Models without indiscernibles
Journal of Symbolic Logic 43 (3):572-600 (1978)
Abstract
For T any completion of Peano Arithmetic and for n any positive integer, there is a model of T of size $\beth_n$ with no (n + 1)-length sequence of indiscernibles. Hence the Hanf number for omitting types over T, H(T), is at least $\beth_\omega$ . (Now, using an upper bound previously obtained by Julia Knight H (true arithmetic) is exactly $\beth_\omega$ ). If T ≠ true arithmetic, then $H(T) = \beth_{\omega1}$ . If $\delta \not\rightarrow (\rho)^{ , then any completion of Peano Arithmetic has a model of size δ with no set of indiscernibles of size ρ. There are similar results for theories strongly resembling Peano Arithmetic, e.g., ZF + V = LMy notes
Similar books and articles
The argument from almost indiscernibles.Gonzalo Rodriguez-Pereyra - 2017 - Philosophical Studies 174 (12):3005-3020.
Haim Gaifman. Models and types of Peano's arithmetic. Annals of mathematical logic, vol. 9, pp. 223–306. - Julia F. Knight. Omitting types in set theory and arithmetic. The journal of symbolic logic, vol. 41 , pp. 25–32. - Julia F. Knight. Hanf numbers for omitting types over particular theories. The journal of symbolic logic, vol. 41 , pp. 583–588. - Fred G. Abramson and Leo A. Harrington. Models without indiscernibles. The journal of symbolic logic, vol. 41 , vol. 43 , pp. 572–600. [REVIEW]J. P. Ressayre - 1983 - Journal of Symbolic Logic 48 (2):484-485.
James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - Jam. [REVIEW]Roman Kossak - 2009 - Bulletin of Symbolic Logic 15 (2):222-227.
James H. Schmerl. Peano models with many generic classes. Pacific Journal of Mathematics, vol. 43 (1973), pp. 523–536. - James H. Schmerl. Correction to: “Peano models with many generic classes”. Pacific Journal of Mathematics, vol. 92 (1981), no. 1, pp. 195–198. - James H. Schmerl. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80. Recursively saturated, rather classless models of Peano arithmetic. Logic Year 1979–80 (Proceedings, Seminars, and Conferences in Mathematical Logic, University of Connecticut, Storrs, Connecticut, 1979/80). edited by M. Lerman, J. H. Schmerl, and R. I. Soare, Lecture Notes in Mathematics, vol. 859. Springer, Berlin, pp. 268–282. - James H. Schmerl. Recursively saturatedmodels generated by indiscernibles. Notre Dane Journal of Formal Logic, vol. 26 (1985), no. 1, pp. 99–105. - James H. Schmerl. Large resplendent models generated by indiscernibles. The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1382–1388. - Jam. [REVIEW]Roman Kossak - 2009 - Bulletin of Symbolic Logic 15 (2):222-227.
Cofinal Indiscernibles and some Applications to New Foundations.Friederike Körner - 1994 - Mathematical Logic Quarterly 40 (3):347-356.
Models and Types of Peano's Arithmetic.Haim Gaifman, Julia F. Knight, Fred G. Abramson & Leo A. Harrington - 1983 - Journal of Symbolic Logic 48 (2):484-485.
Saturated models and models generated by indiscernibles.B. Mariou - 2001 - Journal of Symbolic Logic 66 (1):325-348.
The Identity of Indiscernibles as a Logical Truth.Gerald Keaney - 2007 - Crossroads 1 (2):28-36 Free Online.
Max Black. The identity of indiscernibles. Mind, n.s. vol. 61 , pp. 153–164. Reprinted with minor changes in: Problems of analysis, Philosophical essays, by Max Black, Cornell University Press, Ithaca 1954, pp. 80–92, 292–293. - Gustav Bergmann. The identity of indiscernibles and the formalist definition of “identity.”Mind, n.s. vol. 62 , pp. 75–79. - N. L. Wilson. The identity of indiscernibles and the symmetrical universe. Mind, n.s. vol. 62 , pp. 506–511. - A. J. Ayer. The identity of indiscernibles. Actes du XIème Congrès International de Philosophie, Volume III, Métaphysique et ontologie, North-Holland Publishing Company, Amsterdam1953, and Éditions E. Nauwelaerts, Louvain 1953, pp. 124–129. Reprinted in Philosophical essays by A. J. Ayer, St. Martin's Press, New York 1954, and Macmillan & Co., London 1954, pp. 26–35. - D. J. O'Connor. The identity of indiscernibles. Analysis , vol. 14 no. 5 , pp. 103–110. - Nicholas Rescher. The identity of indiscernibles: A reinterpretation. The. [REVIEW]Charles A. Baylis - 1956 - Journal of Symbolic Logic 21 (1):85-86.
Leibniz's Argument for the Identity of Indiscernibles in his Correspondence with Clarke.Gonzalo Rodriguez-Pereyra - 1999 - Australasian Journal of Philosophy 77 (4):429 – 438.
Una lógica dialógica de orden superior para demostrar la ley de la identidad de los indiscernibles de Leibniz.Mohammad Shafiei - 2017 - Revista de Humanidades de Valparaíso 9:73-88.
Russell and the Identity of Indiscernibles.Michael C. Bradley - 1986 - History of Philosophy Quarterly 3 (3):325 - 333.
Some remarks on initial segments in models of peano arithmetic.Henryk Kotlarski - 1984 - Journal of Symbolic Logic 49 (3):955-960.
Cognitive penetration and the gallery of indiscernibles.Bence Nanay - 2015 - Frontiers in Psychology 5.
Analytics
Added to PP
2009-01-28
Downloads
39 (#301,263)
6 months
2 (#297,972)
2009-01-28
Downloads
39 (#301,263)
6 months
2 (#297,972)
Historical graph of downloads
Citations of this work
On $n$ -Dependence.Artem Chernikov, Daniel Palacin & Kota Takeuchi - 2019 - Notre Dame Journal of Formal Logic 60 (2):195-214.
Characterization of NIP theories by ordered graph-indiscernibles.Lynn Scow - 2012 - Annals of Pure and Applied Logic 163 (11):1624-1641.
Reducts of random hypergraphs.Simon Thomas - 1996 - Annals of Pure and Applied Logic 80 (2):165-193.
The Ramsey theory of the universal homogeneous triangle-free graph.Natasha Dobrinen - 2020 - Journal of Mathematical Logic 20 (2):2050012.
Karp complexity and classes with the independence property.M. C. Laskowski & S. Shelah - 2003 - Annals of Pure and Applied Logic 120 (1-3):263-283.