# FOM: Re: finite sets

Karlis Podnieks podnieks at cclu.lv
Tue May 25 01:24:06 EDT 1999

```-----Original Message-----
From: Stephen G Simpson <simpson at math.psu.edu>
To: fom at math.psu.edu <fom at math.psu.edu>
Date: piektdiena, 1999. gada 30. aprîlis 2:18
Subject: FOM: finite sets

...

>In the absence of the axiom of choice, the most useful of the
various
>inequivalent definitions of finite set is: a set that is
equinumerous
>with a von Neumann ordinal less than the first limit ordinal.
In the
>absence of the axiom of infinity, change that to: not greater
than or
>equal to any limit ordinal.  This is a pretty specific notion
of
>finite set, and it allows you to derive the standard
properties.

KP> You could add some foundational flavour to this definition
by putting it in the following equivalent way:  a set is called
finite, iff it can be doubly-well ordered, i.e. if it can be
ordered in such a way that each non-empty subset contains both
minimum and maximum element. You can introduce this definition
long before you develop the ordinal stuff. This trivial
modification makes set theory more natural for students.

Karlis Podnieks
podnieks at cclu.lv
http://www.ltn.lv/~podnieks/
University of Latvia
Institute of Mathematics and Computer Science

```