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This can be thought of as a continuation of the post Hermis14, which deals with sets of specific structures. Now, the last question that remains to me is when the finiteness of a set really matters.

For nonempty sets $A$ and $B$, consider a surjection $ f: A \to B $.

Consider the sentences:

  • $S1$: $B$ is finite.

  • $S2$: $B$ is infinite.

  • $S3$: There is a function $g: B \to A $ such that $ \forall y \in B: f(g(y)) = y $.

I think both claims $ S1 \to S3$ and $ S2 \to S3 $ require the axiom of choice because though we know that the fiber $f^{-1}[y]$ for each $y \in B$ is nonempty, a specific choice rule cannot be considered since $f$ is unknown.

But Wikipedia says that

In many cases, such a selection can be made without invoking the axiom of choice; this is, in particular, the case if the number of sets is finite, or if a selection rule is available

Does it mean that $S1 \to S3$ does not invoke the axiom of choice? Would you help me figure out what I am missing?

Hermis14
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  • AC asserts the existence of a "choice set"; thus, if we have a finite number od finite sets, we may simply list them: $A_1= { a,b,c }, A_2= { e,f,g }, A_3= {l,m,n }$ and "legislate" that the choice set is $S = { a,f,n }$. No further condition is needed, because ${ A_1,A_2,A_3 }$ is a collection of nonempty sets and $S= (x_i)$ is such that $x_i \in A_i$ for every $i$. – Mauro ALLEGRANZA Nov 29 '21 at 10:07
  • The same for an infinite collection of infinite sets: $A_1= {2,4,8,\ldots }, A_2 = { 3,9,27, \ldots }, A_3 = { 5, 25, 125, \ldots }$ because we have a "rule" that describes them and thus we can always specify the "choice set" with a "derived rule": pick the first one for every set: $S = { 2,3,5,\ldots }$. – Mauro ALLEGRANZA Nov 29 '21 at 10:10
  • In your example, we have that $B$ is finite and thus we can list it: $B = { b_1, \ldots, b_n }$. But how to define $g$? We have that in order to specify $g = { (b_1,a_1), \ldots, (b_n, a_n) }$ we need a rule to pick the $a_i$ (that for sure exist) ... – Mauro ALLEGRANZA Nov 29 '21 at 10:16

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