Re: Anti-Choice

On May 15, 1:25 am, Rupert <rupertmccal...@xxxxxxxxx> wrote:
On May 15, 12:11 am, zuhair <zaljo...@xxxxxxxxx> wrote:

Is the anti-choice axiom equivalent to the following:

Ax ( Ey( y subset_of x & y equinumerous to P(w) ) <->

~ER( R is well ordering on x ) ).

were P(w) is the power set of Omega.

I don't know what you mean by the anti-choice axiom. The statement you
gave is equivalent in ZF to the assertion "the real numbers cannot be
well-ordered".

Not it is not. This is an implication of this axiom and not
equivalence.

Yes, it is. If the real numbers cannot be well-ordered, then no set
with a subset equipollent to the set of real numbers can be well-
ordered. This is very easy to prove in ZF, or Z, probably in much
weaker theories still.

I understand that.

But I'm not clear exactly which sentences are in question here.

As I understand, we have:

~R(R is a well-ordering on the set of reals)
<->
Ax(Ey(y subset of x & y equinumerous to Pw) -> ~ER(R is a well-
ordering on x)).

Okay.

But perhaps the question is as to this?:

~R(R is a well-ordering on the set of reals)
<->
Ax(Ey(y subset of x & y equinumerous to Pw) <-> ~ER(R is a well-
ordering on x)).

So we would be asking about this:

(~R(R is a well-ordering on the set of reals) & ~ER(R is a well-
ordering on x))
->
Ey(y subset of x & y equinumerous to Pw).

And I wonder whether that might involve the continuum hypothesis?

MoeBlee


.

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