## Main Proof (part 2)

February 21, 2011 at 11:34 | Posted in Uncategorized | Leave a comment

Proposition 1Suppose is increasing in and is increasing in . Then

*Proof:* Define

- ,
- , and
- .

Here is where the choice of comes to our rescue. Since for each , it follows that

for each , and therefore

and

Now it follows easily that

Next, we define

Given our definition of , it is not hard to see

- , and
- every proper initial segment of is in .

In particular,

Note that , as it is definable from , , and :

and so

as it can be computed in by taking the Skolem hull of in the model .

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