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Formal mathematics builds on formal logic. It reduces mathematical relationships to questions of set membership. The only undefined primitive object in formal mathematics is the empty set that contains nothing at all.
The standard axioms of set theory are summarized in Figure 4.3. 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 but no finite structure can ever capture all of mathematics.
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.
Completed
second draft of this book
PDF version
of this book
Next: Axioms of Set Theory Up: Mathematical
structure Previous: Formal logic Contents
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