## Consistency of strictly impredicative NF

Centro de Matemática e Aplicações Fundamentais

Universidade de Lisboa

Thursday, 26 January 2012, 14:00

Cybernetica Bldg (Akadeemia tee 21), room B101

Slides from the talk [pdf]

**Abstract**: We present the following our main contribution into the consistency
problem of NF:

An instance of Stratified Comprehension

A x_1 ... A x_n E y A x (x in y <--> phi(x,x_1,...,x_n))

is called *strictly impredicative* iff, under stratification,
*x* receives the minimal type. Using the technology of forcing,
we prove that the fragment of NF based on strictly impredicative
Stratified Comprehension is consistent. A crucial part in this proof,
namely showing genericity of a certain symmetric filter, is due to
Robert Solovay.

As a bonus, our interpretation also satisfies some instances of Stratified
Comprehension which are *not* strictly impredicative. For example, it
verifies existence of Frege natural numbers.

Apparently, this is a new subsystem of NF shown to be consistent. The
consistency question for the whole theory NF remains open (since 1937).

Main reference:

S. Tupailo. Consistency of Strictly Impredicative NF and a little more...
*Journal of Symbolic Logic*, v. 75, n. 4, 1326-1338, 2010.

Tarmo Uustalu

Last update 5.2.2012