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)