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In the book "Handbook of Analysis and its Foundations" by Schechter many weakened versions of the Axiom of Choice are presented.
In particular consider the following:

Axiom of Choice If $\{X_{\lambda}\ \lambda\in\Lambda\}$ is a set of non-empty sets, then the Cartesian product $\prod_\lambda X_\lambda$ is nonempty, i.e. there is a function $f:\Lambda\rightarrow\bigcup_{\lambda\in\Lambda} X_\lambda$ such that $f(\lambda)\in X_\lambda\ \forall \lambda\in\Lambda$.

Ultrafilter Principle Any proper filter is included in an ultrafilter. That is, if $\mathcal{F}$ is a proper filter on a set $X$, then there exists an ultrafilter $\mathcal{U}\supset\mathcal{F}$ on X.

Axiom of Choice for Finite Sets Let $\mathcal{C}$ be a family of finite sets. Then it is possible to choose some member $s=f(S)\in S$ for each $S\in\mathcal{C}$

In the book, the Ultrafilter Principle is introduced as a midway between the other two. Anyhow, this is not obvious to me, and I would like to have some suggestions.
What I notice is that (AC) and (ACF) differ in the cardinality of the sets considered, independently from the cardinality of their collection. In a similar fashion the Axiom of Countable Choice is presented as weakening of the Axiom of Choice, this time acting on the cardinality of the collection of sets, regardless of that of the sets in the collection. Is there more to this 'intuition' ?

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I don't see any precise meaning for "midway" in this context. I suspect that all that was meant is that (1) the axiom of choice implies the ultrafilter principle, (2) the ultrafilter principle implies the axiom of choice for finite sets, and (3) neither of the preceding implications is reversible.

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