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Background

In Henkin's article Are Logic and Mathematics Identical Henkin writes,

But I must mention, also, a second aim of the Russell-Whitehead Principia which also fared ill in the subsequent development of mathematical logic. Russell and Whitehead were very much concerned with the question of consistency. While they hoped to have a complete system, one containing proofs for all correct statements, they were also concerned that their system should not contain proofs of incorrect results. In particular, in a consistent system such as they sought, it would not be possible to prove both a sentence and its negation. ...

...Russell himself produced an even simpler paradox in the intuitive theory of sets, based upon the set of all those sets which are not elements of themselves.

This background sketch will make clear why it was that Russell and Whitehead were concerned that no paradox should be demonstrable in their own system. And yet they themselves never attempted a proof that their system was consistent! The only evidence they adduced was that a large number of theorems had been obtained within their system without encountering paradox, and that all attempts to reproduce within the system of Principia Mathematica the Burali-Forti paradox, and such other paradoxes as were shown, had failed. (my bold)

In his reply to Henkin Russell said,

You note that we [Russell and Whitehead] were indifferent to attempts to prove that our axioms could not lead to contradictions. In this Gödel showed that we had been mistaken. But I thought that it must be impossible to prove that any given set of axioms does not lead to a contradiction, and, for that reason, I had payed little attention to Hilbert's work. (my bold)

My Question

Can someone tell me any reason(s) for Russell's believing that "it must be impossible to prove that any given set of axioms does not lead to a contradiction"? Did Russell have some distinct conception of consistency and/or soundness in mind like Frege (see The Frege-Hilbert Controversy for more details)?

  • "Not" involves all. In order to prove contradiction does not exist, one needs to go through the entire body of propositions implied by the axioms. See [exist] (philosophy.stackexchange.com/a/18042/5116). – George Chen Feb 10 '18 at 5:16
  • @GeorgeChen: But didn't Gödel show just that? Namely that if the system under consideration is complete and if it satisfies certain properties then there exists a contradiction. Not only that he also gave an example of such a sentence. Am I missing something? – user 170039 Feb 10 '18 at 5:27
  • It is possible to prove inconsistency because all you need to do is to find a contradiction.It is very difficult to prove consistency because consistency involves "contradiction does not exist," which entails a thorough examination of the entire system. – George Chen Feb 10 '18 at 5:37
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    @GeorgeChen: I understand now. Sorry for not understanding it earlier. – user 170039 Feb 10 '18 at 5:39
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In a certain sense, Russell was right...

Hilbert's "dream" was to prove the consistency of arithmetic and analisys within a system weaker than arithmetic itself (the so called finitistic fragment).

Russell shared with Frege the idea that the system of logic was all encompassing: logic is a science that applies to any topic whatever.

If so, how can we find a system outside of it to be used to argue about the consistency of the system of logic itself ?

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    I am aware that Russell held the so called "universalist" conception of logic as long as 1901. However, I am not sure that at the time of writing the letter to Henkin (i.e.,1963) he held the "universalist" conception of logic. Can you give some references to confirm this? – user 170039 Feb 9 '18 at 4:01

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