A first constructivist concern is that you must not just assume that a problem is decidable. The answer to a logical question can indeed be "yes" or "no", but it can also be undecidable.https://en.m.wikipedia.org/wiki/Brouwer ... ontroversy
L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism.
Therefore, it is not acceptable to liberally axiomatize the law of the excluded middle (LEM) without first assessing the decidability of the problem. In other words, the LEM is not a valid axiom.
The second constructivist concern is about abstract existence proofs. It is necessary to always give an example of the mathematical object of which you claim the existence, i.e. a "witness".
The foundational war raged on:
Godel, on the other hand, actually recognized the validity of the constructivist concerns:In 1920, Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen.
Therefore, Godel insisted on supplementing his proof with an otherwise poorly chosen example:Gödel was so sensitive to this issue that he took great pains in his 1931 paper to point out that his Theorem VI (the so-called "First incompleteness theorem") "is constructive;45a that is, the following has been proved in an intuitionistically unobjectionable manner ... ." He then demonstrates what he believes to be the constructive nature of his "generalization formula" 17 Gen r. Footnote 45a reinforces his point.
Every logic statement is true or false or undecidable. Hence, "This statement is not provable" is true or false or undecidable.This statement is not provable.
Leading to:
Godel's incompleteness theorem itself claims:"This statement is not provable" is (true and not provable) or (false and provable) or undecidable.
Hence, "This statement is not provable" is indeed a witness for Godel's theorem.There exist statements that are (true and not provable) or (false and provable) or undecidable.
It is indeed quite a far-fetched example. However, according to constructivism, it is preferable to give even a poor example rather than no example at all. Concerning the law of the excluded middle:
By carefully avoiding the law of the excluded middle, as well as offering at least one example, no matter how bad, Godel had indeed done his work in an "intuitionistically unobjectionable manner".But Nagel and Newman note that Gödel's proofs are infinitary in nature,[25] not finitary as Hilbert requested (see Hilbert's second problem) while Gödel insisted that they are intuitionistically satisfactory. These are not incompatible truths, as long as the law of the excluded middle over the completed infinite isn't invoked anywhere in the proofs.
It would take until 1944 for Reuben Goodstein to finally discover a more suitable example for Godel's incompleteness theorem, i.e. Goodstein's theorem.
Hilbert grudgingly accepted Godel's work:
By treading carefully with both Brouwer and Hilbert, Godel judiciously managed to avoid damning criticism of his own work and becoming a victim in the twentieth century's foundational war in mathematics.https://en.m.wikipedia.org/wiki/Hilbert%27s_program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early 1920s,[1] was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies.
Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics.