[FOM] PA with few symbols.
neilt at mercutio.cohums.ohio-state.edu
Mon Jul 19 08:51:37 EDT 2004
On Mon, 19 Jul 2004 W.Taylor at math.canterbury.ac.nz wrote:
> 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.
How about the following method?:
Instead of taking as lexical primitives infinite many distinct variables,
simply *generate* infinitely many variables from two symbols. For example,
take two new symbools, # and *, say, and construct variables as follows:
* #* ##* ###* ####* ...
(i.e. x_0 x_1 x_2 x_3 x_4 ... )
Then every quantified sentence of the language (of Peano arithmetic, say)
would be constructible from a genuinely *finite* lexicon.
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