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- #1

If we do have to invoke AOC, is there any way of `inverting' an injection to get a surjection

*without*using AOC?

- Thread starter Swlabr
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- Thread starter
- #1

If we do have to invoke AOC, is there any way of `inverting' an injection to get a surjection

- Jan 30, 2012

- 2,541

AC is important when when there is an infinite collection of nonempty sets and one has to build a single function that chooses an element of each set. If there is just one or finitely many nonempty sets, then such function can be described in a finite way.

In ZFC, try to use Axiom schema of specification with Axiom schema of replacement (with Axiom schema of collection). I did not check, but it should work. Functions on sets are morphisms in set theory. My educated guess is that ZF (without the Axiom of choice) are strong enough to allow such an algebraically fundamental concept.

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- Feb 15, 2012

- 1,967

i see a problem, here. what if X IS such a collection of non-empty sets? i sincerely doubt that there exists a way to select "any" element from an abitrary single set in a finite way. what if X is uncountable? the problem is: just calling X "a single set" does not mean X has a simple internal structure that can be finitely described.

AC is important when when there is an infinite collection of nonempty sets and one has to build a single function that chooses an element of each set. If there is just one or finitely many nonempty sets, then such function can be described in a finite way.

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- #5

- Feb 7, 2012

- 2,799

Calling X "a single set" means that X is a set. On the other hand, "a collection of non-empty sets" need not be a set.i see a problem, here. what if X IS such a collection of non-empty sets? i sincerely doubt that there exists a way to select "any" element from an abitrary single set in a finite way. what if X is uncountable? the problem is: just calling X "a single set" does not mean X has a simple internal structure that can be finitely described.

One of the basic ingredients in set theory is that the definition of a set has to be constrained (so as to avoid Russell's paradox), so that an arbitrary collection of elements is not necessarily a set. However, if X is a set, then it is legitimate, within ZF without C, to choose an element of X.

Given an injection $f:X\to Y$, it is implicit that the domain $X$ is a set. The OP's argument for constructing a surjection from $Y$ to $X$ (a left inverse for $f$, in fact) is valid in ZF with no need for AoC.

- Feb 15, 2012

- 1,967

with a countable set X, we can leverage the bijection and well-ordering of N to pick a "least" element of X. i am uncertain how one goes about generating a well-defined choice from an uncountable set. this may be sheer ignorance on my part, if so...please, enlighten me.