An uncountable countable set
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From: MoeBlee on 6 Oct 2006 13:15 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > A CONTRADICTION? From what AXIOIMS? Please show a derivation of a > > > > sentence P and ~P from the axioms. Oh, that's right, by "contradiction" > > > > you don't mean a contradiction in the sense of a sentence and its > > > > negation; you mean something that doesn't sit with your personal > > > > intuition. > > > > > > > > > > + 1 Gedankenexperiment: Put 10 balls in A and remove two, one of which > > > is put in B and the other one is put in C. > > > > > > P: At noon all balls are in B. > > > ~P: At noon all balls are in C. > > > > I take it that you mean that "At noon all balls are in C" implies ~P. > > Yes, because they are not in B, if C is not within B (which would be a > special case to be mentioned) and if a normal form of logic is > applied. > > > > Anyway, none of what you mentioned are formulas of set theory > > But the results are necessarily implied by the actual existence of > omega. So, you say. Meanwhile, no one so far has shown a sentence in the language of set theory such that the sentence and its negation are both theorems of set theory. > > nor have > > you stated any axiomatic theory here. Just as I said about the other > > poster, you find a conflict with your intuitions (here, regarding a > > thought experiment), but no actual contradiction in an axiomatized > > theory. > > I am sure, the results "all balls in B" and "all balls not in B" are > not to be interpreted as an actual contradiction of set theory. It is > just counter intuitive. That's pretty much what you can take from my post to Orlow. No contradiction in any theory, but rather just an analog that Orlow and others find counterintuitive in THEIR mathematical interpretation of a a thought experiment. So, following your original post to me on this matter, you've just come around to agree with the premise behind my original question to Orlow. MoeBlee
From: MoeBlee on 6 Oct 2006 13:30 mueckenh(a)rz.fh-augsburg.de wrote: > Of course. The question was: Did Cantor use ordinal subtraction. And > can one define it under certain circumstances. > > k + omega = {-k, -k+1, ..., 0, 1 , 2, 3, ...} is different from > > -k + omega = {k, k+1, k+2, ...} > > I used it as an abbreviation to explain that the set omega (or the set > of digit indexes of a certain number: 0.111...) is not unuiquely > defined. As I said, I haven't read some of the earlier posts. So, I am still not familiar with your notation. Also, we need to be clear that, usually, by 'set theory' we mean one of the axiomatized versions that came well after Cantor, such as Z, ZF, ZFC, NBG, and other variants. So to see if your claims have any import for set theory as currently practiced, it would help if you would give your definitions (in context of the language of, say, Z set theory) and argument together in a single post to which we could refer. Definitions then would be for '-' (as applied to ordinals), 'digit indexes', and decimal notation (e.g., your '0.111...'). MoeBlee
From: MoeBlee on 6 Oct 2006 13:50 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > That axiom of infinity says, in symbols, there is an x such that {} is a > > member of x and FOR ALL y, if y is a member of x so is (y union {y}). > > That is a quantifyer which is not expressed as a word "all" meaning > that all are there. Whatever "all are there" means, so what? The quantifier 'for all' as it is used in mathematics, does appear in the axiom of infinity. > (In text versions we have only: if y is in x then > {y} is in x (Zermelo's version).) "FOR ALL" concerns all those y which > are in x but it does not state that a set of all y did exist. Of course it does not, since it would the set of all sets. The variable 'y' ranges over the entire domain of discourse, which is not itself an object of the theory. > Because > then it would be easy to define this set by: "The set of all y". You're close. It would be the set of all y such that y=y, or using some other property (such as self-identity) held by all objects. But so what? MoeBlee
From: MoeBlee on 6 Oct 2006 13:58 mueckenh(a)rz.fh-augsburg.de wrote: > > What text tells you that part of the axiom of infinity is that n+1 > > exists if n is given? > > If x belongs to the set then {x} belongs to it. From thia n --> n+1 > can be proved. In what language and theory is "n --> n+1" a formula? Maybe you mean: Ex x=n+1. But we don't need the axiom of infinity to prove that. In Z set theory (as currently axiomatized), the existence of nu{n} (von Neumann version) is provided by the axiom of extensionality, the union axiom, and the pairing axiom; and {n} (Zermelo version) is also provided by the axiom of extensionality and the pairing axiom.. > > And the quantifier 'for all' is part of the axiom of infinity: > > > > There exists an x such that 0 is a member of x and, for all y, if y is > > a member of x then yu{y} is a member of x. > > > > In symbols: > > > > Ex(0ex & Ay(yex -> yu{y]ex)) > > I know these symbols. A means "for all" y which are there, it does not > state that "all" y are there. Whatever you mean by "all are there", so what? The point is that the quantifier 'for all', just as that quantifier is understood throughout mathematics, does appear in the axiom of infinity. MoeBlee
From: Virgil on 6 Oct 2006 14:48
In article <ac6c7$45260f70$82a1e228$27946(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > Sure. Simply "define" something that's undefined. And create the self > fulfilling prophecy that suits you best. Beats hell out of defining things that are already defined. Every definition worth having defines something that would be undefined without that definition. |