Re: Definition of finite.

Actually if one a thorough definition of 'finite' set , then it would
be the following:

x is finite <-> ER( Em( (mex&~En(nex&nRm)) ->
Ek(kex&~Ed(dex&kRd) & Ay((yex&~y=m&~y=k) -> EuEw(uRy & yRw & uex & wex
& ~Eb~Ec( bex & cex & uRb & bRy & yRc & cRw)))))).

In words: every set x is said to be finite if and only if there exist
a relation R that arrange the members of x in such a manner that the
existence of a first member m in x implies the existence of a last
member k in x and implies that every member y in x that is neither the
first nor the last member in x , should have two members u and w both
of which are in x, were uRy and yRw,such that no member b in x exist
such that uRb & bRy , nor their exist a member c in x such that yRc
and cRw.

So if such a relation R exist on any set x then x is finite, if
such a relation doesn't exist , then x is infinite.


This is the complete definition of x is finite, and it perfectly
parallel's the intuitive concept of finitude.

Zuhair

.

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