caa a22 4500 388039639 CHVBK 20180307125017.0 cr unu---uuuuu 161130e199803 xx s 000 0 eng 10.2307/2586599 doi S0022481200015449 pii (NATIONALLICENCE)cambridge-10.2307/2586599 Antonelli Gian Aldo Department of Philosophy, Michigan State University, East Lansing, Michigan 48824-1032, USA, E-mail: aantonel@pilot.msu.edu Department of Philosophy, Stanford University, Stanford, CA 94305, USA Extensional quotients for type theory and the consistency problem for NF [Elektronische Daten] [Gian Aldo Antonelli] Quine's "New Foundations” (NF) was first presented in Quine [10] and later on in Quine [11]. Ernst Specker [15, 13], building upon a previous result of Ehrenfeucht and Mostowski [5], showed that NF is consistent if and only if there is a model of the Theory of Negative (and positive) Types (TNT) with full extensionality that admits of a "shifting automorphism,” but the existence of such a model remains an open problem. In his [8], Ronald Jensen gave a partial solution to the problem of the consistency of Quine's NF. Jensen considered a version of NF—referred to as NFU—in which the axiom of extensionality is weakened to allow for Urelemente or "atoms.” He showed, modifying Specker's theorem, that the existence of a model of TNT with atoms admitting of a "shifting automorphism” implies the consistency of NFU, proceeding then to exhibit such a model. This paper presents a reduction of the consistency problem for NF to the existence of a model of TNT with atoms containing certain "large” (unstratified) sets and admitting a shifting automorphism. In particular we show that such a model can be "collapsed” to a model of pure TNT in such a way as to preserve the shifting automorphism. By the above-mentioned result of Specker's, this implies the consistency of NF. Let us take the time to explain the main ideas behind the construction. Suppose we have a certain universe U of sets, built up from certain individuals or "atoms.” In such a universe we have only a weak version of the axiom of extensionality: two objects are the same if and only if they are both sets having the same members. We would like to obtain a universe U ′ that is as close to U as possible, but in which there are no atoms (i.e., the only memberless object is the empty set). One way of doing this is to assign to each atom ξ, a set a (perhaps the empty set), inductively identifying sets that have members that we are already committed to considering "the same.” In doing this we obtain an equivalence relation ≃ over U that interacts nicely with the membership relation (provided we have accounted for multiplicity of members, i.e., we have allowed sets to contain "multiple copies” of the same object). Then we can take U ′ = U/≃, the quotient of U with respect to ≃. It is then possible to define a "membership” relation over U ′ in such a way as to have full extensionality. Relations such as ≃ are referred to as "contractions” by Hinnion and "bisimulations” by Aczel. Copyright © Association for Symbolic Logic 1998 The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 247-261 0022-4812 63:1<247 1998 63 JSL https://doi.org/10.2307/2586599 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.2307/2586599 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Antonelli Gian Aldo Department of Philosophy, Michigan State University, East Lansing, Michigan 48824-1032, USA, E-mail: aantonel@pilot.msu.edu Department of Philosophy, Stanford University, Stanford, CA 94305, USA NATIONALLICENCE 773 0- The Journal of Symbolic Logic Cambridge University Press 63/1(1998-03), 247-261 0022-4812 63:1<247 1998 63 JSL CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge