FOM: Quantifier question

[friedman@mbi.math.ohio-state.edu]

Reply to Franzen 7/12/02. 

It appears that the completeness of the axiomatization that you give for "there 
are infinitely many x such that" without function symbols would imply that this 
obeys the compactness theorem. However, even with = as the only semantic 
symbol, it is not compact. E.g., every finite subset of 

therexists at least 2 things
therexists at least 3 things
therexists at least 4 things
..
not(therexists infinitely many things)

is satisfiable, but the whole subset is not.

Have I misunderstood what you were saying?