# Axiom of Choice/Examples/Russell's Socks and Shoes

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## Example of Use of Axiom of Choice

Suppose we have an infinite number of pairs of socks.

Using the **axiom of choice**, we can simultaneously pick one sock from each pair.

However, if we also have a infinite number of pairs of shoes, we no longer need the **axiom of choice** to pick one shoe from each pair.

We simply choose the left one.

## Proof

This is more formally stated as:

- Given an infinite set of
**unordered**pairs, the**axiom of choice**is needed to pick one element from each pair.

- Given an infinite set of
**ordered**pairs, the**axiom of choice**is**not**needed to pick one element from each of the pairs: we may simply pick the first element of each.

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## Historical Note

This example was devised by Bertrand Russell as a neat illustration of how the **axiom of choice** can be used.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 1$ Introduction to well ordering