An uncountable countable set
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From: MoeBlee on 22 Aug 2006 14:10 Tony Orlow wrote: > I have never said I think there is a largest natural. I have said that > some of your assumptions lead to that conclusion. A while ago you said you don't claim set theory is inconsistent, only that it is not consistent with your own notions. But what you just said is tantamount to claiming set theory to be inconsistent. > Huh? You mean, if I reject your axiom system, and concoct another, the > fact that yours came first means that contradictions between the two are > the fault of mine? Hmmmm..... Nah. If my ideas all fit together and > provide at least as robust a system as the status quo, then seniority > really doesn't count. Seniority should have some weight, but should not be the final arbiter. If your system is markedly better, then it deserves adoption. But you've never come close to presenting a system, so the question is nugatory. > Define "truth". Formal definition is given in a formal meta-theory. Greatly simplified, a sentence S is true in a model M iff the evaluation function per M (definition of this function given courtesy of the defintion by recursion theorem) with the sentence as argument yields the set of all functions on the variables into the domain of the model. MoeBlee
From: Virgil on 22 Aug 2006 15:01 In article <1156231793.095221.243390(a)i42g2000cwa.googlegroups.com>, "Albrecht" <albstorz(a)gmx.de> wrote: > Virgil schrieb: > > > In article <1156163893.101419.232240(a)p79g2000cwp.googlegroups.com>, > > "Albrecht" <albstorz(a)gmx.de> wrote: > > > > > Infinite sets are self contradicting. > > > > Not in ZF or NBG. What are the axioms of Storz's system? > > There is no relevance in which system the axiom is found. > E.g. the Axiom A: "Axiom A is wrong", is self contradicting, regardless > of which other axioms are used, I think. The same holds for the axiom > of infinity. It may be AS's opinion that the axiom of infinity is wrong, but I do not accept opinions about the validity of mathematical statements when unsupported by proofs, or at least strong evidence, and AS has provided nothing but his opinions. In order for AS to declare that a statement is wrong, he must have, a priori, some criteria for the difference he sees between right and wrong and must be able to make such criteria explicit. Absent his having done so, his opinion is worth no more that any other opinion, and if it were put to a vote, he would lose.
From: Tony Orlow on 23 Aug 2006 10:29 David R Tribble wrote: > Dik T. Winter wrote: >>> No that does not make sense. sum{n = 1 .. oo} 1 is not defined, the same >>> holds for sum{n = 1 .. oo} n. If you want to use them you have to >>> provide a definition for them. > > David R Tribble wrote: >>> But Tony thinks he has provided a definition, based on his "Big'un" >>> number. The problem, of course, is that he simply assumes that >>> arithmetic operations on Big'un work "es expected" without providing >>> any proof of that whatsoever. >>> >>> It's one thing to provide a definition, it's quite another to prove >>> that it is a consistent definition. > > Tony Orlow wrote: >> I think the responsibility lies with you to point out an inconsistency >> that arises from my assumptions. > > You're kidding, right? No. You complain that IFR, N=S^L and infinite-case induction lead to some kind of contradiction, but I don't see it. If set theory were "proven" true, then people wouldn't have spent decades searching for some kind of inconsistency in it, would they? > >> Set theory is not "proven true". It >> cannot prove itself consistent. It took years of trying, with some >> succcess in detecting failures, to refine set theory so that it was >> somehow actually consistent. But, it cannot "prove" itself so. > > It's true that in any sufficiently powerful axiomatic system a theorem > cannot be formed that proves the consistency of the that system. > > However, within the framework of the theory (the axioms and theorems > that it comprises), you can prove theorems true and prove false > statements to be false. In that sense, the theorem that, for example, > the set of all naturals is infinite is a provably true theorem within > standard set theory. > Yes, based on the assumed axioms of standard set theory and the set-theoretic definition of infinity. > >> Please >> show where I am being inconsistent, not with set theory, but within my >> own assumptions. Remember, I don't claim to believe in transfinite set >> theory, and don't intend to be consistent with it. > > The problem is that your own assumptions are not consistent with > each other - they don't form a coherent theory. Many of these > inconsistencies have been pointed out to you before, but you choose > to not believe them or simply to ignore them. > It's easy to say that without mentioning any specific inconsistencies. In fact, mapping the naturals in [1,Big'un] to the reals in [Lil'un,1] using the mapping function f(x)=x/Big'un yields Ross' Finlayson numbers, and is perfectly consistent with IFR. Not only do we obviously have Big'un^2 reals on the line because we have the sum of Big'un unit intervals each containing Big'un reals, but the inverse of f(x) is g(x)=x*Bigun, and if we apply IFR over the interval [Lil'un,Big'un], we get |*R|=floor(Big'un*Big'un-Lil'un*Big'un+1) =floor(Big'un^2-1+1) =Big'un^2. I think that's a pretty nice example of consistency between The unit infinity and IFR, which works precisely for both finite and infinite sets of reals mapped from the naturals. I'll not go on about the inconsistencies in transfinitology. TO
From: Tony Orlow on 23 Aug 2006 10:32 David R Tribble wrote: > Tony Orlow wrote: >> But Monsieur, what about the injection from P(N) into N, via the bit >> strings which denote set membership, each of which also corresponds to a >> binary natural? Tsk, tsk. Mustn't forget that one! Remember, the only >> set which doesn't map is the entire set, and that maps to the largest >> natural, that is, ...1111 with all bits in finite positions. > > ... as well as all the infinite subsets of N. You keep forgetting > about those, don't you? > You must be forgetting that, given that all bit positions are finite, even your countably infinite bit strings only can represent finite values, and since they also all have distinct successors and predecessors, your claim that they are not natural numbers is rather unjustified, wouldn't you say? You notice that I called ....1111 the largest natural, don't you? :) Tony
