[FOM] The definition of the natural numbers vs. the axiom of infinity

[Randall Holmes]

This is in response to Dean Buckner.

The definition of the natural numbers in ZF does not require the axiom
of infinity.  A natural number is _defined_ as an object which belongs
to any set which contains the empty set and is closed under von
Neumann successor.  One does not need the axiom of infinity for this
to succeed as a definition.  One does not even need the axiom of
infinity in order to quantify over all "natural numbers" so defined,
because in ZF quantifiers do not need to be bounded by sets.

However, the definition will not make sense unless the axiom of
infinity is true.  If the axiom of infinity is not true (in a
structure satisfying the axiom of separation) it will be the case that
_every_ object is a natural number (because there are no inductive
sets, so everything belongs to all of them).  In fact, an equivalent
formulation of the axiom of infinity is "Not all objects are natural
numbers" -- because from this follows the existence of an inductive
set.

The definition of the set of natural numbers does not have the axiom
of infinity as a prerequisite to be understood, but without the axiom
of infinity holding it will not have the desired effect.

I reiterate that quantification in ZF does not have to be over a set
-- it is perfectly possible to reason about an arbitrary object a
which has a property of interest P which does NOT define a set!  For
example, in ZF we can perfectly sensibly quantify over all sets, and
the totality of sets is certainly not a set.  So your objection (or
comment) on the definition is not correct (though the definition
certainly does have an intimate relationship to the axiom of infinity
in practice).

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.boisestate.edu
not glimpse the wonders therein. | http://math.boisestate.edu/~holmes