An uncountable countable set
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From: Virgil on 24 Aug 2006 15:47 In article <44ed9cc5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> Set theory contradicts with: > >>>> > >>>> (1) E y e N, A x>y, x< 2*x < x^2 < 2^x (y=2) > >>>> > >>>> because: > >>>> > >>>> (2) A y e N, aleph_0>y > >>> I don't know what you intend '<' to stand for. For the domination > >>> relation? The less than relation on ordinals? > >> The standard "less than" operator, commonly used for finite reals. > > > > This is in set theory, right? w is not in the field of the less than > > relation on the reals. Therefore, your (2) is incoherent. Also, if were > > in set theory, 'finite' adds nothing in description of real numbers. > > So, you cannot say that an infinite count like aleph_0 is greater than > any finite count? That's fairly lame. What one can say in cardinality and ordinality one cannot say within the order relatin on the real numbers, nor vice versa. What is extremely lame is TO's deliberate conflation of two different and incompatible ordering contexts. TO makes no more sense that Casey Stengel's famous "Line up alphabetically by height" command. > > > > >>> I don't know what is meant by '(y=2)' in the larger formula. > >> I mean that for x>2, 2*x < x^2 < 2^2. For x=2 they're equal. Is aleph_0>2? > > > > So you mean that y=2 is a counterexample to the claim that ~Ey...etc. ? > > (But then you mean 2^x, not 2^2 just above, I guess.) > Ooops, yes, I mean 2^x, sorry. I meant that 2 is the y greater than > which all x must be for the inequality to hold true. > > > > > And w is not in the less than ordering for real numbers, so it doesn't > > exactly make a lot of sense to ask whether w is greater than 2 in that > > ordering. > > > > You have an infinite value, but can't say whether it's larger than 2? > That's ridiculous. > > >>>> and > >>>> > >>>> (3) aleph_0/2 = aleph_0 = aleph_0^2 < 2^aleph_0 > >>>> > >>>> (1) is trivially inductively provable. > >>> Do you mean (1) is a theorem of set theory, or do you mean it is > >>> provable that (1) is the negation of a theorem of set theory? > >> That (1) contradicts set theory. It's certainly not a standard theorem, > >> in the infinite case, but it should be. However, its incompatible with > >> aleph_0 and the system of limit ordinals. > > > > If you made (1) a rendering of a formula in set theory (without > > ambiguity as to exactly what ordering '<' is to stand for), then we'd > > be in a better position to evaluate whether or not it is a theorem of > > set theory. > > It's obviously not a theorem of set theory, or set theory would have > been rpoven inconsistent a long time ago. It's a simple fact about > finite numbers greater than 2, a fact which I think should be extendable > to infinite numbers greater than 2 as well. But, that's incompatible > with set theory. > > > > >>>> (2)and (3) are from transfinitology. > >>> What is transfinitology? > >> You know, Cantorian religion. That Hollywood stuff. ;) > > > > Right. Just like a Mel Gibson movie. > > Or Tom Cruise's lifestyle. > > > > >>> What is the definition (and in what theory is > >>> this definition?) of '/' where w (omega) is in the numerator? > > > >> Well, I was calling it Bigulosity. The definition of x/y is "how many > >> intervals of length y fit into an interval of length x?". This is > >> constructible using straightedge and compass. Where it's not integral, > >> is as approximable as it is with symbolic reals. > > > > I asked you what theory this is in. So, since you now say that (3) is > > part of set theory, I take it that you intend for the above to be > > something defined in set theory. There are some basic definitions of > > 'interval', 'length', and 'integer' in set theoy, and in certain > > contexts in set theory, but I don't know what 'symbolic reals' are in > > set theory, nor can I fathom how what you claim to be a definition > > works for omega as x. > > It DOESN'T work for omega, otherwise I wouldn't have had to come up with > Big'un as a unit infinity. > > > > > Maybe you're forgetting that you're mixing set theory up with your own > > informal notions again. > > I'm forgetting no such thing. I am pointing out inconsistencies BETWEEN > the two, and what the crux of the difference is. I am not claiming to > have a solid contradiction WITHIN set theory, although the bit positions > in the binary naturals offers a glimmer of hope. > > > > > MoeBlee > > > > :) > > TOny
From: David R Tribble on 24 Aug 2006 15:51 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. > David R Tribble wrote: >> ... as well as all the infinite subsets of N. You keep forgetting >> about those, don't you? > Tony Orlow wrote: > You must be forgetting that, given that all bit positions are finite, > even your countably infinite bit strings only can represent finite > values, ... That's false. It's also irrelevant, since no natural is represented by a countably infinite bitstring. > ... 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 appear to be changing the subject. I was talking about your alleged bijection between N and P(N), which does not exist. It does not exist because the members of N (which are all finite bitstrings) do not map to _any_ of the infinite subsets of N, which comprise the bulk of the members of P(N). > You notice that I called ....1111 the largest natural, don't you? Yes, I did. Calling it that does not make it so, however. All naturals are a finite. Therefore your infinite number ...111 cannot be a natural. QED.
From: David R Tribble on 24 Aug 2006 15:58 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?
From: David R Tribble on 24 Aug 2006 16:14 Tony Orlow schrieb: >> Sufficient for what? > mueckenh wrote: > Sufficient to do mathematics with limits. Sufficient to address all > points. You cannot address more than a countable number of them, > because there are only a finite number of names (including finite > strings of digits). And the points which cannot be addressed do not > exist. In what form should they exist? As carbon atoms from a pencil or > of chalk particles? No. In real mathematics (as opposed to matheology) > we have the valid foundation: What cannot be addressed, that does not > exist. > > Don't call me a finitist. There is no largest number but only a largest > set of less than 10^100 elements. I see now. You allow that for any natural k there is also a k+1, so there is no upper bound on the naturals. Hence there is no largest natural. On the other hand, you place an arbitrary upper bound of 10^100 elements for any set. (Apparently, in order for a set to exist there must exist a corresponding particle in the known physical universe to correspond to each of its members. If we run out of particles, we run out of place-holders in our set, I suppose.) There are no cardinal numbers greater than 10^100. Several conclusions follow from these premises. One is that there is no set that contains all the naturals. Likewise, intervals such as [0,1] cannot be represented by sets of reals (or sets of points) because they are too big to be sets. Another is that the series f(x) = sum{n=0 to oo} (-x)^n/n! cannot exist because we can't make a set from the partial sums of the sequence. Which means, in turn, that irrational numbers cannot exist, because we can only approximate them with a Cauchy sequence of finite length. The list goes on and on.
From: Virgil on 24 Aug 2006 16:17
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}. > > > > > > > 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. > > > > 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 more than TO has read. |