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Strong Logics of First and Second Order

Published online by Cambridge University Press:  15 January 2014

Peter Koellner
Peter Koellner*
Affiliation:
Department of Philosophy, Harvard University, 25 Quincy Street, Cambridge, MA 02138, USAE-mail:, [email protected]
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Abstract

In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 16 , Issue 1 , March 2010 , pp. 1 - 36
Copyright
Copyright © Association for Symbolic Logic 2010

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References

REFERENCES

[1] Bagaria, Joan, Castells, Neus, and Larson, Paul, An Ω-logic primer, Set theory (Bagaria, Joan and Todorcevic, Stevo, editors), Trends in Mathematics, Birkhäuser, Basel, 2006, pp. 1–28.Google Scholar
[2] Boolos, George, To be is to be a value of a variable (or to be some values of some variables), The Journal of Philosophy, vol. 81 (1984), no. 8, pp. 430–449.Google Scholar
[3] Boolos, George, Nominalist platonism, Philosophical Review, vol. 94 (1985), no. 3, pp. 327–344.Google Scholar
[4] Buss, Samuel, Bounded arithmetic, Bibliopolis, Napoli, 1986.Google Scholar
[5] Buss, Samuel, First-order proof theory of arithmetic, In Handbook of Proof Theory [6], 1998, pp. 79–147.Google Scholar
[6] Buss, Samuel (editor), Handbook of proof theory, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1998.Google Scholar
[7] Carnap, Rudolf, Logische Syntax der Sprache, Springer, Wien, 1934, translated by Smeaton, A. as The logical syntax of language, Kegan Paul, London, 1937.Google Scholar
[8] Feferman, Solomon, Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae, vol. 49 (1960), pp. 35–92.Google Scholar
[9] Feng, Qi, Magidor, Menacham, and Woodin, W. Hugh, Universally Baire sets of reals, Set theory of the continuum (Judah, Haim, Just, Winfried, and Woodin, W. Hugh, editors), Mathematical Sciences Research Institute, vol. 26, Springer-Verlag, Berlin, 1992, pp. 203–242.Google Scholar
[10] Henkin, Leon, Completeness in the theory of types, The Journal of Symbolic Logic, vol. 15 (1950), no. 2, pp. 81–91.Google Scholar
[11] Joosten, Joost Johannes, Interpretability formalized, Ph.D. thesis, Universiteit Utrecht, 2004.Google Scholar
[12] Kanamori, Akihiro, The higher infinite: Large cardinals in set theory from their beginnings, second ed., Springer Monographs in Mathematics, Springer, Berlin, 2003.Google Scholar
[13] Koellner, Peter, On the question of absolute undecidability, Philosophia Mathematica, vol. 14 (2006), no. 2, pp. 153–188.Google Scholar
[14] Koellner, Peter, Truth in mathematics: The question of pluralism, New waves in philosophy of mathematics (Bueno, Otávio and Linnebo, Øystein, editors), Palgrave Macmillan, 2009, forthcoming.Google Scholar
[15] Koellner, Peter and Woodin, W. Hugh, Incompatible Ω-complete theories, The Journal of Symbolic Logic, (2009), forthcoming.Google Scholar
[16] Koellner, Peter and Woodin, W. Hugh, Large cardinals from determinacy, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), Springer, 2009, forthcoming.Google Scholar
[17] Larson, Paul B., The stationary tower: Notes on a course by W. Hugh Woodin, University Lecture Series, vol. 32, American Mathematical Society, 2004.Google Scholar
[18] Laver, Richard, Certain very large cardinals are not created in small forcing extensions, Annals of Pure and Applied Logic, vol. 149 (2007), no. 1-3, pp. 1–6.Google Scholar
[19] Lindström, Per, A note on weak second order logic with variables for elementarily definable relations, The proceedings of the Bertrand Russel memorial conference (Uldum, 1971), 1973, pp. 221–233.Google Scholar
[20] Lindström, Per, Aspects of incompleteness, second ed., Lecture Notes in Logic, vol. 10, Association for Symbolic Logic, 2003.Google Scholar
[21] Linnebo, Øystein, Plural quantification exposed, Noûs, vol. 37 (2003), no. 1, pp. 71–92.Google Scholar
[22] Martin, Donald A., Multiple universes of sets and indeterminate truth values, Topoi, vol. 20 (2001), no. 1, pp. 5–16.Google Scholar
[23] Martin, Donald A. and Solovay, Robert M., A basis theorem for sets of reals, Annals of Mathematics, vol. 89 (1969), pp. 138–159.Google Scholar
[24] Montagna, Franco and Mancini, Antonella, A minimal predicative set theory, Notre Dame Journal of Formal Logic, vol. 35 (1994), no. 2, pp. 186–203.Google Scholar
[25] Mycielski, Jan, Pudlák, Pavel, and Stern, Alan S., A lattice of chapters of mathematics (interpretations between theorems), Memoirs of the American Mathematical Society, vol. 84 (1990), no. 426, iv+70.Google Scholar
[26] Nelson, Edward, Predicative arithmetic, Princeton Mathematical Notes, no. 32, Princeton University Press, 1986.Google Scholar
[27] Parsons, Charles, The uniqueness of the natural numbers, Iyyun, vol. 39 (1990), pp. 13–44.Google Scholar
[28] Parsons, Charles, Mathematical thought and its objects, Cambridge University Press, 2008.Google Scholar
[29] Quine, Willard V., Ontological relativity, The Journal of Philosophy, vol. 65 (1968), no. 7, pp. 185–212.Google Scholar
[30] Quine, Willard V., Philosophy of logic, second ed., Harvard University Press, Cambridge, MA, 1986.Google Scholar
[31] Shoenfield, Joseph R., The problem of predicativity, Essays on the foundations of mathematics (Bar-Hillel, Yehoshua, Poznanski, E. I. J., Rabin, Michael O., and Robinson, Abraham, editors), Magnes Press, Hebrew University, Jerusalem, 1961, pp. 132–139.Google Scholar
[32] Szmielew, Wanda and Tarski, Alfred, Mutual interpretability of some essentially undecidable theories, Proceedings of the International Congress of Mathematicians, vol. 1, Cambridge, 1950, p. 734.Google Scholar
[33] Väänänen, Jouko, Second-order logic and foundations of mathematics, this Bulletin, vol. 7 (2001), no. 4, pp. 504–520.Google Scholar
[34] Visser, Albert, An overview of interpretability logic, Advances in modal logic, CSLI Lecture Notes, vol. 87, CSLI Publishing, Stanford, CA, 1998, vol. 1, pp. 307–359.Google Scholar
[35] Visser, Albert, Pairs, sets, and sequences in first-order theories, Archive for Mathematical Logic, vol. 47 (2008), pp. 299–326.Google Scholar
[36] Woodin, W. Hugh, The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and its Applications, vol. 1, de Gruyter, Berlin, 1999.Google Scholar
[37] Woodin, W. Hugh, The continuum hypothesis, Logic Collouqium 2000 (Cori, Rene, Razborov, Alexander, Todorĉević, Steve, and Wood, Carol, editors), Lecture Notes in Logic, vol. 19, Association for Symbolic Logic, 2005, pp. 143–197.Google Scholar
[38] Woodin, W. Hugh, The continuum hypothesis, the generic-multiverse of sets, and the Ω-conjecture, preprint, 2006.Google Scholar