Counterintuitions and the well-ordering theorem
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From: Nam Nguyen on 20 Nov 2009 00:54 Bill Taylor wrote: > We have no doubt about > individual natural numbers, or even N, because we feel that > we know "all about them" in some sense. Though various theorems > still surprise us of course, we feel that there can be no further > *philosophical surprises* from them, if that phrase strikes a chord. The difficulty is what exactly would we mean by "in some sense"? There lurks in any concept as strong as the naturals at least one formula about the concept that could be impossible for us to know its truth value. In other words, we know as much about the naturals as we know about sets in an informal, naive way.
From: Albrecht on 20 Nov 2009 03:23 On 20 Nov., 06:54, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Bill Taylor wrote: > > We have no doubt about > > individual natural numbers, or even N, because we feel that > > we know "all about them" in some sense. Though various theorems > > still surprise us of course, we feel that there can be no further > > *philosophical surprises* from them, if that phrase strikes a chord. > > The difficulty is what exactly would we mean by "in some sense"? > There lurks in any concept as strong as the naturals at least one > formula about the concept that could be impossible for us to know > its truth value. If we can connect such a undecidable formula with an individual natural number in a suitable manner, this number would be not fully determined. If we claim that a set is well defined only in the case that we are, at least potentially, able to know any element, we had to conclude that the set of naturals doesn't exist since it would include at least one such not fully determined element. Best regards Albrecht > > In other words, we know as much about the naturals as we know about sets > in an informal, naive way.
From: Daryl McCullough on 20 Nov 2009 07:12 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. >> We essentially *define* i into existence by that claim. >> It seems to me that the >> power set of omega is similarly defined into existence. > >It is a totally different question in every respect. No, it isn't. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 20 Nov 2009 07:37 Bill Taylor says... > >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >You admit doubts about the O.S. of sets; I presume >(maybe wrongly?) you have no, or at least much lesser, >doubts about the O.S. of natural numbers. I don't see any big difference between the two. The set of naturals and the set of reals are both abstractions. I don't understand in what sense either exists, other than exists as a coherent topic of study. >If this is so, then my own suspicions about powerset etc, >are motivated by similar concerns. I don't see how my concerns motivate suspicion about powerset. I don't know what it *means* to claim that the power set exists or doesn't exist, so neither choice makes any more sense to me than the other. My doubt is not about whether these sets exist, but about what it *means* to say they exist (or DON'T exist). The claim "There is no power set of omega" is just as dubious, ontologically, as the claim "The power set of omega exists". >We have no doubt about individual natural numbers, or even >N, because we feel that we know "all about them" in some sense. I don't feel that the status of N is much different than the status of P(N). Yes, N has the nice feature that every natural has a finite name. But why should that matter? P(N) has a finite name, as well. >What reals numbers (and beyond) are in some sense >INEVITABLE in math, regardless of the framework we choose >to "run" math on top of. I think that given any language for talking about naturals, the "inevitable" reals are those that are definable in that language. Those are "inevitable" in the sense that they exist in every model of the theory of reals and naturals that include that language. >And if it turns out that it is OK, there are still the r.e. reals, >the definable reals, and so forth, all with graver suspicions about >their "reality", their OS, that is their inevitability independent >of the way math is framed in substrate. I think you've drifted off into a different topic. We were talking about whether P(N) exists. You've switched to the topic of which reals exist. I don't see the connection, really. If you want to say that only "inevitable" reals exist, then P(N) is the set of inevitable reals. -- Daryl McCullough Ithaca, NY
From: Herman Jurjus on 20 Nov 2009 08:27
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 |