Philosophy of Mathematics seminar

Keith Hossack (Birkbeck)

The known universe  

According to property realism, properties are entities that are metaphysically real: a property is instantiated by some things just if there exists an appropriate complex whose constituents are that property and those things. Since any plurality of things are collectively self-identical, it follows that given any plurality, there exists the complex whose constituents are the property of collective self-identity and that plurality. If we identify this complex with the set whose elements are the plurality, the axioms of the set theory NFU can then be deduced from the axioms of plural logic. Thus NFU can be arrived at just by philosophical reflection.

NFU is quite a weak theory: it cannot prove the existence of the plurality of natural numbers, so it cannot prove the Axiom of Infinity. However, the combination of NFU with Peano Arithmetic is mathematically more serviceable. Call the known universe the pure sets whose existence can be proved in a finite number of steps by NFU and Peano Arithmetic. The known universe is populous enough to supply all the needs of ordinary mathematics, for it is as big as the first set of non-constructive rank in the cumulative hierarchy. However, if the postulates of a set theory such as ZFC are true, the known universe is only the tiniest speck in the universe as a whole. And how is the question to be answered, whether there really do exist further sets beyond the known universe?