Meditation on « Russell’s response to Godel’s theorem »
Alasdair Urquhart's article « Russell and Gödel » is very helpful for a comprehensive understanding of Russell's views of Gödel's theorem, where Chapitre 3 is « Russell’s response to Godel’s theorem ».
Russell was very careful in his responses to Gödel's theorem, as presented in this article ([1] p.10-14), and Russell never made a simple right or wrong judgement about Gödel's theorem, but was honest in expressing his confusion and doubts : in a 1945 article, Russell described Gödel's theorem as paradox; in a 1950 article, Russell described Gödel's theorem as puzzle; in a 1963 letter to Leon Henkin, Russell expresses his puzzles about Gödel's theorem; and in a 1965 commentary, Russell said that Gödel presented a new difficulty, … Russell's attitude was consistent.
In 1965 Russell was invited by Schilpp to write the following commentary on Gödel's work ([1] p.10-14) :
- Not long after the appearance of Principia Mathematica, G¨odel propounded a new difficulty. He proved that, in any systematic logical language, there are propositions which can be stated, but cannot be either proved or disproved. This has been taken by many (not, I think, by G¨odel) as a fatal objection to mathematical logic in the form which I and others had given to it. I have never been able to adopt this view. It is maintained by those who hold this view that no systematic logical theory can be true of everything. Oddly enough, they never apply this opinion to elementary everyday arithmetic. Until they do so, I consider that they may be ignored. I had always supposed that there are propositions in mathematical logic which can be stated, but neither proved nor disproved. Two of these had a fairly prominent place in Principia Mathematica – namely, the axiom of choice and the axiom of infinity. To many mathematical logicians, however, the destructive influence of G¨odel’s work appears much greater than it does to me and has been thought to requirea great restriction in the scope of mathematical logic. ... I adhere to the view that one should make the best set of axioms that one can think of and believe in it unless and until actual contradictions appear.
It can be seen that Russell was not questioning Gödel's incomplete conclusion that there are undecidable propositions in formal systems, but rather Gödel's proof of the incomplete conclusion, for Russell said, « I had always supposed that there are propositions in mathematical logic which can be stated, but neither proved nor disproved. », yet Gödel took the liar's paradox as a premise and argued that this paradox was a true but unprovable proposition in the formal system. Perhaps this is why Russell says:
- I realized, of course, that Gödel’s work is of fundamental importance, but I was puzzled by it. It made me glad that I was no longer working at mathematical logic. If a given set of axioms leads to a contradiction, it is clear that at least one of the axioms must be false. Does this apply to schoolboys’ arithmetic, and, if so, can we believe anything that we were taught in youth? Are we to think that 2+2 is not 4, but 4.001? Obviously, this is not what is intended.
Russell's response to Gödel's theorem is inspiring : distinguish between the 'conclusion' and the 'proof' of Gödel's theorem in order to clarify where our puzzle lies.
Maybe we can also ask ourselves :
- What do we think of incompleteness conclusion ?
- Are we puzzled by Godel’s proof of the incompleteness conclusion ?
Reference:
[1] Alasdair Urquhart, Russell and Gödel.
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