where
is the successor function.
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