Cantor's Argument Skolemized?

Cantor's Argument Skolemized?

As all on this list will agree, Cantor's argument is entirely valid.
All, however, will also know about the quandaries this has created in
light of the Löwenheim-Skolem Theorem, as stressed by Skolem and known
under the heading "Skolem's "Paradox"".

But it is also a fact that in a ZF-setting one needs a fair amount of
separation to carry through Cantor's argument. Separation restricted
to first-order conditions A(x) with only x free, and no parameters,
will not suffice. For then we may assume that there is a function f
from N onto P(N) such that there is no first order condition F(x) such
that (x)(xef<=>F(x)), and Cantor's argument is blocked. (It is
interesting to see that Skolem in his 1957/58 lectures at Notre Dame
only presupposed a non-parametrized version of separation (available
as chapter 3 here: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197470
). Was he simplifying? When and how did parametrized versions of
separation enter the picture?) But with only separation without
parameters set theory will be seriously hamstrung, and too weak to
serve as a foundation for mathematics.

In the following I offer a couple of ZF-like set theories wherein we
explicitly make use of a domain D in stating the axioms. I call
everything under consideration, in and out of D, "sets". Some places I
make use of the eliminable set-abstract notation. It seems to turn out
that we may here block Cantor's argument in a Skolem-like manner in
the object language itself. The questions I raise are: (1) Am I making
some elementary mistake somewhere? (2) If not (1), (a) is this treated
somewhere, or perhaps folklore; (b) is this, in your opinion, of any
(i) mathematical, (ii) philosophical or other interest?

Axiomatic Sketch I:

A0 Ex(x=D) (Existence of the set D)
A(1) (x)(y)(x,yeD=>((z)(zex<=>zey)=>x=y)) (extensionality relative to
D)
A(2) (x)(y)(xeD & yex => yeD) (transitivity of D)
A(3) Ex(x=Ø & xeD) (existence of empty set in D)
A(4) Ex(x=W & xeD) (omega exists in D)
A(5) (x)(xeD=>Ey(y=Ux & yeD)) (D is closed under union)
A(6) (x)(y)(xeD & yeD => {x,y}eD) (D is closed under pair)
A(8) All universal closures restricted to D of (z)(zeD=>(Ey)(yeD & (x)
(xey<=>xez & A(x))), where A(x) is a first order condition on x that
does not have y free. (D is closed under separation with parameters
from D)
A(9) Replacement is treated analogously with separation.
A(9) (x)(xeD => {y: yeD & y SUBSET OF x}eD) (D is closed under D-
restricted Power)

Axiomatic Sketch II:

As Axiom Sketch I but with A(9) replaced by

A(9)' (x)(xeD => {y:y SUBSET OF x}eD)

Let us now rehearse Cantor's argument in the light of Axiomatic Sketch
I. Suppose f is a function with Dom(f)=W and Ran(f)=P(W) and such that
f is onto P(W). We are then asked to consider the set S={x:xeW & notxef
(x)}. If we assume that feD, a contradiction follows. But then
Cantor's argument here only seems to license the conclusion that there
is no function *in D* which is a bijection from W to P(W) (or from W
onto P(W)). If f is not in D, we do not have enough separation to
assert the existence of S. In light of this, it seems that Axiomatic
sketch I is consistent with an extension stating that there is a
function from W onto P(W), for we need not assume that all sets are in
D. (By analogous reasoning, we may add assumptions to the effect that
there for all sets X in D is a function f from W onto X.)

Some would here be inclined to complain that A(9) does not capture the
"real" power set operation, whatever that might be. Many would e.g.
find it objectionable that the postulated f from W onto P(W) is not a
member of P(WxP(W)). I have therefore offered Axiomatic Sketch II in
order to fend off certain concerns along such lines. Let us say that a
set g is a *dijection* of the sets A and B iff

((x)(xeA=>Ey(yeB & <x,y>eg)) & (x)(xeB => Ey(yeA & <y,x>eg)) &
(x)(y)(z)(xeA & yeB & zeB & <x,y>eg & <x,z>eg => y=z) & (x)(y)(z)(xeB
& yeA & zeA & <y,x>eg & <z,x>eg => y=z))

The idea is that a dijection of A and B may have all kinds of other
stuff in it besides pairs from AxB. All bijections are dijections, but
there are dijections which are not bijections. But it seems fair to
say that if there is a dijection from A to B, then A and B have the
same cardinality. In the light of Axiom Sketch II, we may now again
rehearse Cantor's argument. Suppose there is a dijection f from W to P
(W). Now, if we assume that feD we again fall prey to Cantor's
conclusion. But f may not be in D. Further, we do not even have enough
separation to assert the existence of a bijection f' from W to P(W)
such that Dom(f')=W and Ran(f')=P(W). So we may, it seems, postulate
the existence of a dijection from W to P(W) without worrying that it
should somehow "naturally" be in D.
If this has some merit, it will, it seems, include the one that we may
capture Skolem's intuition, that the notion of "uncountability" is
only relative, in the object language itself. One may, arguably,
maintain the
view that all sets are in fact countable.
.

Relevant Pages

  • Re: Cantors Argument Skolemized?
    ... separation to carry through Cantor's argument. ... explicitly make use of a domain D in stating the axioms. ... Axiomatic Sketch II: ... The idea is that a dijection of A and B may have all kinds of other ...
    (sci.logic)
  • Re: Cantors Argument Skolemized?
    ... separation to carry through Cantor's argument. ... Axiomatic Sketch II: ... is no function *in D* which is a bijection from W to P(or from W ... The idea is that a dijection of A and B may have all kinds of other ...
    (sci.logic)
  • Re: Cantors Argument Skolemized?
    ... separation to carry through Cantor's argument. ... interesting to see that Skolem in his 1957/58 lectures at Notre Dame ... explicitly make use of a domain D in stating the axioms. ... say that if there is a dijection from A to B, then A and B have the ...
    (sci.logic)
  • Re: A brick in Cantors argument.
    ... P_n with n different free variables. ... are to be axioms, where Pis any first order condition, in the ... sense of Skolem, whith x free ). ... history of parameters in separation. ...
    (sci.logic)
  • Re: Cantors Argument Skolemized?
    ... light of the Löwenheim-Skolem Theorem, as stressed by Skolem and known ... separation to carry through Cantor's argument. ... The schema applies TO ALL FIRST-ORDER FORMULA-SCHEMATA ...
    (sci.logic)
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