Wei_Dai comments on Open Problems Related to Solomonoff Induction - Less Wrong

27 Post author: Wei_Dai 06 June 2012 12:26AM

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Comment author: Wei_Dai 30 December 2012 01:57:03AM 0 points [-]

Is there a reason why it'd be different in second-order logic?

Second-order set theory is beyond my expertise too, but I'm going by this paper, which on page 8 says:

We have managed to give a formal semantics for the second-order language of set theory without expanding our ontology to include classes that are not sets. The obvious alternative is to invoke the existence of proper classes. One can then tinker with the definition of a standard model so as to allow for a model with the (proper) class of all sets as its domain and the class of all ordered-pairs x, y (for x an element of y) as its interpretation function.12 The existence of such a model is in fact all it takes to render the truth of a sentence of the language of set theory an immediate consequence of its validity.

So I was taking the "obvious alternative" of proper class of all sets to be the standard model for second order set theory. I don't quite understand the paper's own proposed model, but I don't think it's a set either.

Comment author: Eliezer_Yudkowsky 30 December 2012 02:42:11AM 0 points [-]

I'm not sure I believe in proper classes and in particular, I'm not sure there's a proper class of all sets that could be the model of a second-order theory such that you could not describe any set larger than the model, and as for pinning down that model using axioms I'm pretty sure you shouldn't be able to do that. There are analogues of the Lowenheim-Skolem theorem for sufficiently large infinities in second-order logic, I seem to recall reading.

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