Uncountability of the Rationals?
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From: Transfer Principle on 15 Jun 2010 20:37 On Jun 15, 4:44 pm, Tony Orlow <t...(a)lightlink.com> wrote: > On Jun 15, 5:40 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > I believe that among the sci.math posters who argue against > > ZFC, the most common axioms to reject are Infinity and > > Choice, followed by Powerset (if they accept infinite but > > not uncountable sets). > That's generally the Anti-Cantorians, as opposed to us here Post- > Cantorians. See? Another interesting label, "Post-Cantorian." So here TO calls himself a "Post-Cantorian." I wonder what some of the key differences between these two labels are. > > We notice that in this theory, ~Infinity would be a theorem > > (proved via Deduction Theorem/contrapositives). > Okay. Do you agree wth that axiom, or are you just trying it on for > Halloween? ;) The latter, I guess. I was trying to see what sort of results it leads to, and whether it leads to any interesting results. > > It's doubtful that any textbook discusses sets that have no > > transitive closure, since most textbooks are grounded in ZF, > > which proves that every set does have one. A good starting > > point might be old zuhair threads, since zuhair mentioned > > transitive closures in his theories all the time. > The H-riffics have transitive closure, after some uncountable number > of iterations. Does that count? In that case, the poster whose ideas are represented by my theory isn't TO, at least if we're talking about his set of H-riffics. But there might be another sci.math poster whose ideas might be formulated in this theory.
From: David R Tribble on 15 Jun 2010 21:11 David R Tribble wrote: >> It's obvious that Tony can't and won't answer any of these >> questions, and no one here (except Walker) really sees him >> as capable of going any further in any meaningful sense. >> So except for deriving some entertainment value and learning >> a few new things, I don't see the point in indulging him any more. >> Teaching pigs to sing, and all that. > Transfer Principle (L Walker) wrote: > As expected, I don't believe that working with theories > other than ZFC or set sizes other than standard cardinality > is analogous to "teaching pigs to sing." I definitely > prefer to believe that there is a theory in which infinite > sets work differently from how they work under ZFC, and > perhaps working as TO or another poster would like them to. > > I believe that among the sci.math posters who argue against > ZFC, the most common axioms to reject are Infinity and > Choice, followed by Powerset (if they accept infinite but > not uncountable sets). I suppose I should admit that I was at least partially wrong; Walker has missed the point entirely.
From: David R Tribble on 15 Jun 2010 21:14 Tony Orlow wrote: >> Point-set topology. How do you define each unique point? > Virgil wrote: > When one starts with a set of points, they are all already there. > What need is there of having to replace them with something else? > > Does TO suppose that there was no geometry prior to de Carte? The ancient Greeks used to label unique points as A, B, C, (actually alpha, beta, gamma) and so on. That seemed to work out pretty well for them.
From: David R Tribble on 15 Jun 2010 21:21 Tony Orlow wrote: > Just because an infinite set is defined by the fact that it can be > bijected with a proper subset ... That's only how Dedekind infinite sets are defined. Non-Dedekind infinite sets are not defined that way. Remember? > ... doesn't mean that this bijection implies > the counterintuitive equinumerosity that one would NOT expect from a > "correct" theory. "Correct" as in "derived from its axioms", or something else?
From: Jesse F. Hughes on 15 Jun 2010 21:29
Tony Orlow <tony(a)lightlink.com> writes: > On Jun 15, 2:17 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> Tony Orlow <t...(a)lightlink.com> writes: >> > On Jun 14, 4:49 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: >> >> David R Tribble <da...(a)tribble.com> writes: >> >> >> > Serves me right for trying to think like Tony. >> >> >> > Oh well, this is easily solved! The problem is that TA1 is too >> >> > general, so we need to limit its scope. Instead of any property >> >> > P, let's just consider arithmetic properties and operators. >> >> > So, perhaps something like this: >> >> >> > Tony's Axiom [TA2] >> >> > Given arithmetic function f(x) = y, where x and y are reals, >> >> > then f(w) = u, where w and u are infinite ordinals. >> >> >> > Yeah, I know this really isn't any better when taken any further, >> >> > but Tony really needs to put his money where his mouth is and >> >> > just go ahead and write his assumptions down already. He can't >> >> > proceed with any of his other statements about bigulosity, >> >> > ICI, IFR, or whatever until he starts with an axiom or two. >> >> > And he knows it. >> >> >> It seems to me that he's stated his assumption, but he doesn't get quite >> >> how much it assumes. >> >> >> Given any real valued functions f and g, if lim (f - g) > 0, then f(x) >> >> > g(x) where x is any infinite number. >> >> > That's actually a nice succinct way to state it, however I generally >> > include the condition that finite-case induction applies to f-g as >> > well, to make clear that it is an extension of the finite form of >> > proof to the infinite case. >> >> >> As a consequence of this, I guess, it follows that f and g are defined >> >> on infinite numbers, though we don't know anything about their values >> >> aside from the fact that f(x) > g(x). >> >> > No, that's correct. At best you can say one set is greater then the >> > other, and you can quantify that difference as the ratio of the sizes >> > functionally defined over [1,w]. There is some math that can be done, >> > but it's nothing all that super-special. The special part is that it >> > provides for a countably large spectrum of countably infinite sets >> > which satisfy normal intuitions, such as the proper subset always >> > being smaller, as well as more general notions. That seems like a >> > little bit of an advance to me. >> >> That little bit of math is utterly unclear. >> >> For instance, you claim that sqrt(w) * sqrt(w) = w, but that does *not* >> follow from the above principle, *even if* we assume that the above >> principle entails that sqrt(w) is defined. >> >> The above principle just allows us to conclude that sqrt(w) < w. > > True. So? It doesn't mean there isn't an easy way to envision omega^2 > or sqrt(omega). Envision? Look, we're supposed to be able to prove whatever properties you think are true of sqrt(w) and w^2. We shouldn't have to magically "envision" them. Thus, if sqrt(w) * sqrt(w) = w, there should surely be a proof of this fact. How do you suggest we do this, aside from our magical envisioning powers? -- "It is my opinion that since neither Spight nor Hughes can see or understand their moral trespass [namely, quoting AP in .sigs], that their degrees from whatever university they earned their degree should be annulled." -- Archimedes Plutonium (12/1/09) |