Re: The set of necessary FISONs

On 01.02.2025 15:56, FromTheRafters wrote:

WM was thinking very hard :

On 31.01.2025 18:52, joes wrote:

Am Fri, 31 Jan 2025 11:23:35 +0100 schrieb WM:

>

Because every set of ordinals has a first element.

The empty set doesn’t.
>

The empty set contains no ordinals. Every set of ordinals contains ordinals. "Jeder Inbegriff von verschiedenen Zahlen der ersten und zweiten Zahlenklasse hat eine kleinste Zahl, ein Minimum." [Cantor, p. 332]

 The empty set is order type zero. It comes 'before' order type one.

Anyhow it is not a set of ordinal numbers. Therefore the condition "not empty set of ordinals" shows stupidity of highest level. I quoted Cantor to show that he also did not mention "non-empty". A set of ordinal numbers contains at least one ordinal number!

 Is 'before' the same as 'smaller' in the infinite sense as it is in finite sense?

Yes. At least for the first and second number class, according to Cantor. The numbers are well-ordered by size.
Regards, WM