Uncountability of the Rationals?
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From: MoeBlee on 4 Jun 2010 16:24 On Jun 4, 3:20 pm, David R Tribble <da...(a)tribble.com> wrote: > Tony Orlow wrote: > > One might think there were something like aleph_0^2 rationals, but > > that's not standard theory. > > Actually, there are Aleph_0^2 rationals. And Aleph_0^2 = Aleph_0. Orlow can't be bothered to learn such basics. MoeBlee
From: Tony Orlow on 4 Jun 2010 16:24 On Jun 4, 2:32 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 3, 7:34 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > mathematicians misrepresent the theory, > > claiming that every conclusion they draw "follows logicallly from the > > axioms." This claim is simply not true, as cardinality is not > > mentioned in the axioms, much less anything about omega or the alephs, > > except for a declaration that something homomorphic to the naturals > > exists, which is ultimately not a set, but a sequence. > > No, all of those terms are given by DEFINITIONS, which may take the > form of definitional AXIOMS, which provide a conservative extension of > the theory in its primnitive form. This has been explained to you at > least a hundred times already, but you choose just to ignore. Can you please provide a complete list of your definitional axioms for my inspection? They are not part of ZFC, are they? > > Moreover, we don't just show that there exists a set homomorphic to > the naturals (and their customary ordering) but rather that we define > 'w' (omega) to be the least set that has 0 as a member and is closed > under successsorship. Moreover, in set theory, sequences are sets. Sequences are sets with order. Sets in general have no order. If set theory would like to hide the recursive nature of infinite sets in order to draw precarious conclusions about sequences as if they "just exist", then it behooves the mathematical community to encourage more investigation into the distinction and draw conclusions from that information. Do you disagree? > > > By demanding to > > know with which axioms a non-transfinitologist disagrees, > > mathematicians obscure the question. > > WHAT question? MoeBlee eat some meat or take vitamins. The question is, "Why do you object to set theory, O Crackpot?" Isn't that what you've been asking all along? > > > After some years of this > > discussion have arisen some obvious culprits: the von Neumann ordinals > > as some complete set, > > You're confused. Ordinary set theory does not assert that there is a > set of all the von Neumann ordinals. However, in ordinary set theory > it is proven that there exists a set of all the FINITE von Neumann > ordinals. But fine if you reject that there exists a set of all finite > von Neumann ordinals. Then, if you accept that there is an empty set > and the pairing axiom, then you only (if I'm not missing a detail > here) need reject the axiom of infinity to decline that there is a set > that has all the von Neumann ordinals. Here you have a minor point. I meant the finite von Neumann ordinals as a complete set, and the non-finite limit ordinals as anything but phantoms. You may call N a set, but it is really a sequence that is never complete at any point. Did you miss the word, "complete"? That was a kind of important word. As a mathematician (or whatever) you shouldn't be leaving out symbols, much less whole words. I do not discount the existence of N, but the Axiom of infinite is not about a set, really, but a sequence without end. You do get the distinction, no? Perhaps you're confused. > > > and the simplistic assumption of equivalence of > > "size" based on bijection alone. > > I've presented to you at least a hundred times the answer (in > variation) that instead of the word 'size' we could use the word > 'zize'. Then your protest disappears. You've never dealt with that > point. I used "Bigulosity", remember? Surely you do, you rascal. > > > If bijection determines equal > > cardinality that's fine, but to equate cardinality with set size > > simply does not work for infinite sets to the satisfaction of most > > people's intuitions. > > Most people haven't studied the subject. And, again, everywhere in > discussions about set theory you may substitute 'size' with 'zize'. No > substantive matter to the FORMAL theory. "The" formal theory. I like that. You should really become a Hare Krishna devotee. > > > To claim that they are being illogical by > > objecting to the theory without identifying an axiom at fault is > > disingenuous, since the axioms are clearly not the problem. > > The problem is that you demand that the word 'size' be used in a way > that does not conflict with your ordinary notions about size. But set > theory does not represent that such words are used in a way that > conforms in every sense to such everyday notions. Well, perhaps it satisfies your "everyday" experiences with infinity. Or, perhaps, you don't have any. If you don't call it "size" I won't. You say "cardinality", and I sqay "bigulosity". > > > The von Neumann limit ordinals are > > simply bunk. > > Fine, then probably you want to throw out the axiom of infinity. It says nothing about the non-finite limit ordinals. See above. > > > Everything Jesus said was true and wise, but the Trinity > > and the Immaculate Conception are Christianity's von Neumann limit > > ordinals, thrown in later for Constantine's political goal of state > > religion incorporating pagan beliefs and christian. > > So who is the Jesus of set theory before statist limit ordinals > corrupted the whole thing? Good question. Perhaps Cantor and Dedekind in modern times, though Galileo certainly explored bijection long before, and decided there was no real answer, and Aristotle first made the distinction between potential (countable) and actual (uncountable) infinities. This is some of the ancestry of transfinitology. Good enough? Why do you ask? > > MoeBlee Keep on truckin' ToeKnee
From: Aatu Koskensilta on 4 Jun 2010 16:25 MoeBlee <jazzmobe(a)hotmail.com> writes: > It's not fascinating. Aatu is just reminding of the dry fact that in > conversational mathematics, we do use the word 'more' in a sense not > confined to cardinality. Well, I was also pointing out there are in the logical literature numerous not at all nonsensical attempts at defining a general notion of size on which e.g. the rationals are of a larger size than the naturals. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Tony Orlow on 4 Jun 2010 16:28 On Jun 4, 2:37 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jun 4, 11:16 am, Tony Orlow <t...(a)lightlink.com> wrote: > > > other > > measures of sets can also exist, consistent with the axioms, but > > inconsistent with cardinality. > > It is not ordinarily disputed that definitions may be given > stipulatively. So what? "stipulative - no dictionary results" - dictionary.com > > > > How does TO define 'sequence' without reference to something like the > > > SET of naturals as indices? > > > It is a set wherein every element is either before or after (not > > immediately) every other element. That's one possible definition, > > Yes, the objection is that it's more general than the ordinary > definition, thus loses the special sense obtained from the ordinary > definition. But again, if you don't like the ordinary definition of > 'sequence', then just imagine everywhere 'sequence' appears in set > theory discussion that the word is 'zequence'. > > MoeBlee Do you not remember the thread not long ago where you methamaticians started arguing amonst yourselves? What is the "ordinary" definition, again? Never mind. It's a sidebar, as always with you. Take a deep breath. Love, Tony
From: Tony Orlow on 4 Jun 2010 16:34
On Jun 4, 3:57 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Tony Orlow <t...(a)lightlink.com> writes: > > It's really sad when those who claim to be so smart have sacrificed > > their creativity and imagination to reach such an "esteemed position". > > Perhaps you should partake in some lamb's bread and then reconsider > > the issue. > > I think you should reconsider your choice in rhetoric gambits. I'm sorry Aatu. Do you mean I should choose more carefully which posts to respond to? Or, perhaps, you mean I should watch my tone. In either case, and you are not among them, there are certain folk who seek to correct and dominate rather than discuss and think, and sometimes I feel they must have proper feedback, for their good and that of the world. Perhaps that makes me one of those that seeks to correct and dominate? > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus Please advise. Love, Tony |