Re: Continuum hypothesis

On Thu, 16 Aug 2007 djrt20@xxxxxxxxxx wrote:
On Aug 16, 10:25 am, William Elliot <ma...@xxxxxxxxxxxxxxxxxx> wrote:
On Thu, 16 Aug 2007 djr...@xxxxxxxxxx wrote:

OK, I was reading a paper by Woodin on CH and he said something to the
effect that whether one adopts CH or ?CH does not affect the
arithmetic statements true in the theory. I took this to mean
"arithmetic" statements as statements which quantify over numbers, or
sets of numbers, and so on. i.e. statements of any order number
theory.

What does ?CH meant? 'not CH' sometimes written '~CH'?
What is an arithmetic statement? A statement of FOL
with Peano's axioms included as also appropiate function
(successor) and predicate (is natural number) symbols?- Hide quoted text -

- Show quoted text -

What's that about?

Yeah it means not CH, screwy keyboard. What is an arithmetic
statement? Well that was what I was wondering. I guessed he (Woodin)
meant a statement of number theory of some order, first order or
second order or whatever. Then if it makes sense to say CH is not a
statement of any order number theory, then what he said would make
sense.

CH is a statement about transfinite cardinal numbers.

Arithmetic is about integers and fractions and adding, substracting,
multiplying and dividing them.

Number theory is about integers, prime and composite integers and how to
find integer solutions for equations.

Though I've heard about analytic number theory that using analysis to
solve number theoric problems, I've never heard the term 'first order
number theory'. What does it mean?

I'm not accquainted with Woodin's work. Are you asking how to read or
understand a passage from his paper?
.