An uncountable countable set
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From: Tony Orlow on 20 Sep 2006 10:42 Mike Kelly wrote: > Han de Bruijn wrote: >> Virgil wrote: >> >>> In article <1158489723.269348.27860(a)e3g2000cwe.googlegroups.com>, >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>> >>>> What's wrong with mathematics ?! >>> Nothing!! >> "Mathematics should be a science" is the answer. > > Why? > Because we are discovering truths, not making up games, in real math.
From: Tony Orlow on 20 Sep 2006 10:49 Virgil wrote: > In article <450d5597(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <450c87cc(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Virgil wrote: >>>>> In article <450c71a1(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> Aatu Koskensilta wrote: >>>>>> Given the axioms and rules of inference, the conclusions are provably >>>>>> true or false. >>>>>> >>>>>> Soundness is another issue, regarding the fundamental justification for >>>>>> the logical axioms themselves, and whether they are "correct", meaning >>>>>> "objectively verifiable". >>>>> If axioms were ever objectively verifiable they would not need to be >>>>> assumed in the first place, but would be objectively verified. >>>>> >>>> In the mathematical world, the greater framework can be considered >>>> relatively objective. >>> Greater than what? If one wnats something in one's system, either it is >>> provable in terms of other things in the system or it must be assumed >>> without being provable in terms of other things in the sysem, and just >>> like with having to have undefined terms, at some point you have to have >>> unproven assumptions. >>> >>> In mathematics, when you get to that point, you call those unproven >>> assumptions axioms. >>> >>> TO seems to want to do without any axioms by some sort of daisy chain >>> circle of proofs lifting the whole mess up by its bootstraps. >> How on Earth do you read all that from what I said. The "greater >> framework" is mathematics in general. If a particular axiom or theory >> contradicts enough other math, then it's trouble. There's no reason that >> all of mathematics can't be consistent. That's the greater framework. > > The axioms system of Euclidean geometry is inconsistent with that of > various non-Euclidean geometries. and there are a lot of other places > where one system contradicts another. What happened is that the parallel postulate was downgraded to a postulate, rather than a law. Whether two parallel lines meets at 0, 1 or two places determines the type of space we are discussing. To say that the parallel postulate applied and also DIDN'T apply to a GIVEN space would contradictory. > > What mathematics allows is any system of axioms which does not appear to > contain any self-contradictions, at least for as long as it maintains > that appearance. If it claims to be a generalization, the it should not contradict those particulars which it generalizes. > >> I understand that axioms are necessary, but they should not be arbitrary. > > Which axioms in which mathematical systems does TO think have not been > judicially chosen? The axiom of choice, for one. That's optional even in your system.
From: Tony Orlow on 20 Sep 2006 10:50 Virgil wrote: > In article <450d5757(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Without the additional >> structure which Aatu suggests, bijection may show some kind of >> equivalence, but it cannot be considered any kind of exact analog for >> the size of finite sets. You're trying to extract measure from something >> with no measure in it, like blood from a stone. > > On the contrary, Cantor was trying to devise a measure which was > entirely independent of every property of the members of each set other > than their being distinguishable from each other. > > TO is measuring order relations, not sets. I am measuring sets USING order relations in a standard metric space. Otherwise, there is no measure.
From: Tony Orlow on 20 Sep 2006 10:51 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.
From: Randy Poe on 20 Sep 2006 10:52
Tony Orlow 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, It isn't. What does "successor to N" even mean? > it is > greater than all finite naturals, This isn't even meaningful using the ">" symbol as defined on natural numbers. However it can be made meaningful if you define ">" in terms of bijections. You are working with an undefined term, with undefined operations, and then trying to draw conclusions as if you'd defined those things. > as any successor is greater then all > those that precede it. It isn't a successor of any particular natural number. So given that it is NOT a successor, how do you think a property of "any successor" is relevant? > It is certainly a positive number, No it's not, as it does not fit the definition for any positive number. > if it is a > count or size of anything. A T-Axiom? > If it is a positive natural greater then 1, It isn't a positive natural. > then its reciprocal is a real in (0,1). To say that some count which is > greater than any finite count does not obey this general rule is a > kludge, like all the transfinite "arithmetic". It doesn't obey this general rule because it doesn't fit the definitions of the things that do. That's not a kludge. It's pointing out the "broccoli" does not necessarily have the properties of things in the set of mammals. - Randy |