Question: is it possible for me to ask for a proof, but the sheet doesn't have enough domain axioms (or, enough Boolean identities) for you to successfully prove it? The technical way to phrase this question is ``Is our system complete?''

In the late 1800s and early 1900s, mathematicians became more concerned with logic and proofs, realizing it was the foundation of all mathematics. A couple of disturbing problems had been posed, with almost childish ease: If a set is defined simply by being able to say, for any item, whether or not the item is in the set, then we can talk about UU, the set of everything. (When asked if something is in UU, the answer will be yes.) Thus, UU contains itself. Okay, but what about the set RR, defined as the set of: all sets which don't contain themselves. Does RR contain itself? There is no consistent answer to this question, called ``Russell's Paradox'', after Bertrand Russell. This caused people to ask, is mathematics sound? That is, can it be used to prove things which aren't true? (Such as a set which both does and doesn't contain itself.)

This led to the development of a set theory which avoided the possibility of paradoxical sets, and at the same time led people to wonder what is a proof system which both sound and complete, and can we prove it so? Russell, along with Alfred North Whitehead, spent years writing the Principia Mathematica, a multi-volume tome of formal proofs, trying to secure the foundation of all mathematics. Unfortunately, they were unable to prove the soundness of mathematics, using those same proofs.

Some light reading: Hilbert's 23 problems: Here's a Scientific American article which provides some nice context, and a straight listing of Hilbert's 23 problems and their status.

Frege-Lukasiewicz introduced an inference system for Propositional Logic (not first-order logic) which has only three rules:

After Hilbert's 1900 address, many talented mathematicians worked hard on proving the completeness of first-order logic. It was not a problem that turned out to be easy, and it became an entire research program, many people investing their entire career in this. The answer came in two surprising parts,

This latter was a particular rude shock to many mathematicians; overnight their life's work of incremental progress was shown to have been advancing down a wrong alley. In fact, the idea that something can be true yet not provable is disturbing all by itself.

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This work is licensed by Ian Barland, John Greiner, Phokion Kolaitis, Moshe Vardi, and Matthias Felleisen under a Creative Commons Attribution License (CC-BY 1.0), and is an Open Educational Resource.

Last edited by Ian Barland on Jun 15, 2006 4:21 pm -0500.