Re: Finite : A definition

On Wed, 19 Mar 2008 16:01:29 -0700 (PDT), Zaljohar@xxxxxxxxx
<Zaljohar@xxxxxxxxx> said:
Hi all,

The following is my definition of 'x is finite' and I think it is
equivalent to the standard definition:

x is finite <-> E!y (y is a V.N. ordinal & Ef(f:y->x,f is bijective))

Obviously, as an ord is finite iff it is equinumerous to no other ord.

V.N. stands for Von Neumann ordinal.

In words: x is finite if and only if there exist a unique Von Neumann
ordinal that is bijective to x.

One can see that all Dedekindian infinite classes are bijective to
multiple V.N. ordinals.

Are you assuming ZFC? Without Choice, there might be Dedekind infinite
non-well-orderable sets (hence Dedekind infinite sets that are
equinumerous to no V.N. ords).

.

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