On 12 Jun 2013, at 20:49, meekerdb wrote:

On 6/12/2013 1:57 AM, Bruno Marchal wrote:
Arithmetic is large, and I do not know of any theorem in math which is not a theorem in arithmetic, except in mathematical logic, and universal algebra, which are typically "meta-"mathematics.

What about theorems in calculus and topology?

Most, if not all, are theorems of arithmetic in disguise. Few math go really beyond PI-1 or Pi_2 arithmetical complexity, when you study their logical complexity. Only category and set theory go much beyond, and genuinely go beyond arithmetical complexity.

I could say more on this when I have more time, but it requires some amount of mathematical logic. Macintyre wrote papers on this, and Torkel Franzen made a similar point in his book "inexhaustibility".

Some other people makes this points trivial, but only by a *misuse* of comp, note.

Bruno




Brent

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http://iridia.ulb.ac.be/~marchal/



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