Bourbaki and foundations

[martdowd at aol.com]

FOM:
Jodmos Horon writes: 

It's based on sets. Meaning it's based on elements. Meaning that, implicitly, all sets are equivalent to the boolean algebras they induce. Essentially, set theory has atoms. I believe it's not necessary and may even be flat out wrong.

 
 The sets at level $\alpha$ of the cumulative hierarchy become the elements at level $\alpha+1$.  This stratification has such multifarious properties that it can be argued to constitute a natural phenomenon.  As such, it is no accident that any mathematical object can be taken as an element of the cumulative hierarchy.  The choice has varying degrees of arbitrariness, depending on the object, with some choices having nice, convenient, properties for some objects.  The von Neumann ordinals are a good example of this.  Those less than $\omega$ provide a set which can be adopted as the natural numbers, with useful properties in some discussions.
Whatever choice is made for $N$, the construction of $Q$ is so straightforward it possesses some degree of naturality.  The same can be said of Dedekind cuts.
Thus, it is not surprising that a group should be considered as a set when the situation demands this (a fortiori also its elements).  That the group is an object in a category is a "bonus", which in no way detracts from the utility of set theory.  Further, the category is a predicate on V, singling out some sets to provide a realization of the category in V.  This is a logical and foundational convenience, having an incidental bearing on category theory itself, but providing a point of reference for discussions of the foundational significance of category theory.

Martin Dowd
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