O'Reilly logo

The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

V.15 Gödel’s Theorem

Peter J. Cameron

In response to problems in the foundations of mathematics such as Russell’s paradox (“consider the set of all sets which are not members of themselves; is it a member of itself?”), HILBERT [VI.63] proposed that the consistency of any given part of mathematics should be established by finitary methods that could not lead to a contradiction. Any part for which this had been done could then be used as a secure foundation for all of mathematics.

An example of a “part of mathematics” is the arithmetic of the natural numbers, which can be described in terms of FIRST-ORDER LOGIC [IV.23 §1]. We begin with symbols, both logical (connectives such as “not” and “implies,” quantifiers such as “for all,” the equality ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required