Don't know of a name, but it's equivalent to the abelianization having order not divisible by $p$.

Per Dieter's comment, these are precisely $x$ such that $x+1$ is a Riesel number.

At least in the ZF(C) context, the term "class" is commonly used for a collection of sets which is definable (with parameters). A proper class is any such definable collection which is itself not a set. Separation then asserts that any subclass of a set is itself a set.

Some answers here may apply.

You don't even need the "local choice axiom" - if U is bijective to On, then any proper class is bijective with a subclass of On, and any proper subclass of On is in bijection with On itself.

I wouldn't say we can "almost prove this for $1093n+1$". We only know it doesn't reach 1, we can prove nothing regarding divergence.

@C7X I'm uncertain what I meant 10 years ago but I think replacing it with "PTO of ... is $\leq\gamma$" would match what I was after. But then again by now I am aware this is very sensitive to how the well-orders are coded and just speaking of ordinals on their own doesn't make sense here, so I'm not sure how meaningful that question is.

$C_2$ acts on $\mathbb C^2$ by swapping the coordinates, and acts transitively on the standard basis, but is reducible.

An easy way to see this sequence is not periodic is that the density of $k$ for which $\xi(k)=1$ is equal to $1/\sqrt{2}$.

"Germs that form a field" does not specify the field uniquely. The ring of germs has many distinct maximal subfields. And the elements whose derivatives are eventually nonzero does not form a ring, let alone a field.

But a Hardy field cannot contain all of those germs. I didn't downvote the question, though it would do with some clarifications. I downvoted your answer since without those clarifications it is simply wrong, the ring of all smooth germs at infinity doesn't embed into surreals.

That's not a field. It's also not an integral domain so definitely doesn't embed into surreals.

Apologies, I meant germs of rational functions. They form the smallest Hardy field containing polynomials.