Skolem's Paradox (First Order Logic)

[Matterwave]

(b) Suppose we are given some arbitrary model $M$. In some specific cases it may be the case that all the ordinals in $M$ are also in $Ord$ (class of ordinals). However, it is worth noting that there may be cases where this does not hold [if we have a $V$ with standard $\mathbb{N}$ then I think perhaps an example of this could be an $M$ with non-standard $\mathbb{N}$].

I must admit that when ordinals are brought up the topic usually moves beyond me. The only thing I "know" is that given the Von Neumann encoding, a number is the "set of all lower numbers" (e.g. $2 := \{\emptyset, \{\emptyset \}\}=\{0,1\}$) so then we can define infinite ordinals $\omega:=\mathbb{N},\, \omega+1=\mathbb{N}\cup\{\mathbb{N}\},...$ and then go from there. Beyond that.. nada.

 

[SSequence]

What I was trying to describe in post#25 seems to be described better in this image. Also if you just read the paragraph just below the image (in previous sentence) in the link it seems to be fairly instructive (and readable) ..... the other stuff though is a bit advanced for me.