A paper by Hintikka about the modified Ramsey theorem

[Richard Kimberly Heck]

On 7/24/21 11:48 AM, Giovanni Lagnese wrote:
> Is this paper by Hintikka correct? 
> https://iphras.ru/uplfile/logic/log19/LI19_Hintikka.pdf 
> <https://iphras.ru/uplfile/logic/log19/LI19_Hintikka.pdf>
>
> "Yet f is obviously computable by a mechanical process, for we can 
> simply by going through for a given m all the possible relations R 
> (different 'colorings') for n = m, m + 1, m + 2, . . . Hence MFRT 
> appears to be highly interesting even if it is not a Godel sentence. 
> It is a counterexample to Church's Thesis: in a pretheoretical sense 
> computable, but not general recursive nor therefore a Turing machine 
> computable function." (!!)

It seems to me that this is not correct, though plowing through the 
details looks unpleasant. As usual, Hintikka's main interest here is in 
'independence friendly' quantifiers and, as usual, his presentation is 
not exactly clear. But, as I read him, part of the point is supposed to 
be that the quantifier ∀R is genuinely second-order, which is what is 
supposed to prevent MFRT from being a first-order sentence (which is why 
f is supposed not to be recursive). But if so, then we can't just 'go 
through' all the R.

Indeed, the remarks quoted make it seem that R depends upon n. But it's 
a large part of Hintikka's point that it doesn't. That's where the 
'independence friendly' stuff comes in.

Riki


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Richard Kimberly (Riki) Heck
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Brown University

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