[FOM] Existential Incomopleteness of Set Theory
laureano luna
laureanoluna at yahoo.es
Mon Apr 14 11:45:50 EDT 2008
Set theory is incomplete by Gödel's theorem in the usual well-known sense.
But it is also incomplete in the sense that it cannot prove the existence of all "reasonably existing" sets; it cannot even define all "reasonably definable" sets.
Take any of the usual axiomatizations of set theory; call it "A"; call its language "L(A)". Consider the set S of all sets/classes definable in L(A) (under its standard interpetation). S seems well-defined and is enumerable. But S cannot be defined in L(A).
Suppose it can. Then R = (x: xeS & ~xex) is also definable in L(A). But ~ReS. So L(A) can define a set R that L(A) cannot define.
Say that a concept C1 is extensible wrt a concept C2 iff there is a function f such that for any set S falling under C2 and containing only C1's:
1. f(S) is a C1
2. ~f(S)e S
Then there is no set falling under C2 of all C1's.
The concept of definable set is extensible wrt the concept of definable set of formally definable sets. So, there is a function (Rusellian diagonalization) that for any set S of definable sets that is a definable set of formally definable sets, gives a definable set not in S.
No formalism can aspire to define all definable sets, let aside to prove the existence of all existing sets.
Surely someone knew of this. Any links or literature?
Thanks,
Laureano Luna
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