Talk:Set of Natural Numbers Equals its Union

Before throwing around {{rename}} templates: Wouldn't it be better to have these theorems with "Finite Ordinals" instead of "Natural Numbers"? That would make them more universally applicable. Thoughts welcome. — Lord_Farin (talk) 18:23, 26 April 2022 (UTC)

I am deliberately using the terminology used by Smullyna and Fitting, where they specifically refer to $\omega$ as the set of natural numbers.

Their approach is both straightforward and thorough, and they are very careful to keep everything as accessible and simple as possible. In consequence, they don't even mention the term "ordinal", and at this stage they haven't even defined the term "finite".

They construct the set of natural numbers by the Von Neumann construction, and identify them with the natural numbers, having briefly mentioned Peano's axioms.

As a consequence, I think it a good idea to keep this page as is, without trying to fit it into an existing conceptual framework that we are perhaps more familiar with because we "know better" -- bearing in mind the fact that at present, the $\mathsf{Pr} \infty \mathsf{fWiki}$ journey to the ordinal set is the Takeuti and Zaring route which as we know is somewhat problematic.

I have high hopes for the Smullyan and Fitting approach, because it looks and feels a lot more coherent than what we've currently got. Now, I don't know if this is genuinely because S&F tread a more solidly-founded route than does T&Z, or because the rendition of T&Z as presented on $\mathsf{Pr} \infty \mathsf{fWiki}$ has cut corners, but I believe it's worth continuing the S&F work in its current shape. I shouldn't have stopped where I did, I should have soldiered on with it, but I broke off to do something else and never got back to it. I will get back to it, I just wanted to consolidate Warner first, to make sure its approach is equally consistent and complete. And I'm getting there, but because of stress of the day job and other stuff going on at the moment, I am unable to bend a braincell in quite the right direction this last week or so, and I'm getting bogged down in the work concerning inductive semigroups, which I believe is actually more profound than it looks on the surface: uniqueness of certain algebraic structures related to $\N$ may offer significant insights into this whole area of foundation theory. --prime mover (talk) 20:47, 26 April 2022 (UTC)

Alright, that sounds fair enough. No rush. Nonetheless, my hands are itching to get proper coverage of ordinals done on PW, as indeed having only T&Z is quite lacking for such an important aspect of mathematics. However I cannot commit the time and dedication to set this up properly at this moment, so I am left to come to terms with my disappointment. Such is life. — Lord_Farin (talk) 19:13, 27 April 2022 (UTC)

I'll take another look at Devlin in due course, and hope I don't get piled on for addressing the subject incorrectly. --prime mover (talk) 05:22, 28 April 2022 (UTC)