Re: Definition of finite.

On May 5, 7:32 am, zuhair <zaljo...@xxxxxxxxx> wrote:
On May 4, 9:11 am, "Dave L. Renfro" <renfr...@xxxxxxxxx> wrote:

1. Every nonempty family of subsets of F has
a subset-maximal element.

How a subset-maximal element is defined in first order logic?
Can you write thisdefinitionin first order logic, please.

m is a subset-maximal element of S <-> (meS & Axes ~ m proper subset
of x)

Get Suppes's set theory book.

2. Every nonempty family of subsets of F,
ordered by inclusion, has a subset-maximal
element.

The formula please.

Here's Tarksi's definition ['P' stands for 'power set of']:

x is finite <-> AS((S subset of Px & ~S=0) -> Em m is a subset-maximal
element of S)

Get Suppes's set theory book. There's a bunch of this stuff in there.

You say they are distinct, then thedefinitionI gave at this thread
which was said to be equivalent to Dedekind's is in reality not!
I have always been suspecious regarding this 'equivalence'.

Just get a good set theory textbook.

In Z set theories:

The Tarski definition I just gave is equivalent to the 'well ordered
by an R and its converse' definition.

The Dedekind definition is equivalent to the 'incremental'
definition.

The Tarski condition entails the Dedekind condition ('not equinumerous
with a proper subset of itself') (indeed, this is called the
'pigeonhole principle') but the Dedekind condition does not entail the
Tarski condition.

Adding the axiom of choice (even just the axiom of countable choice),
the Dedekind condition entails the Tarski condition so that with the
axiom of choice (or even just the axiom of countable choice), the
Tarski and Dedekind definitions are equivalent.

MoeBlee

.