*
where is the successor function.
*

- , that is, induction with parameters for free variables in .

*The first three axioms are sometimes called weak Peano Arithmetic.
*

*We might have first guessed that the induction axiom should have been
. But this is not how we do induction in real life.
*

*The induction axiom is in fact a different axiom for each . An
axiom like this specifying an infinite set of axioms is sometimes
called an axiom scheme.
*

*PA has an infinite model (
) so by the Upward-Löwenheim-Skolem
theorem PA has an uncountable model which is therefore not
. But
we would like
to be characterized uniquely by these axioms. The
problem is that the induction axiom is not powerful enough - it only
refers to countably many subsets of
(those defined by a )
whereas normal induction refers to all subsets.
*

*Therefore induction is not a first order property.
*

John Fremlin 2010-02-17