[FOM] PA with few symbols.

[Neil Tennant]

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.

Neil Tennant