nLab New Foundations

Contents

Context

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axiom

Removing axioms

Contents

Idea

New Foundations (NF) is a version of axiomatic set theory in which there is a set of all sets, but the axiom of comprehension is restricted by a “stratification” condition.

The category of sets and functions in NF is known to not be cartesian closed, so it is not a topos, but can be equipped with finite limits and a subobject classifier. One can even construct a set of all functions between two sets, but there is no evaluation map A×B ABA\times B^A \to B making this an exponential in the category.

There are at least three proofs of this fact: one, due to Randall Holmes, doesn’t seem to have been made public; the others are due to McLarty and Forster, and derive contradictions (different in the two proofs) from assumptions on the behaviour of a pairing function used to construct binary products.

For a long time the (relative) consistency of NF was an open problem, although NF with urelements (NFU) was known to be consistent (relative to ZF plus some large cardinals). Recently there may be a proof of its consistency.

References

Last revised on May 21, 2022 at 14:30:30. See the history of this page for a list of all contributions to it.