Wednesday, March 19, 2008

The crowning glory of modern logic

Quoting Paul Halmos on Goedel's incompleteness theorems in their algebraic form:
"What has been said so far makes the Goedel Incompleteness theorem take the following form: not every Peano algebra is syntactically complete. In view of the algebraic characterization of syntactic completeness this can be rephrased thus: not every Peano algebra is a simple polyadic algebra. ... What follows is another rephrasing of this description... Consider one of the systems of axiomatic set theory that is commonly accepted as a foundation for all extant mathematics. There is no difficulty in constructing polyadic algebras with sufficiently rich structure to mirror that axiomatic system in all detail. Since set theory is, in particular, an adequate foundation for elementary arithmetic, each such algebra is a Peano algebra. The elements of such a Peano algebra correspond in a natural way to the propositions considered in mathematics; it is stretching a point, but not very far, to identify such an algebra with mathematics itself. Some of these 'mathematics' may turn out to possess no non-trivial proper ideals, i.e. to be syntactically complete [since ideals represent refutable propositions]; the Goedel theorem implies that some of them will certainly be syntactically incomplete. The conclusion is that the crowning glory of modern logic (algebraic or not) is the assertion: mathematics is not necessarily simple."
That was from his "Basic Concepts of Algebraic Logic," available on JSTOR. Two comments: (1) It'd be nice if more logic articles made me laugh. (2) I must figure out how to work a pun like that into some future article. (As is probably clear from those comments, I have a weak spot for puns.)

2 comments:

Duck said...

I'd never heard of Kleene, but my logic prof told us that in logic, Kleene-ness is next to Gödeliness.

Shawn said...

Kleene was one of the first major American logicians of the twentieth century. His wikipedia entry is pretty good.