[FOM] First Order Logic

[Harvey Friedman]

The primary crucial property of first order logic and first order theories
for the foundations of mathematics, is that they demand certificates.
Second order and weak second order theories generally don't require
certificates. In order for them to be modified to require certificates, one
goes to an associated first order theory.

If my memory serves me, if we restrict to countable models only, then the
only logic satisfying non controversial conditions plus "validities are
r.e." is first order logic. Therefore compactness also follows.

Harvey Friedman

On Sun, Sep 8, 2013 at 1:20 AM, Panu Raatikainen <
panu.raatikainen at helsinki.fi> wrote:

> Let me ask a clarificatory question:
>
> Is it Harvey's (or anyone's) view that compactness in itself is a
> desirable property of a logic? And if so, why more exactly?
>
>
> All the Best
>
> Panu
> --
> Panu Raatikainen
>
>
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