Last Updated：2022/12/24
For the arithmeticity of the axioms of PA—i. e. the claim that they can be seen as true on the basis of our basic grasp of the structure of the natural numbers—is motivated by Isaacson by appealing to the categoricity of PA 2, the second-order theory which provides us with a categorical characterization of the natural numbers as the smallest structure closed under a one-to-one successor operation and containing an element which is not the successor of any element.
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For
the
arithmeticity
of
the
axioms
of
PA—i.
e.
the
claim
that
they
can
be
seen
as
true
on
the
basis
of
our
basic
grasp
of
the
structure
of
the
natural
numbers—is
motivated
by
Isaacson
by
appealing
to
the
categoricity
of
PA
2,
the
second-order
theory
which
provides
us
with
a
categorical
characterization
of
the
natural
numbers
as
the
smallest
structure
closed
under
a
one-to-one
successor
operation
and
containing
an
element
which
is
not
the
successor
of
any
element.
