[FOM] How much of math is logic?

joeshipman@aol.com joeshipman at aol.com
Wed Feb 28 20:48:26 EST 2007

>My suggestion is that, in order to avoid arguments about contentious
>topics that are tangential to your (first) main question, you rephrase
>your question as follows:
>For suitable "X", one can say that ZFC = logic + AxInf + X.  Just how 
>can "X" be made to be?

I do want to avoid tangential discussions, and I like this suggestion. 
To clarify, it is also the case that one can say PA = logic + Y; and we 
also want to know how weak Y can be; what I was really driving at is, 
for the weakest such Y, how close logic + AxInf +Y comes to ZFC.

This deals with the deductive side of mathematics. For the semantic 
side, where I don't care about proof calculi but just expressive power, 
my question is "what mathematical X are not interpretable in 
second-order logic?"

>I am still interested in a summary of what Russell did.  I can't 
>that I'm the only one on FOM who doesn't know exactly what degree of
>strength each of the assumptions in PM buys you.

I don't know this exactly either; I believe Russell could recover 
elementary number theory without needing his reducibility axiom or 
Choice, but I am not familiar with the details. Can anyone else help 

-- JS
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