An uncountable countable set
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From: Tony Orlow on 25 Aug 2006 12:55 David R Tribble wrote: > Tony Orlow wrote: >> If set theory were "proven" true ... > > How would we do that? Would it be the same way you would prove > your theories to be true? > Yes, by contradicting Goedel's Incompleteness Theorem. It can't be done, so don't ask me to "prove" my system is conistent. If you detenct an inconsistency, please let me know, but I have to say that, much as I'm accused of mixing apples and oranges when it comes to my ideas and transfinitology, most of the objections to my system are based, like your last one, on assumptions of transfinitology, such as the set of reals in [0,1] is equinumerous with the set of reals overall. So, don't ask me to "prove" my ideas are correct. Find inconsistencies between them and point them out, please. Otherwise, consider their viability, independent of standard treatments of oo. Tony
From: MoeBlee on 25 Aug 2006 13:02 Tony Orlow wrote: > > Indeed. Not step by step. And so? I have said this before. I am quite > > sure others have said that before. You can not get the set of naturals > > by adding the one by one. It is the axiom of infinity that asserts that > > that set does exist. > > And that it's infinite. Why do you need an axiom for that? WHy is it not > derivable logically? What do you mean "why"? It is not derivable since there are finite domains of discourse that satisfy Z-I (Z set theory without the axiom of infinity). There is NO system that will give you any kind of even minimal mathematics without adopting axioms that are not true in all domains of discourse. > If Ross and I each have our linearizations of the reals, then they > exist, independent of the axiom of choice (which might as well be called > the axiom of free will). We prove existence of objects in a theory from axioms. You posit the existence of objectw without any system of logic, primitives, or axioms. > > Yes, but in that case you are not using standard mathematical terminology. > > And I have yet to see a *definition* of you about *infinite*. > > Larger than any finite. The set of naturals is as large as, but no > larger than, every natural. Define 'larger'. Define 'finite'. Actually, nevermind, since you have no logicistic system, primitives, or axioms from which to make defintions. Moreover, you have no idea what are correct forms for defintions, why such forms are correct and others are incorrect. You even resort to circular definitions. > > You are using words without definition, using words from standard terminology > > but with (apparently) not the standard definition. That is what makes you > > on occasion not understandable. > > I thought it was clear that I was using a notion of infinite, like WM, > from a quantitative standpoint, rather than set-theoretic. Oh, yes, the "quantitative standpoint". Have you read Zerbernieskoskywoskyozerlichmanosty's paper on the standard quantitative geometric induction? It's right up your alley. MoeBlee
From: MoeBlee on 25 Aug 2006 13:08 Tony Orlow wrote: > Given a set x, can we always determine card(x)? I'm guessing, but I'm pretty sure that there is not an algorithm that will tell you the ordinal index of the aleph that is the cardinality of any given set. > >> However, the set of bit positions > >> required to list the naturals in binary defies classification in this > >> system. > > > > Whatever you mean by that, based on remark that you know of no proof of > > a contradiction in set theory, it is a problem of your own > > misconception and not a contradiction in set theory. Fortunately, we > > don't obligate ourselves to the burdens of your own misconceptions. > > No, but you are obligated to define a cardinality for this set which is > consistent, if you claim the theory is consistent. You can't. First you need to define the set in set theory and prove that it exists in set theory. Then, set theory would not be inconsistent for there not being an algorithm to determine the ordinal index of the aleph that is the cardinality of any given set. Again, you're just yapping without regard for the specific definitions of such things as 'consistent'. MoeBlee
From: Tony Orlow on 25 Aug 2006 13:12 Virgil wrote: > In article <44ed9fcb(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Dik T. Winter wrote: >>> In article <ecig8i$ta$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: >>> > Dik T. Winter wrote: >>> ... >>> > > The first part of his first proof shows that a complete ordered field >>> > > has cardinality larger than the natural numbers. In his proof he did >>> > > not rely an any properties of the reals other than that they form a >>> > > complete ordered field (he uses reals to exemplify). >>> > >>> > Specifically, he demonstrates that there will always exist an >>> > unaccessible real in any finite interval, given that we are only allowed >>> > a finite (natural, that is) number of iterations. >>> >>> Sorry, you have got that wrong. We are allowed an infinite number of >>> iterations indexed with finite numbers (whatever that may mean). >> Thagt means that at no time have we ever completed an infinite number of >> iterations. That's why we cannot arrive at that intermediate value. > > There is nothing in Cantor's first proof that is time dependent, and any > "number" of iterations are "completed" merely by defining