# [FOM] Real Division Algebras + Tarski/Lefschetz Principles (typo fixed)

Robert M. Solovay solovay at math.berkeley.edu
Wed Sep 11 02:19:23 EDT 2002

```On Tue, 10 Sep 2002, Adam Epstein wrote:

>
>
> ---------- Forwarded message ----------
> Date: Tue, 10 Sep 2002 14:03:09 +0100 (BST)
> To: fom at cs.nyu.edu
> Subject: Real Division Algebras + Tarski/Lefschetz Principles
>
>
> First, here is Joe Shipman's response to my question about CH and the Law
> of the Excluded Middle:
>
> On Mon, 9 Sep 2002 JoeShipman at aol.com wrote:
>
> >
> > Choice is essential here, you need AC both to get a non-Borel set and to
> > get a set of reals of cardinality aleph-one.
> >
This is wrong. One can prove the existence of a non-Borel set in
ZF [by essentially the Cantor diagonal arguement.] It is true that one
can't prove [without AC] that there is a set of reals of cardinality
aleph_one. More precisely the theory "ZF + DC + there is no set of reals
of
cardinality aleph_one' is equiconsistent with "ZFC + "There is an
inaccessible cardinal".

[For one way, use my old model for "All sets Lebesgue measurable";
for the other if aleph_1 is the aleph_1 of L[x] for some real x, then we
can get aleph_1 reals. So aleph_1 is a limit cardinal in L; DC shows its
regular.]

OTOH if I recall correctly,  Con(ZF) implies Con(ZF + "aleph_1
does not inject into R"). One just imitates my model using aleph_omega in
place of an inaccessible cardinal.

> >
> > -- JS
>
? I know that Classical Analysis is conservative over
> PA.

This is not correct. "Classical analysis" = "Second-order number
theory" proves Con(PA).

--Bob Solovay

```