Talk:Minimal Infinite Successor Set is Minimal

I would have to say this is not ideal naming/conception. There are at least three concepts running around this particular zone of natural number definitions:


 * Finite ordinal/natural number


 * Natural numbers


 * Minimal infinite successor set

There are two approaches one can take within this zone:

1. Define a "finite ordinal"/"(von Neumann) natural number" as an ordinal $n$ having one of the following "finiteness" properties: (a) $n$ is strictly well-ordered under $\ni$ as well as $\in$. (b) $n$ is either $\varnothing$ or a successor ordinal whose elements are themselves either $\varnothing$ or successor ordinals. Then define $\omega$ as the set of all finite ordinals.

This approach has the advantage that it produces a concept of "natural number" without needing the axiom of infinity.

2. Define $\omega$ as a "minimal infinite successor set", use the axiom of infinity to prove that $\omega$ exists, and define a natural number to be an element of $\omega$.

We currently have only approach 2. I would like to add approach 1b (source: Gödel). --22:44, 27 May 2013 (UTC)

What this is for, I think
OK, maybe the point is that the "minimal" of "minimal infinite successor set" can be taken literally. i.e., that if $Q$ is an infinite successor set that does not have a proper subset which is an infinite successor set, then $Q = \omega$. (so that a minimal inf succ set is SMALLEST). Maybe. --Dfeuer (talk) 22:49, 27 May 2013 (UTC)