[FOM] Rv: Is Linnebo’s system MS inconsistent?
laureano luna
laureanoluna at yahoo.es
Thu Oct 5 09:56:00 EDT 2017
Dear Editor,
on September 21, I sent you the message below about Linnebo's modal set theory, as a contribution to FOM. For all I know, it hasn't been published.
As I am really interested in receiving opinions about the topic, I dare ask you why it hasn't been published.
Cordially yours,
Laureano Luna.
El Jueves 21 de septiembre de 2017 17:54, laureano luna <laureanoluna at yahoo.es> escribió:
Oysten Linneboin 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 modalset theory MS. MS’s languageincludes FOL, set membership, the usual modal operators, and plural variablesand quantifiers. The underlying logic is FOL extended by the usual inferencerules for plural quantifiers and Necessitation. MS’s axioms are those of themodal system S4.2 (K, T, 4, .2) plus some axioms and axiomatic schematareferring to pluralities and sets. Among them: (P-Comp) Exx (y) [y is one of the xx iff phi(y)], where ‘xx’ isnot free in phi, and this axiom relating pluralities to sets: (C) nec (xx) posEy nec (u) [u in y iff u is one of the xx], In Linnebo’sintended interpretation, the possible worlds are the stages or levels of thecumulative hierarchy. So (C) is meant to say something like: “for anyplurality 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 frombeta, u is a member of y iff u is one of the xx”. The caveat inthis interpretation is that Linnebo doesn’t intend the modal operators to beread as quantifying over the possible worlds but as primitive symbols. Now, itseems to me that {P-Comp, (C)} is inconsistent in MS because P-Comp allows toobtain the plurality rr of all non self-membered sets (just set phi(y) = ynotin y) and then, applying (C), we obtain at some level the set R of all nonself-membered sets, which leads to Russell’s paradox. I sketch a proof: Instantiatingthe existential quantifier into rr in an instance of P-Comp with phi(y) = ynotin y, we get: 1. (x) [x is oneof the rr iff x notin x] Instantiatingthe universal quantifier in (C) into rr: 2. nec pos Eynec (u) [u in y iff u is one of the rr]. That is, by 1.: 3. nec pos Eynec (u) [u in y iff u notin u]. Applying axiom T: 4. pos Ey nec (u) [u in y iff u notin u] Instantiatingthe existential quantifier in 4. into R: 5. pos nec (u) [u in R iff u notin u].But, incontradiction with 5., “nec (u) [u in R iff u notin u]” is not possiblebecause, via T, it would yield “(u) [u in R iff u notin u]” and the Russellparadox. // I guess Linnebointended (C) to behave this way: given any leveland any plurality of sets AVAILABLE AT THAT LEVEL, the corresponding set can beformed at the next level; as rr isavailable at no level, (C) would not permit to form R. Surely, Linnebo intendedP-Comp and (C) to act at each level but not across them all. But, if I amright, that is not the way they behave in MS, essentially because P-Comp is notmodal and axiom T permits to eliminate the first necessity operator from (C). However, I haven’tbeen able to find any mention of the possible inconsistency of Linnebo’s MS inthe literature. So, I guess I’ve slipped up somewhere. But where? Laureano Luna.
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