The generalized Axiom of Dependent Choice (DC) - is this a valid generalization? After studying the axiom of dependent choice, I tried to think of a possible generalization of the axiom that would work in a similar way on infinite uncountable sets: by replacing the binary relation with a general set membership relation over ordinal-indexed elements, I finally obtained the axiom below.
Remark: in order to make the axiom easier to read, I use the notation $(\underbrace{y_0, y_1, \dots}_{\alpha})$ to denote a map of elements indexed over an ordinal $\alpha$.
Generalized Axiom of Dependent Choice: given a set $X$, an element $x_0\in X$, a limit ordinal $\beta$, and a set $R$ containing elements of $X$ indexed over any $\alpha\in\beta$, suppose that the following two relations hold:
$
(a)\quad \forall \alpha\in\beta\,:\,(\underbrace{y_0, y_1, \dots}_{\alpha})\in R\,\,\Rightarrow\,\,\exists z\in X : (\underbrace{y_0, y_1, \dots, z}_{\alpha+1})\in R
$
$(b)\quad$ if $\sigma$ is a sequence of elements of $X$ indexed by a limit ordinal $\gamma\in\beta$ such that each initial segment of $\sigma$ is an element of $R$, then $\sigma$ is an element of $R$.
Then there exists a function $f\,:\,\beta\,\mapsto\,X$ s.t. $f(0)=x_0$ and
$$
\forall\alpha\in\beta\,:\,(\underbrace{f(0), f(1), \dots}_{\alpha})\in R
$$
Do you think this is a valid and sound generalization? I found simple proofs of the following facts:


*

*if $\beta=\omega$ (i.e. the first infinite ordinal), then the generalized version is equivalent to the axiom of dependent choice with fixed first element.

*the generalized axiom of dependent choice implies the axiom of choice on the ordinals (although it is not clear if they are also equivalent)

*although inherently non-constructive, the generalized axiom hints to a 'method' for building the function $f\,:\,\beta\,\mapsto\,X$: choose $x_0$ as the first element, then randomly select the following elements $x_1, x_2,\dots$ in such a way that the sequence $(x_0,x_1,x_2,\dots)$ is still in $R$. This seems to be an intuitive process

*if we replace the condition $\cdots\Rightarrow\,\,\exists z\in X\cdots$ with the condition $\cdots\Rightarrow\,\,\exists ! z\in X\cdots$ (where the existing element $z\in X$ is also unique) then the function $f\,:\,\beta\,\mapsto\,X$ is provably unique. Although the changed version may seem to be provable in ZF, it is not a consequence of the axiom schema of replacement, nor it is definable via transfinite recursion - in other words, I suspect it is independent of ZF.
I would like to receive some feedback from other logicians: does this look like a valid generalization? Do you think it may be worth of a publication, and in case, in which journal?
Please share your thoughts about this, thank you in advance and have a great day!
Kind regards,
Gianluca Calcagni  
 A: The axiom you propose is inconsistent: take $\beta=\omega2$, and consider the set $R(\langle y_\eta\rangle_{\eta<\alpha})\iff \alpha<\omega$. Then certainly $R(\langle x_0\rangle)$ (regardless of what $x_0$ is), and $R(\langle y_0, y_1, . . . \rangle)$ implies $R(\langle y_0, y_1, . . . , z\rangle)$ (since appending an additional term to a finite string yields a finite string).
The problem is that the property $R$ above is sensitive to the "gap" between the finite ordinals and the infinite ones; in general, this is why Dependent Choice is defined as it is. 
A: Noah gave a nice example why this generalization does not work. But there is in fact a generalization of $\sf DC$ to larger [$\aleph$] cardinals, due to Azriel Levy.
$\sf DC_\kappa$ is the statement that, whenever $S$ is a non-empty set, and $R$ is a relation on $S^{<\kappa}\times S$ with domain $S^{<\kappa}$, then there is a function $f\colon\kappa\to S$ such that $f\restriction\alpha\mathrel{R}f(\alpha)$ for all $\alpha<\kappa$.
Namely, if we have a relation $R$ such that for every $\alpha<\kappa$ and every $f\colon\alpha\to S$, there is some $z\in S$ such that $f\mathrel{R}z$, then there is a function from $\kappa$ which satisfies the wanted conclusion.
As a side note, this is equivalent to saying that every $\kappa$-closed tree without leaves has a branch of length $\kappa$; and that Zorn's lemma holds for chains of order type $<\kappa$ (namely, if $P$ is a partial where every chain of order type $<\kappa$ has an upper bound, then $P$ has a maximal element).
Let me finish by saying that this sort of gap is the reason that these two proofs cannot be generalized immediately to accommodate $\sf DC_\kappa$ for uncountable $\kappa$:


*

*If for every ordinal $\alpha$, $\sf AC_\alpha$ holds, then $\sf DC$ holds.

*If $M$ is a structure for a finite language, then $M$ has an elementary submodel of size $\leq\kappa$ if and only if $\sf DC+AC_\kappa$.


Both proofs work because for $\kappa=\aleph_0$ we can readily approximate things with finite steps, but we cannot bridge the obvious gap of limit ordinals.
