Counterintuitions and the well-ordering theorem
in [Logic]
|
Prev: Solutions manual to Entrepreneurship 1e Bygrave Zacharakis
Next: Solutions manual for Engineering Mechanics Statics (12th Edition) by Russell C. Hibbeler
From: Daryl McCullough on 17 Nov 2009 09:18 Bill Taylor says... > >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> But what does that really mean? What does it mean to say that >> the collection of all reals > >They would say there is no such beast. What does it *mean* to say that there is no such beast? Clearly, the term "the subsets of omega" exists as a meaningful term, a meaningful collective noun. What does it mean to say that it does not exist "as a collection"? It seems to be completely empty words. If you have an unambiguous, consistent membership criterion, then you have a collection, it seems to me. If not, then what does it take to have a collection "really exist"? What does "really exist" mean, when referring to abstract objects? It's funny that you should feel that the collection of all naturals exist as a set, but not the collection of all reals. To me, we can understand collections as types, which are just terms built using type-formation operations. For example: If A and B are types, then so are 1. A (+) B, the disjoint union of A and B, 2. A (x) B, the set of ordered pairs <x,y> where x is of type A, and y is of type B. 3. A -> B, the set of functions f which when given an object of type A, return an object of type B. To get things started, we also allow explicit types, where all the elements are given explicitly. For example, boolean = { true, false } What we *can't* get out of our type language is the type of all naturals. No finite construction can yield the naturals. So we just have to posit the type N, and posit that 0 is in N, and that S is a one-to-one function in N -> N such that 0 is not in the image of S. But after we have posited N, the reals and sets of reals are completely unproblematic. We identify "the reals" with the type N -> boolean. We identify "the power set of the reals" with the type (N -> boolean) -> boolean. As a *type* (which is just an expression in a type language), the reals certainly exist if the naturals do. What does it mean to claim that some types "form collections" and some do not? I really don't have any idea what that claim means. -- Daryl McCullough Ithaca, NY
From: WM on 17 Nov 2009 11:04 On 17 Nov., 09:44, Virgil <Vir...(a)home.esc> wrote: > In article > <734ce777-8e88-4030-9a79-4fe211ced...(a)w19g2000yqk.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 17 Nov., 05:00, Bill Taylor <w.tay...(a)math.canterbury.ac.nz> wrote: > > > > Naturally, to one committed, perhaps unwittingly, to a certain > > > particular all-embracing point of view, anything that goes counter > > > to it will seem incomprehensible. I well recall how, in my younger > > > days, I was astounded that there could be anything controversial > > > about AC - it just seemed so OBVIOUS! Decades later, I began > > > to acquire suspicions, as greater familiarity with math came. > > > Similarly, now even more decades later, and still slowly, it is > > > finally occurring to me that there are similar suspicions about > > > the completion of the entity of (all) the reals. > > > It is rather simple. > > WM is what is rather simple > > > A "number" that cannot be identified or addressed > > so that we can talk about it > > And which reals are ones which we cannot talk about? Those which have no finite definitions like the algebraics or some transcendentals like pi. Obviously (for those who can think simply - those who simply can think) it is impossible to name a number by an infinite lawless sequence of digits or symbols. By the way, this has been forgotten by Cantor and his followers. There is no finished infinite sequence. Therefore Her Majesty, the divine "diagonal number", is not a number and does not prove anything about countability or uncountability. Divinitely! Regards, WM
From: WM on 17 Nov 2009 11:07 On 17 Nov., 12:53, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > I don't see how that is relevant. We can certainly talk about > definability relative to a language (for example, relative to > the language of ZF). That is not problematic. The collection > of all reals *is* definable relative to this language. But the elements of this collection are not all definable. You can define the the straight line between two points. But you cannot define all points of that line. Regards, WM
From: Daryl McCullough on 18 Nov 2009 10:16 Jesse F. Hughes says... > >stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > >>>If one has NO feeling for the concerns of suspicion involved, >>>it hardly seems likely that mere debate can produce a change. >> >> That's a cop-out, it seems to me. > >Indeed. At the least, he should make these concerns of suspicion >explicit. I can't see that he has presented any such concerns that >cast the slightest doubt on the power set axiom. I guess I'm confused about the ontological status of abstract objects such as sets. It seems silly to think of them realistically, as if there exists, in some Platonic heaven, the universe of sets, in which all questions, such as the existence of measurable cardinals, is resolved. On the other hand, it also doesn't seem right to treat them as pure fictions, where the question of "Does there exist a cardinality between N and P(N)?" has no more of an answer than "What's the name of Sherlock Holmes' grandmother's dog?" In any case, I can understand being skeptical about the *consistency* of a statement such as "every set has a power set", but beyond consistency, I don't really understand what it *means* to doubt its truth. Does it really make sense to doubt the truth of the claim that there is a number i such that i*i = -1? We essentially *define* i into existence by that claim. It seems to me that the power set of omega is similarly defined into existence. I'm not 100% sure, though. Is introducing P(omega) and introducing square-root(-1) the same sort of mathematical act, or is there some subtle difference? -- Daryl McCullough Ithaca, NY
From: George Greene on 18 Nov 2009 10:49
On Oct 12, 7:24 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > the usual explanations of > the supposed counter-intuitiveness depend on the baffling idea that > non-measurable sets corresponds to "cuttings" in some physical > sense. That idea IS NOT "baffling". Much MORE baffling would be the notion that partitioning something into sets DIDN'T constitute "cutting". |