**Authors:** Andrew Banks

This paper will demonstrate a diagonal argument by listing all non-empty finite ordinals in a table according to their ε order using their subset representation, meaning {0,1,2…n-1} is listed for the ordinal n. Next, the axiom of choice is applied to all of these ordinals and selects the maximal element. This selection process forms a diagonal which satisfies the axiom of infinity, hence, the diagonal is a limit ordinal. However, it will also be shown for the nth choice made by the choice function, the diagonal is the successor ordinal number n = {0,1,2…n-1} and this is true for all n. So, at the n+1 choice, the diagonal is the ordinal n+1 and so on. Therefore, based on all the actions of the choice function, it is provable from ZFC on one hand that this diagonal cannot ever be anything other than a successor ordinal and on the other hand, the diagonal is a limit ordinal.

**Comments:** 8 Pages.

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[v1] 2012-06-09 09:13:06

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