I just had a weird thought
in [Logic]
|
Prev: Please, DO NOT forget the only thing that matters on this planet........................……..
Next: Topos theory: axiom of choice implies a topos is Boolean
From: George Greene on 31 May 2010 12:26 On May 29, 9:31 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > Can't define "It is a finite set."? ~(all X)(eY)P(Y) ^ LT(X,Y) > > C-B That 's not even grammatical. You haven't said what P() means. Nobody knows what language you're speaking. The rest of us were speaking ZFC, in which "LT" simply does not occur. Trust me, you have no hope of defining this in any simpler language. In your string above, which letter is supposed to represent the set that may or may not be finite? Is it X ? If it is, then a definition of finite can't possibly begin with ~(all X). Maybe you better learn how to define stuff. A definition of "finite", if you abbreviated "finite" as F(.), would look like Ax[F(x) <--> some other stuff about x ]
From: George Greene on 31 May 2010 12:27 On May 29, 9:32 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > Or you mean a finite set in general? Who said it can't be defined? > (What a great challenge!) It can be defined at least 5 or 6 different ways in ZF. In ZFC, these different ways are not different after all. There is a famous post about it that I re-cite from time to time, just so it doesn't get lost in antiquity. But it is all in rather abstruse ZF, which you of course can't be bothered to bother with.
From: George Greene on 31 May 2010 23:20
On May 29, 9:32 pm, Charlie-Boo <shymath...(a)gmail.com> wrote: > Or you mean a finite set in general? Yes. I mean, if you have a set theory, where you are talking about an infinite number of sets, some of which are finite, how do you tell specifically which are finite and which are not?? > Who said it can't be defined? > (What a great challenge!) AK was the one who told me it couldn't be defined IN FIRST-ORDER LOGIC FROM A RECURSIVE AXIOM-SET. If you use more powerful machinery then (of course) more things are possible. My point in the original argument was that if something NEEDS more than first-order logic, then that something cannot be "trivial". Here are 5 different meetings of your "challenge" (you need to start here): This was originally from Herman Rubin in sci.math. If X and Y are sets, then we define X <= Y if either of the equivalent definitions hold: (1) There is a one-to-one function from X into Y. (2) There is a one-to-one function from a subset of X onto Y. If X and Y are sets, write X == Y (actually written with a single pair of wavy lines, and read X is equipollent to Y) if either of the two equivalent conditions below hold: (1) X <= Y and Y <= X (2) There is a 1-1 function from X onto Y. If X and Y are sets, then write X < Y if X <= Y and not X == Y. If X is a set, define "X+1" to be any set Y containing X such that Y\X is a singleton. (As used, it won't matter which set is used, and the axiom of choice will not be required in any of the proofs, but this is NOT obvious. There is, however, a formal definition that will work, but I don't need to go into that much detail at this time.) If X is a non-empty set, define "X-1" to be any subset of X with only one element removed. (Again, the axiom of choice will not be required in any of the proofs, but, this time, there is no suitable formal definition.) Define X <=_* Y if any of the equivalent definitions below hold: (1) X is empty, or there is a function from Y onto X. (2) There is a function from a subset of Y onto X. (3) There is a function from Y+1 onto X+1. If X is a set, define P(X) (usually written with a script P) to be the power set of X, the collection of all subsets of X. "omega" will be the first infinite ordinal. (Ordinal numbers are defined in such a way that "n" = {0, 1, 2, ..., n-1}. I don't think I need to go into the details of the definition.) X is n-finite (n for normal) if any of the following equivalent definitions hold: (1) X is equipollent to a finite ordinal. (2) X < omega (3) P(X) is PD-finite (4) P(X) is D*-finite (5) P(P(X)) is D-finite X is A-finite if for any subset Y of X, Y is n-finite or X\Y is n-finite. (A stands for amorphous.) X is PD-infinite if P(X) is D-infinite (defined below). X is D*-infinite if any of the following equivalent conditions hold: (1) There is a proper subset Y of X such that X <=* Y (2) X <=* X-1 (3) X+1 <=* X (4) omega <=* X (oops, that's not quite the same but 1-3 imply 4) X is D-infinte (Dedekind-infinite) if any of the following equivalent conditions hold: (1) There is a proper subset Y of X such that X <= Y (2) There is a proper subset Y of X such that X == Y (3) X+1 <= X (4) X+1 == X (5) X <= X-1 (6) X == X-1 (7) omega <= X It can be seen that, for any set X, with the following definitions, (1) -> (2) -> (3) -> (4) -> (5) -> (6) (1) X is D-infinite (2) X is D*-infinite (3) X is D*-infinite (definition 4, but I don't have a separate name) (4) X is PD-infinite (5) X is A-infinite (6) X is n-infinite (BTW, some of the lemmas used are: (A) If X <=_* Y, then P(X) <= P(Y). (B) If X is n-infinite, then P(X) is D*-infinite (C) If X is D*-infinite, then P(X) is D-infinite.) With more effort and a lot of knowledge of set theory, one can show that none of these implications is reversable. For some background, including the definitions of <=, and <=_*, I recommend _Set Theory for the Mathematician_, by Jean E. Rubin. I have a vague memory of a paper by one or more Rubin's about relationships between definitions of finiteness, but I can't find a reference at the moment. |