[FOM] PA with few symbols.

W.Taylor@math.canterbury.ac.nz W.Taylor at math.canterbury.ac.nz
Sun Jul 18 22:37:29 EDT 2004

Often, and very recently here, one sees comments to the effect that
1st-order theories, and PA in particular, can be simply written with
a very small number of connectives, quantifiers, arithmetic symbols...
...and an *infinite* (though very simply formed) number of variable symbols.

This last point seems somewhat jarring, and though it is of cosmetic
value only, I have wondered if it can be gotten around in some way. 
Perhaps in the case of PA the effect of an unlimited number of quantifiers
can be cunningly handled by using only a small number of them and
"coding up" the rest in much the same way that all PR functions
can be coded up using just + and * .

Is this so?  Does anyone know of any work done on representing PA
with a fixed finite number of variables, as well as the finite number
of everything else?

        Bill Taylor                     W.Taylor at math.canterbury.ac.nz
        Ultra-finitism: Math done on a computer with limited CPU space

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