An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Albrecht on 24 Aug 2006 08:03 Tony Orlow schrieb: > Albrecht 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. > > > > Best regards > > Albrecht S. Storz > > > > Hi Abrecht - > > From that statement I would guess you ascribe to the notion of > universal consistency within mathematics, that not only must each theory > be internally consistent, but the axioms of each theory must not > contradict those of any other, since math describes one universe. How > far off am I from understanding your position? > Yes, I think you meet my intention nearly. But it's not because math is an universe, it's because our world is an universe. You may study the structure of an strange universe; but the heart of math should be dedicated to our real universe and the structures of it. And set theorists like to claim their field as to be math itself. With concepts like "infinity" we are near to the boundaries of our understanding. So we have to reason very carefully. I'm just going on holiday for some weeks. Have a good time and So long! Albrecht
From: Tony Orlow on 24 Aug 2006 08:07 MoeBlee wrote: > Tony Orlow wrote: >> If set theory claims to have a cardinality >> which fits every set, then this set stands out as the counterexample. >> Any finite number of bit positions produces a finite set of strings. Any >> countably infinite set of bit positions produces an uncountable set of >> strings. And, there's nothing in between. This number doesn't exist. It >> can't be classified. So, yes, I think you have an actual flaw within. > > Please say what sentence and its negation you believe are both theorems > of set theory. I'm not sure how this would be proven in set theory (I don't think it is), but it appears to be a belief, anyway, that all sets can be classified through cardinality. However, the set of bit positions required to list the naturals in binary defies classification in this system. > >> I have presented a system > > No you haven't. You've posted disconnected pieces of undefined > terminology. No, I've shown how IFR works with the notion of Big'un. That is a bit disconnected from symbolic sets, which are dealt with using N=S^L, but one answer for all questions is not likely to be a very specific answer, is it? > >>>> 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. >> Maybe that's a little simplified. Not sure what you're saying. Sorry. > > My statement is too compressed and not pinpoint accurate given the > limitations of a one paragraph answer. See Enderton's mathematical > logic textbook. The full formalization is culminated in the exercise in > which you are asked to make sets of functions from the variables into > the universe the values of a certain recursively defined function. > > MoeBlee > Okay. That's not a very fundamental definition, as far as I can tell. Tony
From: Tony Orlow on 24 Aug 2006 08:15 David R Tribble wrote: > Tony Orlow wrote: >>> Because it's a set of consecutive naturals starting at 1. > > Virgil wrote: >>> It doesn't matter where it starts, the issue is whether it ends with a >>> "largest natural". In standard mathematics it does not. >>> TO seems to switch postions erratically on the issue. > > Tony Orlow wrote: >> Let's put it this way. The max and the size of the set are equal. If one >> exists, the other exists as well, since it's the same. If one does not >> exist, then neither does the other. > > A more logical conclusion would be that if a set has no largest member, > then its size is not equal to any finite number. Which happens to > be the case. > > Just because a set doe not have a largest (or smallest) member > does not mean it mysteriously does not have a cardinality. Logic > dictates that every set must have a cardinality, either zero, a > finite cardinality, or an infinite cardinality. > "Logic dictates"? Is this a theorem? If so, it is contradicted by the non-existence of a cardinality for the set of bit positions required to list the naturals in binary. That can be neither finite nor infinite, without producing a contradiction. > >> I have never said I think there is a largest natural. I have said that >> some of your assumptions lead to that conclusion. > > You assume that the axioms of set theory lead to that conclusion, > but you've never proved it. > I don't assume it. Your assumption of a smallest infinite leads to that conclusion. Logic dictates it.
From: Tony Orlow on 24 Aug 2006 08:21 Dik T. Winter wrote: > In article <1156363640.845840.187460(a)75g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > In article <ec8if0$rtd$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > > > > Indeed, I agree with WM's logic concerning the identity relationship > > > > between element count and value in the naturals. He's quite correct in > > > > that regard. > > > > > > Well, you and he are not. The logic is flawed. > > > > For 1, 2, and 3, order and value are identical, even according to your > > logic, I suppose. Where does the first deviation happen? Which one does > > deviate first, i.e., which one does first be larger than the other? And > > why? > > It is the case when the set size is not a natural number. And, with the addition of which natural number to the set does the set achieve this unnatural size? > > > The cardinal number aleph_0 is infinite while the ordinal number > > remains finite in order to have infinitely many finite numbers. > > This makes absolutely no sense. The set of all natural numbers (all > finite) has cardinal number aleph-0 and the ordinal number of that > set is w. > > > > And his proof is not a proof. > > > > O course not, because every proof which has an unpleasant result is not > > a proof. > > Then show that the set of all natural numbers does not have cardinal number > aleph-0 and ordinal number w. A proof please. I have already shown how the set of bit positions in the binary naturals has no cardinal or ordinal that can be assigned to it. > > > > > For my part, I agree that the set of finite > > > > naturals is finite, though unbounded, > > > > > > In that case you are not using standard mathematical terminology. I > > > have no idea what a finite but unbounded set is. > > > > That's why you cannot understand mathematics. You fall back behind > > Cantor. He knew it. > > Oh, perhaps *you* do not understand mathematics? Earlier I have already > written that a few things that Cantor has written do not conform with > current thinking. But see <http://mathworld.wolfram.com/FiniteSet.html>: > A set X whose elements can be numbered from 1 to n, for some positive > integer n. Yes, given the current understanding, any unbounded set is infinite. > A set is infinite if it is not finite. A set is Dedekind infinite if > it can be mapped to a proper subset of itself, and it is Dedekind > finite if it is not Dedekind infinite. When we assume the axiom of > choice, the two notions are identical. Without that axiom there > can be infinite sets that are Dedekind finite. Do you want to know > more about set theory? > > Now, using, this terminology (pretty standard), what is a "finite but > unbounded" set? Using that system, the phrase is senseless, but that system is not the real universe. It's a concoction.
From: Tony Orlow on 24 Aug 2006 08:34
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. > >>> 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 |