Re: Definition of finite.

On May 1, 6:10 pm, s...@xxxxxxxxxxxxx wrote:
On 1 May, 15:28, "Dave L. Renfro" <renfr...@xxxxxxxxx> wrote:

zuhair wrote:

[snipped]

On the off-hand chance that anyone getting to this thread
is interested in seriously pursuing various notions of
"finite", a very complete survey (but dated) is:

Alfred Tarski, "Sur les ensembles finis", Fundamenta
Mathematicae 6 (1924), 45-95.http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=6http://matwbn.icm.edu.p...

Tarski introduces many notions of "finite" (over 20,
I think) and studies the logical relationships between
them.

I had no idea that there was more than one. Is it possible to explain
in terms that an amateur might understand what is inadequate about the
definition that a set x is finite iff there exists a bijection between
x and an element of |N?

-Rotwang

This last definition is Tarsky's, it is essentially a circular
definition , that doesn't tell us what 'finite' means, it is a
definition by reference, which is not adequate here.

I have presented two definitions here, in a previous post I have
presented a definition which looks similar to this:

x is finite <-> ER( R is asymmetric & Ay( y subset_of x -> ( Em(mey &
~En(ney&nRm)) -> Er(rey & ~Es(sey&rRs)))).


Em(mey & ~En(ney&nRm) <-> m is first member in y as arranged by R.
Er(rey & ~Es(sey&rRs)) <-> r is last member in y as arranged by R.

So the above definition can be simplified to

x is finite <->ER( R is asymmetric & Ay( y subset_of x ->( Em( m is
first member in y as arranged by R) -> Er( r is last member in y as
arranged by R ) ) ) ).

This is a simpler one than the one I have presented earlier with R in
it.

However it would be interesting to see the equivalence between
this definition and the increamental definition John Maurera trying to
prove that it is equivalent to Dedekind's.

Zuhair


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