Date: Apr 4, 2013 5:15 PM
Author: fom
Subject: Re: Matheology § 224

On 4/4/2013 3:45 PM, WM wrote:

<snip>

>
> But an inductive set contains elements that are not subject to
> induction?
>


Yes.

The definition in Jech corresponds with Dedekind's
simply infinite set.

An inductive set is a set having an injection
into itself such that at least one element
has no pre-image under the injection and
such that no chain of elements under successive
application of the injection closes in the
sense that a pre-image is also a successor.

It need not be "thin" in the sense of the
ordinal numbers defined to be transitive
and well-ordered by membership.

For purposes of representing arithmetic within
formal set theory, the empty set is taken as
that element with no pre-image in the formal
statement of the axiom of infinity. Then,
some transfinite initial segment of the ordinals
is necessarily a subset of the set whose
existence is assumed by the axiom.

Of course, no elements are subject to induction
since the purpose of the existence axiom is to
provide a domain for inductive arguments to
be applied.

WM probably thinks the platinum bar in Paris
is a meter long (by definition, if another bar
of different length replaced the current bar,
a great deal of "empirical data" would become
untrue instantaneously -- it is just one of
those things that cannot be falsified)