Uncountability of the Rationals?
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From: MoeBlee on 4 Jun 2010 14:32 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. 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. > By demanding to > know with which axioms a non-transfinitologist disagrees, > mathematicians obscure the question. WHAT question? > 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. > 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. > 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. > 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. > The von Neumann limit ordinals are > simply bunk. Fine, then probably you want to throw out the axiom of infinity. > 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? MoeBlee
From: MoeBlee on 4 Jun 2010 14:37 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? > > 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
From: MoeBlee on 4 Jun 2010 14:47 On Jun 4, 12:03 pm, Tony Orlow <t...(a)lightlink.com> wrote: > seasoned mathematicians don't even agree on what a sequence is, > some considering it isomorphic to the naturals, and others to the > entire class of von Neumann ordinals. No, you're confused, in Z set theory we don't adopt any such difference. We don't speak of isomorphism with the class of ordinal numbers. In Z set theory there is no such thing. In such context, it is only in an informal sense that we may speak of a sequence on the entire class of ordinal numbers. Rather, we take a sequence to be a function on AN ordinal number (not the entire class). In any given context, we simply make clear what sense we mean: A sequence as a function whose domain is a natural number. A sequence as a function whose domain is either a natural number or w. A sequence as a function whose domain is an ordinal number. A sequence as a function whose domain is some well ordered set. And, in an INformal sense, the ordinal sequence or a sequence (such the levels in the universe V) on the ordinals. That different texts use different defintions does not signal that there is confusion over the matter, but rather only that in a given context we need to make clear which definition is being used. MoeBlee
From: Tony Orlow on 4 Jun 2010 15:50 On Jun 4, 1:12 pm, Pubkeybreaker <pubkeybrea...(a)aol.com> wrote: > On Jun 4, 12:20 pm, Tony Orlow <t...(a)lightlink.com> wrote: > > > > > > > On Jun 3, 11:40 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > > > > kunzmilan <kunzmi...(a)atlas.cz> writes: > > > > The uncountability of the reals is simply based on the fact, that > > > > there are more rational numbers than there are the natural numbers. > > > > This is, of course, utterly butt-wrong. > > > > There are not more rational numbers than naturals -- that is, |N|=|Q|. > > > > Even Tony knows that. > > > > -- > > > Jesse F. Hughes > > > > One is not superior merely because one sees the world as odious. > > > -- Chateaubriand (1768-1848) > > > Hi Jesse - > > > Just because I concede that both sets are countably infinite and > > therefore of the same cardinality, nevertheless the sparse proper > > subset of the rationals called the naturals should not be equated in > > size with its dense proper superset. > > You have been given an explicit injection from Q to N. > > If you think it is wrong, explain why. > > Otherwise, explain this idiocy in which you believe that there are > more rationals than > integers.- Hide quoted text - > > - Show quoted text - Oy Vey! I have stated clearly that I understand there is a bijection between the countable infinities of N and Q. Are you daft? Why correct something which isn't wrong? Sure, they have the same cardinality. So, what? You cannot disagree with the fact that the rationals are a dense set whereas the naturals are sparse, with a countably infinite number of rationals lying quantitatively between any two given naturals, can you? Do you not see that in some sense there appear to be more rationals than naturals? Are you incapable of considering any other system of set measurement besides what was drilled into you in class? 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. Peace, Tony
From: Aatu Koskensilta on 4 Jun 2010 15:57
Tony Orlow <tony(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. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |