Best known for his Incompleteness Theorem, Kurt Gödel (1906-1978) is considered one of the most important mathematicians and logicians of the 20th century. By showing that the establishment of a set of axioms encompassing all of mathematics would never succeed, he revolutionized the world of mathematics, logic and philosophy.
Sir Roger Penrose is known worldwide for his work in mathematics and mathematical physics, in particular general relativity and cosmology. Currently Emeritus Rouse Ball Professor of Mathematics at the University of Oxford and Emeritus Fellow of Wadham College, he has been awarded numerous honors for his scientific contributions. His many books include, The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics; Shadows of the Mind: A Search for the Missing Science of Consciousness; The Nature of Space and Time (with Stephen Hawking) and most recently The Road to Reality: A Complete Guide to the Laws of the Universe.
Q: Kurt Gödel’s 1931 Incompleteness Theorem disrupted German mathematician David Hilbert’s agenda for the 20th-century mathematical research and rocked the very foundations of mathematics in general. What was this pivotal insight that turned the foundations of mathematics on its head?
A: Hilbert was hoping to be able to formalize mathematics in a completely clear way so that the issue of whether a result was to be considered to be “proved” could be made completely unambiguous. This desire had been prompted by the appearance of “paradoxes,” such as Bertrand Russell’s "set of all sets that are not members of themselves." If some area of mathematics could be formulated in such a way that the proof procedures are completely unambiguous and clear cut (in a sense that I shall come to below), one should be able to make sure that contradictions, such as Russell’s paradox, didn’t occur (i.e. were not part of the accepted proof procedures), then that area of mathematics would be put on a sound basis.Gödel’s Incompleteness Theorems showed that Hilbert’s program was unachievable—at least for sufficiently broad areas of mathematics (such as the ordinary number theory of the integers). Gödel showed that for such an area of mathematics, for any proposed formal system F (a “formalization,” in the above Hilbertian sense) which intended to describe it would always fail to be able to establish some result (that could be explicitly constructed in terms of the rules of F)—let us call this result G(F)—even though G(F) could be seen to be necessarily true, by methods outside the rules of F, provided that the rules of F could themselves be trusted as yielding only true results. In the form of Gödel’s result that is most commonly referred to is his “second” Incompleteness Theorem, in which G(F) effectively asserts that F is consistent, so the argument tells us that the consistency of F cannot be proved within the rules of F itself.
In my view, this has the appearance of somewhat downgrading the significance of Gödel’s theorem because it gives it perhaps a somewhat circular appearance, “consistency” being a somewhat internal matter of concern.
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