corresponding question about real numbers, whether there exist numbers a,b such that a=e^a cos(b) and b=e^a sin(b). This is an issue for me with the axiom of choice. I've made a small stab at coming up with a satisfactory interpretation of set theory with the axiom of choice. Let's say, with a set of representatives for R/Q. Now I have a model that would be adequate, L, if all I wanted was to make the usual axioms and AC true in the model. But it interprets statements wrong. It converts the existence of a measurable cardinal into an odd and provably false statement about ordinals; but that's not what I mean by the existence of a measurable cardinal. It converts the existence of 0# into an odd and provably false statement, but one which doesn't mean what I mean if I were to talk about the existence of 0#. The Goedel completeness theorem supplies us with models of recursively enumerable sets of axioms that are themselves arithmetically definable, which means that in general the natural numbers in them are wrong and interpret other arithmetic statements A into arithmetic statements A' having only a certain number of quantifiers in them, where one happens not to be able to prove the inequivalence of A and A' with your axioms. It seems possible on the other hand that a suitable interpretation of the axiom of choice could exist. There are constructivists who've shown that certain results provable from AC can be interpreted constructively by looking at the contrapositive. AC says that under suitable conditions a subset of a set having some property exists. The contrapositive says that if every subset of a set has some property, then it's because certain conditions hold. This may not be entirely palatable constructively, but we can at least use it as a metamathematical tool: if you have proven that every subset of a set has some property, we can say something about what else you must've proven along the way. Suppose you've proven that no ideal of a ring R containing {x,y,z} is maximal. Then it surely follows that you've actually proven that 1 is contained in the ideal generated by {x,y,z}, and hence that 1=ax+by+cz for some a,b,c in R, even if you don't realize that. I think there might be a general way to translate statements about these purported choice sets like a set of representatives of R/Q that would seem faithful to me, but I don't know how to do it yet. With the powerset axiom, I have the feeling that it's not so much the axiom on its own as its combination with certain others that is at issue. Or to put it another way, it seems like we can cook up interpretations that fit with it, but maybe not the ones that fit with everything else. One philosopher, and I wish I could remember for sure which one, suggested that the cumulative hierarchy motivating ZFC was a kind of chimera, a mix of two motivations that worked better separately than together: on the one hand, we have the idea that the problem with the Russell set and so on has to do with its being "too big" in some sense. This idea is good for motivating the replacement axiom, since the replacement axiom creates something that at least is not bigger than the set it's being applied to. On the other hand, we have the idea that the problem had to do with circularity of membership, which more directly motivates the cumulative hierarchy. I think in fact the combination is O.K., but it's a potential concern. I think impredicativity is more of a meaningful concern here. If you have a set of natural numbers, in the usual context, then you also have all the subsets definable by formulas quantifying over sets, which include all such possible subsets of the natural numbers. Hence you have that kind of impredicativity. If you have a set of all subsets of natural numbers (or equivalent if you have a real line), then you have also all of the subsets produced by selection, where the formula can now quantify over all of those subsets. That's a further kind of impredicativity. (But seriously, it hardly seems likely that the one kind would be O.K. while the other is not.) I think it's a valid concern that impredicative definitions might be consistent without being entirely meaningful. You could have a model containing a phony real line that merely had enough of an appearance of completeness to make all of your impredicative definitions be satisfied. One wants to make sure that we're not just lucking out in this way. It should be, as most of us expect, that for each such definition, there is a bona fide set of natural numbers, the one that the definition gives you if you apply it to the model with *all* the sets of sets of natural numbers included. I just find it very difficult to imagine what sort of argument could be used to further justify the sort of vague intuition most of us seem to have that this is okay. Keith Ramsay
From: Bill Taylor on 22 Nov 2009 06:57 Well, I really think this debate, between me and Daryl at least, has gone beyond the point of usefulness. I write quite long articles, attempting to explain my concerns, and get replies niggling at small points, while major themes are ignored. No doubt it seems exactly the same the other way round. So this is really a sign-off, unless some other worthwhile points are raised, as yet unforeseen. I just append a few trivialities, for the sake of some actual content. stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > I don't feel that the status of N is much different than > the status of P(N). Oh, I completely misunderstood your last post then. You said (something like) "the ontological status of sets is suspect" (or words to that effect, I can't call up the exact quote right now), so I thought, by mentioning sets specifically, that you meant there was something MORE suspect about them than about other abstract objects. But now it seems, from the above, that you regard all abstract objects on the same footing, in this respect at least. Sorry for the misunderstanding. No doubt the fault is mine, but I can't escape the feeling that I've been wriggled away from... > The claim "There is no power set of omega" is just as dubious, > ontologically, as the claim "The power set of omega exists". This SURELY can't be right! The default option is for things NOT to exist, e.g. fairies etc. So the claim that something *doesn't* exist is much safer than the opposite. For a thing to exist needs the definite, firm evidence! Please don't feel the need to reply if there is nothing new to add. -- Beaten Bill
From: Daryl McCullough on 22 Nov 2009 09:12 Bill Taylor says... >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> The claim "There is no power set of omega" is just as dubious, >> ontologically, as the claim "The power set of omega exists". > >This SURELY can't be right! The default option is for things NOT >to exist, e.g. fairies etc. That default is part of the scientific method for discovering the truth about the *physical* world. What does it say about abstract objects? *WHY* should there be a default? You haven't said what it *means* for an abstract object to exist or not exist. >So the claim that something *doesn't* exist is much safer than >the opposite. Why? What does "safe" mean here? You want to apply the scientific method to abstract objects, as if they were part of the natural world, as opposed to things created by the human mind. I don't see how that makes any sense. You are taking a principle that is justified in one domain (physical science) and applying in another domain where there is no justification. -- Daryl McCullough Ithaca, NY
From: Frederick Williams on 22 Nov 2009 10:04
Bill Taylor wrote: > This SURELY can't be right! The default option is for things NOT > to exist, e.g. fairies etc. Why? -- Which of the seven heavens / Was responsible her smile / Wouldn't be sure but attested / That, whoever it was, a god / Worth kneeling-to for a while / Had tabernacled and rested. |