An infinite number of axioms like in an axiom schema doesn’t really hurt anything, but you can’t have infinitely long single axioms.
∀x((x = 0) ∨ (x = S0) ∨ (x = SS0) ∨ (x = SSS0) ∨ ...)
is not an option. And neither is the axiom set
P0(x) iff x = 0 PS0(x) iff x = S0 PSS0(x) iff x = SS0 ... ∀x(P0(x) ∨ PS0(x) ∨ PS0(x) ∨ PS0(x) ∨ ...)
We could instead try the axioms
P(0, x) iff x = 0 P(S0, x) iff x = S0 P(SS0, x) iff x = SS0 ... ∀x(∃n(P(n, x)))
but then again we have the problem of n being a nonstandard number.
An infinite number of axioms like in an axiom schema doesn’t really hurt anything, but you can’t have infinitely long single axioms.
is not an option. And neither is the axiom set
We could instead try the axioms
but then again we have the problem of n being a nonstandard number.