An uncountable countable set
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From: MoeBlee on 24 Aug 2006 13:07 Tony Orlow wrote: > > 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. No, I did not say that. I said that w is not in the field of the standard ordering on the set of real numbers. What's lame is you, for not taking the least steps to understand even the most basic things about the mathematics upon which you so adamantly opine. > You have an infinite value, but can't say whether it's larger than 2? > That's ridiculous. I said no such thing. See above. > > 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. So what theory is (1) a theorem of? In what theory is w a member of the field of the standard ordering of the set of real numbers? > 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. Oh, okay, it's something that you think "should be". Not in any theory. Just some notation to stand for something that you think should be. Nevermind that your notation is undefined, let alone that you haven't said from what axioms you derive this statement that you think should hold. > Or Tom Cruise's lifestyle. I'm sure you know more about that than me. > > 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. Again, you said (3) was a formula of set theory. But your definition of the operation symbol '/' used in (3) requires a bunch of undefined terminology outside of set theory. That's incoherent and is not in set theory. Here's (3) again: (3) aleph_0/2 = aleph_0 = aleph_0^2 < 2^aleph_0 '/' used that way is not in set theory. > > 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. When you claim (3) is part of set theory, you are mixing up set theory with your own jumble of informal notions and undefined terms. MoeBlee
From: MoeBlee on 24 Aug 2006 13:27 Tony Orlow wrote: > Dik T. Winter wrote: > > Tony Orlow wrote: > > > I don't disagree that the first is infinite while the second is > > > unbounded bt finite, and therefore smaller. > > > > Pray, for once, provide a definition. A set is either finite or infinite. > > And if a set is finite, by the definitions there is a largest element. > > 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. The man is PLEADING with you for a definition, and you give him circularity: "A truly INFINITE set must have elements INFINITELY many position beyond other elements" [all caps added] And this circularity has been pointed out to you time and time again. So OF COURSE people get fed up with your twaddle. > I wonder how many books on set theory Cantor read? Cantor was well educated in mathematics. MoeBlee
From: MoeBlee on 24 Aug 2006 13:37 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. > > > > 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. Then logic dictates that set theory is inconsistent. And I mean 'inconsistent' in the usual mathematical sense. In one post you say that you don't claim that set theory is inconsistent and in another post you say that the statement (I'm making the statements more precise) "there exists a least infinite ordinal" entails the statement "there exists a greatest natural number"; but that is to say that set theory is inconsistent, since set theory has both "there exists a least infinite ordinal" and 'there does not exist a greatest natual number" as theorems. So you contradict yourself. MoeBlee
From: MoeBlee on 24 Aug 2006 13:43 Tony Orlow wrote: > Logic dictates it. What logic? We have first order predicate logic with identity, which we prove to be sound and complete (in the sense of the completeness theorem), as well as that proof is algorithmically checkable, as well as that definitions are proven to satisfy the criteria of eliminability and non-creativity. You, on the other hand, have your personal freewheeling mental associations. MoeBlee
From: Virgil on 24 Aug 2006 14:55
In article <44ed9670(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > 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. In ZF it is NOT the case that every set must have a cardinality comparible with that of other sets, but it is a thereom in both ZFC and NBG. > However, the set of bit positions > required to list the naturals in binary defies classification in this > system. On the contrary, that set of bit positions has cardinality equal to that of N. The fact that the same set of bit positions is capable of more does not mean that there need be any smaller set which is sufficient. There are exact analogies in finite cases. E.g., the number of bits required to ennumerate all the naturals up to, say, seventy is sufficient to ennumerate considerably more than seventy. In fact the only cases in which the same set of binary digits will NOT enumerate more is when you want them to enumerate every natural up to 2^n-1 for n=some natural n. As these cases become ever more rare as n increases, they are in effect infinitely rare for infinite n, so the result that TO object to is quite normal . and any other would be infinitely unusual. > > > > >> 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. You have made all sorts of unsupported claims, but one of them work in any extant system. > >>>> 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. Let's see if TO can do any better, then. So, TO, what is YOUR definition of TRUTH? |