[2] **viXra:1206.0106 [pdf]**
*submitted on 2012-07-01 00:39:37*

**Authors:** Pierre-Yves Gaillard

**Comments:** 6 Pages.

This is the beginning of an attempt at rewriting the book "Categories and Sheaves" by Kashiwara and Schapira without using Grothendieck's universes axiom.

**Category:** Set Theory and Logic

[1] **viXra:1206.0030 [pdf]**
*submitted on 2012-06-09 09:13:06*

**Authors:** Andrew Banks

**Comments:** 8 Pages.

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.

**Category:** Set Theory and Logic