[FOM] Rv: Is Linnebo’s system MS inconsistent?

[Christopher Menzel]

> Oystein Linnebo in his The Potential Hierarchy of Sets, 2013,
> (https://www.cambridge.org/core/journals/review-of-symbolic-logic/article/the-potential-hierarchy-of-sets/334647199574BFA9ADBCEFE379CDDB14)
> presents a modal set theory MS.
>  
> MS’s language includes FOL, set membership, the usual modal operators, and plural variables and quantifiers. The underlying logic is FOL extended by the usual inference rules for plural quantifiers and Necessitation. MS’s axioms are those of the modal system S4.2 (K, T, 4, .2) plus some axioms and axiomatic schemata referring to pluralities and sets. Among them:
>  
> (P-Comp)  Exx (y) [y is one of the xx iff phi(y)],
>  
> where ‘xx’ is not free in phi, and this axiom relating pluralities to sets:
>  
> (C) nec (xx) pos Ey nec (u) [u in y iff u is one of the xx],
>  
> In Linnebo’s intended interpretation, the possible worlds are the stages or levels of the cumulative hierarchy. So (C) is meant to say something like:
>  
> “for any plurality xx at any level alpha, there is a set y at some level beta, accessible from alpha, such that, for any set u at any level accessible from beta, u is a member of y iff u is one of the xx”.
>  
> The caveat in this interpretation is that Linnebo doesn’t intend the modal operators to be read as quantifying over the possible worlds but as primitive symbols. Now, it seems to me that {P-Comp, (C)} is inconsistent in MS

Just to note: Linnebo proves that MS is interpretable in ZF, so if you are right, ZF is inconsistent.

> because P-Comp allows to obtain the plurality rr of all non self-membered sets

Well, yes, the plurality of all the non-self-membered sets of some given level.

> (just set phi(y) = y notin y) and then, applying (C), we obtain at some level the set R of all non self-membered sets,

All non-self-membered sets *of some earlier level*. You appear to be leaving out this crucial qualification in your proof.

> which leads to Russell’s paradox.
> 
> I sketch a proof:
>  
> Instantiating the existential quantifier into rr in an instance of P-Comp with phi(y) = y notin y, we get:
>  
> 1. (x) [x is one of the rr iff x notin x]
>  
> Instantiating the universal quantifier in (C) into rr: 
>  
> 2. nec pos Ey nec (u) [u in y iff u is one of the rr].
>  
> That is, by 1.:
>  
> 3. nec pos Ey nec (u) [u in y iff u notin u].

This inference does not follow. Just from a purely formal standpoint, 1 only expresses a material equivalence so you can't substitute one side for the other in arbitrary modal contexts. From the standpoint of Linnebo's theory, 1 just says that the rr are all the non-self-membered sets of some given level (intuitively, the non-self-membered sets in the "actual" world, the world over which the non-modalized quantifiers range). Since pluralities are inextensible (they neither "shrink" nor "grow" in accessible worlds), none of the "new" non-self-membered sets that come to be in accessible possible worlds are among the rr. Hence, it is false that

	nec (x) [x is one of the rr iff x notin x]

But that is the principle you need to infer 3 from 2. (Linnebo discusses exactly this case in his paper "Pluralities and Sets", Journal of Philosophy 107, 144-164.)

Chris Menzel

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