Thursday, February 23, 2012

Why Gödel induced no despair among mathematicians

The Hilbert program aims to formalize (axiomatize) all mathematics on a secure core foundation of “finitistic” (excluding infinite sets, etc) mathematics whose “truth” is beyond all doubt.

Gödel's incompleteness theorems prove that any such formalization that is powerful enough to embed (encode) Peano arithmetic is incomplete (i.e. there is a statement of Peano arithmetic that can neither be proved nor disproved). This shows the impossibility of the Hilbert program, and is said to have induced a “crisis” in the foundation of mathematics.

Despite the occasional sensational claim that Gödel's incompleteness theorems in effect destroyed both the mathematical enterprise, and society’s and scientists’ faith in mathematics, mathematics has in fact progressed apace since 1931.

Other than logicians and theoretical computer scientists, mathematicians are, in general, blissfully uneducated in Gödel's incompleteness theorems. An algebraist or a topologist would happily pursue her/his research without any regard to the foundational “crisis“. Far from being disillusioned, mathematicians are as motivated and enthusiastic as ever.


Case 1

If one is trying to prove or diaprove a statement M in a theory T (such as
set theory), that is powerful enough to embed Peano arithmetic, then, as T is incomplete (as proved by Gödel), there are statements S in T such that both S and (not S) are not provable. M  may be just such a statement. If so, trying to prove or disprove M is doomed to fail.

This then is the extent of one’s psychological “crisis” induced by the foundational “crisis“.

However, even if the theory were complete, so that either M or (not M) is provable, proving either may well be beyond one‘s intellect. I would contend that knowing that the theory is complete does not make one's attempt to prove either M on (not M) any easier or reassuring.

Case 2

If one is working on a theory incapable of embedding Peano arithmetic, then the theory may in fact be complete, and perhaps there is no crisis at all.

Some examples of such complete theories are Presburger arithmetic , Tarski's axioms for Euclidean geometry , the theory of dense linear orders, the theory of algebraically closed fields of a given characteristic, the theory of real closed fields , every uncountably categorical countable theory, and every countably categorical countable theory. (see Complete theory)

In general, Gödel makes no practical difference to a research mathematician. Mathematics has continued to develop despite the failure of the over-ambitious Hilbert program.

For the less ambitious successors to the Hilbert program, see
Wikipedia and Partial Realizations of Hilbert’s Programs, by Stephen G. Simpson.

(For a brief introduction to mathematical logic, and the foundation of mathematics, see Logic and Mathematics, by Stephen G. Simpson.)




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