LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

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AC supplies countable selections and prescribed serial paths

Statement

Assume AC. Then countable choice holds. Moreover, if $R \subseteq X \times X$ is serial on a nonempty set X and $a \in X$ , there is a sequence $(x_{n})_{n \geq 0}$ with $x_{0} = a$ and $x_{n} R x_{n + 1}$ for every n. Thus the countable and dependent choices required by the probability-product suppliers are available under AC.

Facts & Assumptions

[F1]

AC supplies a choice function on a set of nonempty sets. The Axiom of Choice.

[F2]

Countable choice is selection from an omega-indexed nonempty family. The Axiom of Countable Choice ( ${A C}_{ω}$ ).

[F3]

DC requires a serial path starting at a prescribed point. The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain.

[F4]

A given self-map and initial point have a uniquely specified natural-number iterate sequence. The recursion theorem.

Proof

Given: Assume AC. Then countable choice holds. Moreover, if $R \subseteq X \times X$ is serial on a nonempty set X and $a \in X$ , there is a sequence $(x_{n})_{n \geq 0}$ with $x_{0} = a$ and $x_{n} R x_{n + 1}$ for every n. Thus the countable and dependent choices required by the probability-product suppliers are available under AC.

1.1

For an omega-indexed family $(A_{n})$ of nonempty sets, its image $S = {A_{n} : n \in N}$ is a set of nonempty sets. By [F1] choose $c$ with $c (A) \in A$ for every $A \in S$ . Then $b (n) = c (A_{n})$ is a function and $b (n) \in A_{n}$ , which is [F2]. Repeated sets use the same selected value and cause no ambiguity. For an empty index family the empty function already suffices; singleton fibers force their unique value.

F1F2

1.2

For the stated serial R, every fiber $R [x] = {y \in X : x R y}$ is nonempty. AC applied to the set of all these fibers gives c with $c (R [x]) \in R [x]$ . Define the self-map $s (x) = c (R [x])$ on X. No recursively changing choice is being assumed: s is now one fixed function chosen from a fixed set of nonempty fibers. It satisfies $x R s (x)$ for every x.

F1F3

2.1

Apply [F4] to X, the prescribed a and s to obtain $x_{0} = a$ and $x_{n + 1} = s (x_{n})$ for every natural n. The defining property of s gives $x_{n} R x_{n + 1}$ , precisely [F3]. If X has one element and R is serial, this is its constant sequence. Empty X is excluded by the prescribed a. The uses of AC are exactly the two fixed-family selections in steps 1.1 and 1.2; recursion itself requires no further choice. This proves the two implications from AC, not either converse.

step 1.1step 1.2F3F4∎

Depends on

Used by

De Moivre-Laplace central limit theorem Corollary
Infinite variance can defeat square-root-n CLT scaling Counterexample
Multivariate normal law, including singular covariance Definition
A Lindeberg array with no identically distributed row Example
CLT for sums of uniform random variables Example
Lyapunov condition for nonidentical summands Example

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jech, The Axiom of Choice, §2.4, pp22–23; elementary AC-to-DC restriction proof (standard reference, not scraped)