From: Tony Orlow on 23 Aug 2006 11:01
David R Tribble wrote: > Tony Orlow wrote: >>> ... it's inductively provable likewise that the size of a successor >>> ordinal is one greater than its max element. That would make aleph_0 1 >>> greater than the max finite natural. > > David R Tribble wrote: >>> Except that Aleph_0 is not a successor ordinal. It's not even an >>> ordinal. And the fact that there is no maximum finite natural. >>> >>> So your "inductive proof" has a couple of holes in it. > > Tony Orlow wrote: >> Apologies. I mean omega. The concept with the limit ordinals is that >> they are the first after the last of something that doesn't end, the >> next biggest thing. But, when all successor ordinals are exactly one >> larger than their max element, then limit ordinals are something else >> entirely. The declaration of omega as that thing right after all the >> finites itself leads to silliness. > > omega is defined as the least ordinal greater than every finite > ordinal. > > It has well-defined properties, e.g., it is an ordinal, it obeys > ordinal arithmetic including ordering (<) and set arithmetic > (+, x, ^). > > In contrast, your Big'un does not have any such well-defined > properties. You define it as "the number of reals in [0,1]", which > of course is c, but then you tack on "and it operates arithmetically > just like a finite value" without proof or elucidation. Which is just > silly. > It's not silly, because it works. If it didn't work, I wouldn't push it. What does it break? > > Tony Orlow wrote: >>> The only way around this is the declaration of the limit ordinals, >>> but that's just a philosophical monkey wrench. > > David R Tribble wrote: > Tony Orlow wrote: >>> Ordinals and cardinals are necessities if we want to talk about >>> set "order" and "size" in any kind of logical, well-defined way. > > Tony Orlow wrote: >> Not limit ordinals and transfinite cardinalities, and in the finite >> case, a count's a count. You don't need to call 1 an ordinal sometimes >> and a cardinal at others. It's just a natural, a count. > > 1 is: > - a natural > - an integer > - a rational > - a real > - an ordinal > - a cardinal > - a multiplicative identity > - not a prime, not a composite > etc. > > Why exclude a couple of properties you don't like? > There is no real use for them. The theories based on them are counterintuitive, because they're, uh, wrong. > > David R Tribble wrote: >>> What do you call the least ordinal that is greater than all finite >>> ordinals? You don't have a name for it, do you? > > Tony Orlow wrote: >> obviously, that's omega, if you entertain such a useless idea. I don't. > > Proving my point. That my philosophy doesn't include declaring some point where infinity begins? Okay. > >> There is no least infinity, any more than there's a greatest finite, >> where addition or subtraction changes the value. It always should. > > In standard arithmetic and standard set theory? Then prove it. No, obviously standard tranfinitology holds that omega is the first limit ordinal and aleph_0 the first transfinite cardinality, but that follows from the definition of the von Neumann ordinals and their acceptance as a model of the naturals. Like I said, I don't consider that model to be correct. > > I you mean in your theory, you still have to prove it. But first you > have to define your system consistently. > It follows directly form infinite case induction. Since x-1<x can be proven inductively and proplery, it applies to infinite x. This means for every infinite x, there is a distinct x-1 which is less than x but still infinite. Thus there is no smallest infinite, and more than there isa largest finite,. > > David R Tribble wrote: >>> What do you call the "size" of a countable set with no end? >>> You don't have a name for it, do you? > > Tony Orlow wrote: >> No, I find focused concentration on the Twilight Zone, the "boundary" >> between finite and infinite, to be a rather fruitless exercise. There is >> no such distinct boundary. > > Correct. The finite and the infinite ordinals do not "meet" at any > point. And yet they are still ordered with respect to each other. > Yes, but omega is considered the smallest infinity, which notion itself leads to inconsistencies, such as the ability to classify the number of bits required to list the naturals. It's neither finite nor countably infinite. It's a direct result of omega's status as "littlest giant". > > David R Tribble wrote: >>> What do you call the "size" of an uncountable set? You don't >>> have a name for it, do you? > > Tony Orlow wrote: >> Uh, yeah. Infinite. Infinite sets can be finely ordered formulaically. > > Do you have different names for those different formulaically > ordered infinite sets? Or do you agree with Ross that all infinite > sets are the same, so one name is enough? > Different names? I specify infinite set sizes using formulas on Big'un, which can be ordered by applying infinite case induction, of course. I like to call the big one Moe, though. > > David R Tribble wrote: >>> If you are going to keep taking this political approach to >>> mathematics, condemning what's already established and functional >>> but not providing any workable alternatives, you might want to >>> consider running for office instead. > > Tony Orlow wrote: >> It's not functional, and I have provided functional alternatives. > > But not consistently defined alternatives. > (sigh) Where are the inconsistencies? > >> But, maybe I should run for office anyway. Maybe we can outlaw this thing.... >> ;) haha > > Indiana might have a special docket for that kind of legislature, right > next to their tabled proposal for their exclusive official value of pi. > Yeah, pi is 22/7 and there are 364 days in a year. Transfinitology is like spontaneous generation. TO |