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Sets and ontology

One of the most powerful claims for the relevance for set theory has to do with the notion that all mathematical objects can be reduced to pure sets. Steinhart points out that von Neumann, for example, reduces natural numbers to pure sets by arguing that "0 is the empty set {} and each next number n+1 is the set of all lesser numbers. That is, n+1 = {0,...n}"(p. 19). Thus:

0 = {};

1 = {0} = {{}};

2 = {0, 1} = {{}, {{}}};

3 = {0, 1, 2} = {{}, {{}}, {{}}} (Steinart, p. 19)

"Similar techniques let mathematicians identify all the objects of pure mathematics with pure sets. So it would seem that the whole universe of mathematical objects - numbers, functions, vectors, and so on - is in fact just the universe of pure sets V"(Steinhart, p. 19).

Zermelo, on the other hand, reduced all natural numbers to sets according to the following method:

0 = {};

1 = {0} = {{}};

2 = {1} = {{{}}};

3 = {2} = {{{{}}}} (Steinhart, p. 20).

Are both ways of reducing natural numbers to sets equally good? Some say yes and others say no. Many argue that Neumann's method has advantages over Zermelo's. Some deny that natural numbers can be reduced to pure sets at all. Others, like Quine, argue that literally everything (not only mathematical objects) can be reduced to pure sets. For Quine, all objects are reducible to pure sets; and because all things are reducible to pure sets, they therefore, at base, are pure sets. Steinhart explains:

"...material things can be reduced to the space-time regions they occupy; space-time regions can be reduced to sets of points; points can be reduced to their numerical coordinates; and, finally, numbers can be reduced to pure sets. Quine's ontology is clear: all you need is sets"(Steinhart, p. 20).

Steinhart, Eric. "More Precisely: The Mathematics You Need To Do Philosophy." Broadview Press, 2009.