__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.

Why?

**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__) are not provable.

*S**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__) 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**M*on (

__not__) any easier or reassuring.

*M***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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