Is it possible to define a family of fundamental sequences for all countable (limit) ordinals? (Without AC)

Question

A fundamental sequence for a limit ordinal $\lambda$ is a sequence of ordinals who converge to it.

A family of fundamental sequences for $\lambda$ is a function $F$ that assigns a fundamental sequence for all limit ordinals below $\lambda$.

At least, that is the definitions I'm working with. I don't mind if you have others. I think it is known that you may define a family of fundamental sequences for all ordinals below $\omega_1^{CK}$, and that without AC you can't do it for $\omega_1$.

I was wondering whether you can define famlies for all countable ordinals or if this also stops before $\omega_1$.

Answer 1

Assuming "define" just means "prove it exists", this is possible for any countable ordinal in ZF. Indeed, if $\alpha$ is a countably infinite ordinal, there exists a bijection $f:\omega\to\alpha$. For any limit ordinal $\beta<\alpha$, we now define $F_\beta:\omega\to\beta$ recursively as follows. Having defined $F_\beta(m)$ for all $m<n$, we define $F_\beta(n)$ as $f(N)$ for the least $N\in \omega$ such that $f(N)<\beta$ and $f(N)>F_\beta(m)$ for each $m<n$.

It is then easy to see by induction on $n$ that $F_\beta$ is increasing and $F_\beta(n)\geq f(n)$ for all $n$ such that $f(n)<\beta$. It follows that $F_\beta(n)$ converges to $\beta$, since it is eventually greater than every ordinal less than $\beta$. So, letting $F$ be the function $\beta\mapsto F_\beta$, we have a family of fundamental sequences for $\alpha$.

Answer 2

In a strong sense, the answer is no:

$(1)\quad$ It is consistent with ZF that there is no function $f$ which assigns to each countable limit ordinal a fundamental sequence for that ordinal.

This is a consequence of the simpler result that countable unions of countable sets need not be countable without choice:

$(2)\quad$ It is consistent with ZF that $\omega_1$ has countable cofinality - that is, that there is an increasing sequence of infinite countable ordinals $(\alpha_n)_{n\in\mathbb{N}}$ whose supremum is $\omega_1$.

$(1)$ follows from $(2)$ as follows. Working in ZF, from a putative function $f$ assigning fundamental sequences to each limit ordinal we could by transfinite recursion build a function $g$ which assigns to each infinite countable ordinal a bijection between that ordinal and $\omega$. But in any model where $\omega_1$ has countable cofinality such a $g$ can't exist: if $(\alpha_n)_{n\in\mathbb{N}}$ is a sequence of infinite countable ordinals cofinal in $\omega_1$, then from a sequence $h_n$ of bijections from $\alpha_n$ to $\omega$ we would get a bijection between $\omega_1$ and $\omega$, and obviously this can't happen.

Answer 3

If by "define" it is meant "prove to exist", then the answer is yes, but if by "define" you mean to give an explicit notion of describing a fundamental sequence system for all ordinals $<\omega_1$, then no. For any (limit) ordinal $<\omega^{CK}_1$ you can always assign a family of computable fundamental sequences for all ordinals up to it, and you can do the same for up to any ordinal $\omega^{CK}_1\leq\alpha<\omega_1$ but it will necassarily be uncomputable.

You can prove a family of fundamental sequences up to $\omega_1$ exists, and it is consistent with ZFC and follows from the axiom of choice, but you can never give one explicitly, nor can you prove that the one you give actually works as a fundamental sequence system. In short, you can't give a formula which provably defines such a fundamental sequence system.