Tuesday, November 25, 2008

Chwistek on the axiom of reducibility and completeness

So, I finally found some time to read Chwistek's 1912 paper. As it turns out, after a few pages about Lukasiewicz's criticism (where Chwistek defends Aristotle, but, to my mind, quite superficially), he moves on to a description of R&W type theory with the axiom of reducibility.

The most interesting part is where he actually argues that it is inconsistent: it turns out that the argument that was in Ueber die Antinomen der Prinzipien der Mathematik, translated in McCall's volume was already (pretty much) shaped in 1912. I haven't compared those arguments in details, but prima facie, it's the same argument.

The last two sections of 1912 have quite intriguing titles:

iv. Contradictions in Russell's system. The possibility of a system free of contradictions.
v. The pseudo-problem of a logic without the principle of contradiction

In section iv he basically :

(i) claims type theory without the axiom of reducibility is consistent (needless to say, no proof is given;P )
(ii) is very careful about completeness: he says it's possible there are true sentences that can't be proven in this type theory:

The system allows to prove many true sentences. These, however, won't suffice for determining the fertility of the system. To answer this question, tremendous further work is required [...] for now, I'll limit myself to the claim that the task isn't fully delusional, that is, that even if many true claims are left out, we sill have a system that does possess certain scientific value. [p. 326]

(iii) insists that after leaving out the axiom of reducibility the system is consistent, because it was because of the axiom that the contradiction arose. (but he is quite consious that this isn't a proof).

In section vi he criticizes the idea of a logic which admits contradiction (the idea occured in Łukasiewicz's book). I'll write about this when I have more time...

Interestingly, there is no mention of Lesniewski's work whatsoever.

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