An uncountable countable set
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From: Tony Orlow on 20 Sep 2006 12:13 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> Virgil wrote: > > <snip> > >>> 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. > > Would you care to quantify who the "you" refers to in "your system"? Do > you imagine that the four horsemen of the apocalypse [g] have > constructed our own "system"? > > Brian Chandler > http://imaginatorium.org > Who's that, you, Stephen, Randy and Mike? In your dreams. Tony
From: Randy Poe on 20 Sep 2006 12:15 Tony Orlow wrote: > Randy Poe wrote: > > 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? > > Ask von Neumann. I consider that a crackpot response. First the crackpot misconstrues some paper they're reading (by Cantor or Einstein, say), then when pressed they respond "are you arguing with Cantor/Einstein?" I'm not asking von Neumann, I'm asking you. You used a phrase. Did you have any idea what you meant by that phrase? > It is the set of all naturals, N is the successor to N? > and any ordinal being > the set of all preceding naturals, omega is the set of all preceding > naturals. It's larger than all naturals. DO you disagree with that > simple statement? It contains terms whose meaning we do not agree on. What is the test for "largeness"? > >> it is > >> greater than all finite naturals, > > > > This isn't even meaningful using the ">" symbol > > as defined on natural numbers. > > Sure it is. We have a set N. We have an operation given meaning on elements of the set N. It is not meaningful on things not in set N. Even if you say "sure it is". Asserting "sure it is" does not actually assign a meaning to it. It gets a meaning only when you actually go through the process of GIVING it one. > > However it can be made meaningful if you define > > ">" in terms of bijections. > > If you introduce a metric, sure. Again you say "sure" too quickly. Is that an agreement that for ordinals, we will use bijection to define ">"? Specifically, that "A > B" means that any mapping from B to A will leave elements of A unmapped? > > You are working with an undefined term, with undefined > > operations, and then trying to draw conclusions > > as if you'd defined those things. > > If an infinite number is not greater than a finite number, then it's not > a number at all. Precisely. It doesn't fit the definitions for natural number, real number, or complex number. It's a new kind of thing, with new properties. The operators like ">" which apply to natural numbers don't apply here. You have to DEFINE them. They don't automatically get meaning without new definitions, because the old definitions don't apply. > >> 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. > > No, but if I say I have a science of all life, it should apply equally > to broccoli and mammals, Poor analogy, since the theory of natural numbers is not analogous to a "science of all numbers". Not even of all positive numbers. > and if I have a science of animals, it applies > to mammals, but NOT broccoli. So, if I have a rule for numbers, which > aleph_0 doesn't obey, I don't consider it a number. Right. Stop applying the rules of naturals and reals to aleph_0. It isn't one of them. - Randy
From: MoeBlee on 20 Sep 2006 12:44 Tony Orlow wrote: > >> Actually infinite(S) <-> E seS A neN index(s)>n > >> Potentially infinite(S) <-> A seS index(s)eN ^ A neN E seS index(s)=n. So this is what you have now ('w' stands for 'N', which stands for?): S is actually infinite <-> Es(seS & An(new -> index(s)>n)) S is potentially infinite <-> As(seS -> (index(s)ew & An(new -> Es(seS & index(s) = n)))) Okay, now I see that your 'actually infinite' is something like what set theory would describe as 'S has a member greater than any member of w'; and your 'potentially infinite' is what set theory would describe as 'S has only finite members but S has a denumerable number of finite members'. Neither of those correspond to the usual senses of 'actually infinite' and 'potentially infinite'. Morevover, if you are going to allow the existence of a T-potentially infinite set (I'm going to use 'T' since your definitions don't correspond to usual senses) and just the basic set operations, then you are still going to have sets bijectable with proper subsets of themselves. And, you still will not have contradicted that an unbounded set must be infinite ('infinite' given the usual definition). > > Next, what is the logistic system? > > First order logic Okay, classical first order logic with identity, I take it. > What are the non-logical primitives? > e > > > index Good. > > What are the non-logical axioms? > > These are definitions of potential vs. actual infinity. Consider them > axiomatically stated if you wish. They're definitions now that you fixed them. But now you still have 'index' and 'w' to define. And you need more axioms, so that 'e' and 'index' can work for you. > What is the definition of 'index'? > > The count of elements up to and including a given element. The number of > steps between and element and the first element in a linear order, plus 1. Now you just need to define 'count', 'elements', 'up to and 'including' or 'number of steps', 'between', 'first', 'linear order', 'plus', and '1'. MoeBlee
From: MoeBlee on 20 Sep 2006 12:51 Tony Orlow wrote: > 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. Then, since that is not the defininition of |S|, there's nothing to argue about. Probably the most common definition is this: |S| = the least ordinal that has a bijection with S. MoeBlee
From: Randy Poe on 20 Sep 2006 12:51
Tony Orlow wrote: > Given that all evens are natural, but only every other natural is even, Yes, the evens are a proper subset of the naturals. > is is logically deducible No, that's not a deduction. There's no theorem or axiom you use to make this jump. > that the naturals are twice as numerous in any > range as the evens. Because you have an intuitive notion of range and numerous, neither of which applies a priori to either the set of naturals or of evens. The problems start when one tries to pin down a meaning of "numerous" for such sets. Then the way things are no longer follows "the way Tony wants them to be". But I never worried about what you were thinking when I learned mathematics, so it's not overly bothersome to me that the deductions lead to a different conclusion than the desires of some stranger named Tony on the internet. > Intuition is a wholistic application of logic and > association, No, intuition is guessing in the absence of logic. > not to be rejected unless proven contradictory. Which it has been, over and over and over... - Randy |