Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

164 concept calculus: much better thancan use an elimination of quantifiers for small ordinals. This provides an interpretationof B + VSE + SSE + UI + SUI within EFA = exponential function arithmetic, orequivalently I 0 (exp). In fact, we obtain a consistency proof of B + VSE + SSE +UI + SUI within EFA.To obtain a fragment that is mutually interpretable with P = Peano Arithmetic, weneed only use >, =. Drop the last axiom of B, and useDiverse Exactness (in >, =). y>ϕ→ (∃z)(z > ex ϕ ∧¬, =) in which y,z are not free.If we strengthen MBT by replacing L(>, =) with L(>, ≫, =) in SUI, then weobtain an inconsistent system. If we strengthen B + SDE + UI by replacing L(>, =)with L(>, ≫, =) in UI, then we also obtain an inconsistent system.There are substantial connections between these systems and mereology. For backgroundon mereology, see Simons (1987), Varzi (2007), and Hovda (2009).Specifically, we can read x>yas “y is a proper part of x” and read x ≫ y as “y isan infinitesimal part of x.”AcknowledgmentsThis research was partially supported by Templeton Grant #15400.ReferencesEnayat, A., Schmerl, J., and Visser, A. 2008. ω-models of finite set theory, www.phil.uu.nl/preprints/lgps/, #266, Logic Group Preprint Series, Department of Philosophy, Utrecht University.Feferman, S. 1960. Arithemetization of metamathematics in a general setting. In Fundamenta Mathematicae49, 35–92.Friedman, H. 2007. Interpretations, according to Tarski. Lecture 1 of the 19th Annual TarskiLectures, Department of Mathematics, University of California, Berkeley. www.math.ohiostate.edu/%7Efriedman/manuscripts.html,#60.Friedman, H. In press. Forty years on his shoulders. In Horizons of Truth: Gödel Centenary. CambridgeUniversity Press.Friedman, H. In preparation. Concept Calculus, www.math.ohio-state.edu/%7Efriedman/manuscripts.html.Friedman, H., and Visser, A. In preparation. Interpretations between Theories.Hovda, P. 2009. What is classical mereology? Journal of Philosophical Logic 38: 55–82.Kaye, R., and Wong, T. L. 2007. On interpretations of arithmetic and set theory. The Notre DameJournal of Formal Logic 48 (4): 497–510.Simons, P. 1987. Parts: A Study in Ontology. Oxford: Oxford University Press.Varzi, A. C. 2007. Spatial reasoning and ontology: Parts, wholes, and locations. In Handbook ofSpatial Logics, M. Aiello et al. (eds.) pp. 945–1038. Berlin: Springer-Verlag.