[FOM] 176:Count Arithmetic

[Neil Tennant]

On Tue, 10 Jun 2003, Harvey Friedman wrote:
 
> FIRST ORDER COUNT ARITHMETIC WITHOUT INDUCTION
> ...
> The vocabularly of COA (count arithmetic) is as follows.
> ...
> v. the special symbol #.
> ...
> We inductively define the formulas.
> ...
> d. If A is a formula, k >= 1, x1,...,xk are distinct variables, and y 
> is a variable, then #(A,y1,...,yk;y) is a formula.
> 
> The idea in d is that #(A,x1,...,xk;y) is read:
> 
> the number of k-tuples x1,...,xk such that A, is y.

Does not this formation clause show that there is not just ONE symbol #,
but rather a countable infinity of them (#1, #2, #3, ...) with increasing
arities, to be used as follows:

#1(A,x1;y)
#2(A,x1,x2;y)
#3(A,x1,x2,x3;y)
:
#k(A,x1,...,xk;y)
:

...?

Neil Tennant