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Formal mathematics builds on formal logic. Set theory has become the standard way to formalize the foundations of mathematics because of its simplicity and power. The only undefined primitive object in set theory is the empty. Its only property is that it contains no subsets. There is no clearer illustration of the absence of content or intrinsic nature in mathematics and physics than the construction of all of mathematics assuming the empty set as the only primitive entity.
The standard axioms of set theory are summarized in Figure 6.4. This figure references the sections where the axioms are explained. These axioms are adequate for all of conventional mathematics. Almost every mathematical abstraction that has ever been investigated can be derived as a set that these axioms imply exists. Almost every mathematical proof ever constructed can be made assuming nothing beyond these axioms. These axioms are less than a page long.
It is straightforward to program a computer to output all the theorems that can be deduced from these axioms. This is not a practical way to derive mathematics because most of the theorems are trivial and of no interest. Interesting theorems are extremely rare. It would take a long time before such theorems occur and it would be very difficult to select them out.
PDF version
of this book
Next: Axioms of Set Theory Up: Mathematical
structure Previous: Formal logic Contents
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