them. > Also, there is a necessity that none of the "intervals" is ever a single > point. > > TO misses the point once more. > >> Using the axiom of choice, it is provable that every set can be well >> ordered, which entails a linear order, but this proof gives no clue as >> to how this can be accomplished. > >> But, well-ordering aside, the H-riffic >> numbers > > For which there is no known system in which they exist. > > >> The >> other approach is Ross, which relies on the notion of infinitesimally >> different reals, sequential in their natural order. > > Ross depends on having x < y and x = y simultaneously, which > simultaneoulsy contradicts the requirements of every known order > relation. > >>> > It is related to the powerset through |P(S)|=2^|S|, but he did it in >>> > decimal, no? >>> >>> No. Nowhere in his papers did Cantor use decimals. >> Perhaps it is just the explanations of his second proof of >> uncountability which I have seen. He did use SOME digital representation >> of his list of reals. Was it binary? I have not read his original papers >> like you have. > > Cantor's second proof was never applied to the reals, at least by Cantor > himself. What he actually said in the second proof as essentially that > the set of all binary strings is not countable, proving it by the famous > (or infamous) diagonal argument. Though I do not think he used 0 and 1, > but a pair of letters, as his binary symbols. >>> > In any case, the powerset relation boils down to the >>> > symbolc equation N=S^L, where S is the number of logical states allowed >>> > (there can be more than two in various systems) and where L ultimately >>> > corresponds to the size of the root set. So, the two are related. >>> >>> You make here as much sense as in much earlier articles. >> What part of does not make sense? > > In what system is S^L defined for either S or L not finite numbers? > > As far as I can see, only in that as yet not on-existent system that TO > has promised often and delivered never. > > >>> So, how do you define "unbounded but finite"? >> Not with the Dedekind definition. A truly infinite set must have >> elements infinitely many position beyond other elements, the way I see it. > > To has not been asked what do not constitute definitions, but what do. > > According to TO's non-definition above, a set must be ordered in order > to be infinite. Thus, among other things, the set of points on a circle > must be finite because it is not an ordered set, but removing one point > makes it infinite? > > We need a less self-contradictory definition that that, TO. >>> > > His second proof is about sets of sequences, and his diagonal proof >>> > > shows that the cardinality of the set of infinite sequences with >>> > > two possible elements is strictly larger than the cardinality of >>> > > the natural numbers. Zermelo has indicated how that can be converted >>> > > to a proof that the reals have cardinality larger than that of the >>> > > natural numbers. But Cantor had *no* intention to prove that at that >>> > > point at all, and did nowhere write that it could be applied for that >>> > > purpose. >>> > >>> > Yes, the second is really a proof about power set and/or symbolic >>> > representations of quantites. >>> >>> Not at all. There is no question about "representation of quantities". >> In the second proof of uncountability of the reals, there are no digital >> strings? What list is he deriving the antidiagonal from, a shopping list? > > Lists whose entries are taken from any set of two distinct objects. I > think Cantor used two distinct letters, but he could have as easily used > the set of binary digits, 2 = {0,1}. Oh, okay, so he did it in binary, as Dik said with "w" and "m" (for "Wolfgnag" and "Meuckenheim"? ;) It essentially, then, boiled down also to a power set proof, as well as a real proof. Like I said, power set and digital number system are the same, even in other bases, when set membership can have more than two distinct states. :) >>> > > Cantor's one and only purpose was a proof of the theorem: >>> > > There are sets with cardinality larger than that of the natural >>> > > numbers. >>> > >>> > Considering that the set of finite naturals is ultimately finite, as >>> > Wolfgang has been trying to express, that's not a surprising result. >>> > What's surprising is Dr. M's reluctance to admit the actually infinite >>> > in the real interval. >>> >>> What does not surprise me at all is that apparently you do not understand >>> the discussions at all. >> If you say so. It seem that you're the one saying you don't understand >> what I'm saying, but perhaps that's a matter of perspective. > > What things seem like to TO and what things are in actuality bear little > resemblance. To those who view reality form over here, there are only slight differences. ;) > >>> Quite a lot. When was the last time you did read a book about set theory? >> Quite a while back. I wonder how many books on set theory Cantor read? > > In any case, it appears that Cantor wrote mor
From: MoeBlee on 25 Aug 2006 13:13
Tony Orlow wrote: > Yes, by contradicting Goedel's Incompleteness Theorem. It can't be done, > so don't ask me to "prove" my system is conistent. If you detenct an > inconsistency, please let me know, You don't HAVE a system! For a system you need a logistic system, axiom, and primitives. Then defintions can be added and theorems derived. MoeBlee |