An uncountable countable set
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From: Mike Kelly on 20 Sep 2006 16:55 Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <450d5f76(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>> > >>>> Mike, you haven't responded to my use of IFR > >>> An IFR, being dependent on order relations, at best measures order > >>> relations, not their underlying sets. > >> Funny how it DOES measure the sizes of sets perfectly in all finite cases. > > > > So... how do we use IFR to tell us the size of the set {sqrt(17), Pi, > > e, {}, 42 } ? > > > > That's a finite set. You don't need it. So IFR doesn't even work on finite sets? What sets *does* it work on? -- mike.
From: Virgil on 20 Sep 2006 16:56 In article <45114cfe(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <45102b5d(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> I am well aware my position is "provably false", but I am > >>>> also aware that that depends on the axioms assumed and the rules > >>>> regarding logical inference. > >>> No, your "position" is not coherent enough to be addressed as either > >>> true or false. A bunch of gibberish doesn't admit of examination for > >>> truth or falsehood. > >> You are wrong. If I assert that S = proper_subset(T) -> |S|<|T|, then > >> this is a basic axiom that one can accept which makes all of > >> transfinitology untenable. > > > > Depends on how one defines |S| and |T|. > > Given TO's assertion above, Card(S) and |S| are not the same, but that > > in no way interferes with the validity of cardinality. > > If |S| is "the size of S, in count of elements", then it contradicts the > use of cardinality as an analog for size in infinite sets. Why does defining |S| to be something other than cardinality cause any problems with cardinality? Cardinality is only concerned with whether injections, surjections or bijections exist between a given pairi of sets. Anything else is irrelevant, at least as far as cardinality is concerned.
From: MoeBlee on 20 Sep 2006 16:57 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Omega is successor to N > > > > No, it is not not. Why do you keep repeating your error? > > In the limit ordinal sense, it's what comes after the set of finite > naturals. No it doesn't come RIGHT AFTER or AFTER the set of finite naturals. It IS the set of finite naturals. Again, WHY do you keep repeating your error? > > w is larger than any natural number where 'larger' is taken in the > > sense of the standard ordinal ordering. But w is not larger in the > > sense you just mentioned, since w is not any number of successions away > > from any natural number. > > No, there is no such number in any theory including omega. You'd need > uncountable sequences. I don't know what YOU mean by 'a theory including omega'. Anyway, my point was made: w is not any number of successions "away" from any natural number. > >> Can you really not say that an infinite count is greater than any finite > >> count? That seems to be the basis for omega. > > > > What does "basis for" mean? > > I mean the justification for the limit ordinals as being the next > greater thing than any of the finite successor ordinals, considered a > closed set. The "justification" is just in a proof, given a definition of 'next greater'. But, again, w is not a NEXT greater than any particular natural number. > > 'w' has a definition. That definition is not that of "having infinite > > count greater than any finite count". > > That's a theorem based on the logic that if there were any finite size > to the set, then one could come up with a finite natural not in the set, > and so the set size must be greater than any finite size. Yesno? As usual, no. Why don't you just read a damn book already? MoeBlee
From: Virgil on 20 Sep 2006 17:02 In article <45115046(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > >> Mike Kelly schrieb: > >> > >>> mueckenh(a)rz.fh-augsburg.de wrote: > >>>> Mike Kelly schrieb: > >>>> > >>>>>> Any set that can be established is a finite set. > >>>>> Why? > >>>> Look: If aleph_0 were a number larger than any natural number, then for > >>>> any natural number n we had n < aleph_0. "For all" means: even in the > >>>> limit. > >>> OK so far. Every cardinal number which is a natural number is less than > >>> aleph_0. > >>> > >>>> So lim [n-->oo] n/aleph_0 < 1 > >>> Division is not defined for infinite cardinal numbers. > >> Is that your only escape? If you dare to say that aleph_0 > n for any > >> n e N, then we can conclude the above inequality. > > > > No, because division is not defined on infinite cardinal numbers. The > > above inequality is meaningless. > > > >> But remedy is easy. > >> Take lim [n-->oo] aleph_0 / n > 1 and reverse the following fractions > >> analogously. > >>>> If aleph_0 counted the numbers, for instance the even naturals, then we > >>>> had for all of them > >>>> > >>>> lim [n-->oo] |{2,4,6,...,2n}| = aleph_0. > >>>> > >>>> This would yield lim [n-->oo] (2n/|{2,4,6,...,2n}|) < 1 > >>>> > >>>> But we have lim [n-->oo] (2n/|{2,4,6,...,2n}|) = 2 > >>>> > >>>> Therefore aleph_0 does not exist as a number which could be compared > >>>> with other numbers. > >>>>>> No. Just this is the point! The series 1 + 1/2 + 1/4 + ... is 2 (or at > >>>>>> least as close to 2 as we like), not by definition and not by any > >>>>>> axiom, but by rational thought. > >>>>> Prove that to be the case without using any definition of what a series > >>>>> is and without any axioms. > >>>> Archimedes did so when exhausting the area of the parabola. In decimal > >>>> notation 2 + 2 = 4, and in any system we have II and II = IIII. > >>> In airthmetic modulo 3, 2+2 = 1. > >> If you say "in arithemtic mod 3", then you imply that you subtract 3 > >> from the true result as often as possible. It does not invalidate II + > >> II = IIII, if you subsequently tale off III. > > > > Huh? The "true" result is that 2+2 = 1, if you are working in > > arithmetic modulo 3. Or if it's 10 o'clock now and I wait 5 hours then > > it is 3 o'clock. > > > > Your position seems very inconsistent. You claim that numbers have no > > existence outside their representation. And now you are claiming there > > exists a "true" arithmetic. > > > >>>> For self-evident truths you don't need axioms. Only if you want to > >>>> establish uncertain things like "There exist a set which contains O and > >>>> with a also {a}" then axioms may be required. > >>>> > >>>> Don't misunderstand me: I do not oppose the principle of induction but > >>>> the phrase "there exists" which suggests the existence of the completed > >>>> set. > >>> Why do you object to this? > >> Because of he proof above. > > > > It is not a proof. Division is not defined where either operand is an > > infinite cardinal number. > > > > If omega is the successor to the set of all finite naturals, Omega IS the set of all finite naturals, as ordinals, but NOT the successor to the set of all finite naturals in any way whatsoever, nor it is the successor to anything at all. TO seems to be regressing.
From: Virgil on 20 Sep 2006 17:06
In article <45115232(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Han is saying that transfinitology has > NOTHING to do with everyday arithmetic. That's the point. It doesn't fir > into mathematics. The conclusions are absurd. To quote George Boole, > inventor of the system which allows you to confirm the deductive > consistency of your axiom systems, in his "An Investigation Into The > Laws Of Thought": > > "Let it be considered whether in any science, viewed either as a system > of truths or as the foundation of a practical art, there can properly be > any other test of the completeness and fundamental character of its > laws, than the completeness of its system of derived truths, and the > generality of the methods which it serves to establish." TO seems to be unaware the "system of derived truths" means precisely those things that can be derived from a set of axioms. |