Uncountability of the Rationals?
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From: Jesse F. Hughes on 24 Jun 2010 10:01 Brian Chandler <imaginatorium(a)despammed.com> writes: >> So what gives? Earlier in this thread, I mentioned a schema which >> might fit TO's theory: >> >> If phi doesn't contain the symbol "tav," then all closures of: >> >> ((En fin. nat. (phi(n)) & Ax (phi(x) -> phi(xu{x}))) -> phi(tav) > >> are axioms. In other words, if all but finitely many naturals share a >> property phi, then so does tav. > > OK, so if phi(n) is the property "n is not tav", this means that given > an infinite set of naturals, none of which is 'tav', this implies that > tav is not tav. Doesn't sound likely to me. In Walker's defense, he said that the formula phi should not contain the symbol tav. -- Jesse F. Hughes "[Iota]'s the smallest infinitesimal, Russell, there are smaller infinitesimals." -- Ross Finlayson
From: Brian Chandler on 24 Jun 2010 10:34 Jesse F. Hughes wrote: > Brian Chandler <imaginatorium(a)despammed.com> writes: > > >> So what gives? Earlier in this thread, I mentioned a schema which > >> might fit TO's theory: > >> > >> If phi doesn't contain the symbol "tav," then all closures of: > >> > >> ((En fin. nat. (phi(n)) & Ax (phi(x) -> phi(xu{x}))) -> phi(tav) > > > >> are axioms. In other words, if all but finitely many naturals share a > >> property phi, then so does tav. > > > > OK, so if phi(n) is the property "n is not tav", this means that given > > an infinite set of naturals, none of which is 'tav', this implies that > > tav is not tav. Doesn't sound likely to me. > > In Walker's defense, he said that the formula phi should not contain the > symbol tav. Oh, right. Oh, golly. Well, perhaps that's OK then. I really don't know, but it sounds awfully fishy to me. Brian Chandler
From: Brian Chandler on 24 Jun 2010 10:41 Jesse F. Hughes wrote: > "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: > > Transfer Principle <lwalke3(a)lausd.net> writes: <snip> > > If not, the usual construction of w suffices to define your set of > > pofnats. > > Well, this last claim of mine is not so clear, since it's possible that > w defined this way still has infinite elements (because, perhaps, every > inductive set in this theory has infinite elements). Note, however, .... > Finally, > (3) even if w has infinite elements, w serves as a canonical choice for > Tav, since it is the least inductive set. Why doesn't Tony just use w? I think I can answer this bit. Tony tells us that "ordinals are schlock" (sorry no reference, Google search up spout again; try sci.math 2006 passim), and Tony has no use for omega. Anyway I think you are still stuck in set theory, and omega is a set; Tony uses numbers, and infinities, and "sizes" expressed in "unit infinities". 'Tav' was my attempty to coin a non-confusing name for a declared "unit infinity". Brian Chandler
From: Jesse F. Hughes on 24 Jun 2010 11:21 Brian Chandler <imaginatorium(a)despammed.com> writes: > Jesse F. Hughes wrote: >> "Jesse F. Hughes" <jesse(a)phiwumbda.org> writes: >> > Transfer Principle <lwalke3(a)lausd.net> writes: > <snip> >> > If not, the usual construction of w suffices to define your set of >> > pofnats. >> >> Well, this last claim of mine is not so clear, since it's possible that >> w defined this way still has infinite elements (because, perhaps, every >> inductive set in this theory has infinite elements). Note, however, > ... >> Finally, >> (3) even if w has infinite elements, w serves as a canonical choice for >> Tav, since it is the least inductive set. Why doesn't Tony just use w? > > I think I can answer this bit. Tony tells us that "ordinals are > schlock" (sorry no reference, Google search up spout again; try > sci.math 2006 passim), and Tony has no use for omega. Yes, but as far as I can tell, omega is definable in his theory. Of course, I can't be certain, since I don't know what his theory is, but if it is definable, then it seems to be a perfectly good choice for tav, Tony's opinions notwithstanding. > Anyway I think you are still stuck in set theory, and omega is a set; > Tony uses numbers, and infinities, and "sizes" expressed in "unit > infinities". 'Tav' was my attempty to coin a non-confusing name for a > declared "unit infinity". Tony will have to explain why I can't declare tav=omega, if this is his opinion. -- Jesse F. Hughes "A factor is simply something that multiplies against another factor to produce a 'product'." -- James Harris offers a definition.
From: David R Tribble on 24 Jun 2010 11:30
Transfer Principle (Walker) quoted: >> Tony Orlow wrote, crica 2005: >> Each pair of naturals has a difference of 1. The largest member >> of a set of n distinct naturals will be at least n. >> Therefore, a set of an infinite number of natural numbers will contain >> infinite values. If there is no largest finite number, is there a largest >> finite set of naturals? Why do we consider the naturals to be finite >> and the set to be infinite, when the members are a finite constant >> quantity apart frome each other? > Brian Chandler wrote: > Well, there you go. It's the loop of string argument. But this > argument is totally bogus. (Surely Walker can see that?) You left out the part where Tony decides that there are "finite but unbounded" sets, like N. That is, they contain only finite elements (numbers), but since they have no "identifiable" largest element, they are "unbounded". The situation is, and has always been since Tony started posting, that he wants everything to act like "numbers" including set "sizes", and specifically that they can be mapped to points on the real number line and obey the usual rules of (finite) arithmetic. Thus his constant mention of properties "without regard to finiteness", "infinite induction", "formulaic" infinities, infinite set members, "unit infinities", "bogus" bijections, and so forth. Look, Tony has demonstrated that he's a Crank many times over, with all of the usual symptoms: unwillingness to change his ideas even in the face of blatant illogic, abusive replies to disbelievers in his faith, proofs by declaration, assumed authority on terminology and topics he has no clue about, expectation of the impact of his ideas on the world, etc. I admit to finding the last few years occasionally entertaining, but lately it's become, well, boring. Nothing new is being offered, and more importantly, I'm not learning much new from the posters who actually know something about set theory. |