Skip to main content
×
×
Home

REFERENCE IN ARITHMETIC

  • LAVINIA PICOLLO (a1)
Abstract

Self-reference has played a prominent role in the development of metamathematics in the past century, starting with Gödel’s first incompleteness theorem. Given the nature of this and other results in the area, the informal understanding of self-reference in arithmetic has sufficed so far. Recently, however, it has been argued that for other related issues in metamathematics and philosophical logic a precise notion of self-reference and, more generally, reference is actually required. These notions have been so far elusive and are surrounded by an aura of scepticism that has kept most philosophers away. In this paper I suggest we shouldn’t give up all hope. First, I introduce the reader to these issues. Second, I discuss the conditions a good notion of reference in arithmetic must satisfy. Accordingly, I then introduce adequate notions of reference for the language of first-order arithmetic, which I show to be fruitful for addressing the aforementioned issues in metamathematics.

Copyright
Corresponding author
*MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY LMU MUNICH MUNICH, GERMANY E-mail: Lavinia.Picollo@lrz.uni-muenchen.de
References
Hide All
[1] Boolos, G., Burgess, J. P., & Jeffrey, R. C. (2007). Computability and Logic (fifth edition). Cambridge: Cambridge University Press.
[2] Carnap, R. (1937). Logical Syntax of Language. London: Routledge.
[3] Cook, R. T. (2006). There are non-circular paradoxes (but Yablo’s isn’t one of them!). The Monist, 89, 118149.
[4] Cresswell, M. J. (1975). Hyperintensional logic. Studia Logica, 34, 2538.
[5] Gödel, K. (1931). Über formal unentscheidebarre Sätze der Principia Mathematica und verwandler System I. Monathshefte für Mathematik und Physik, 38, 173198.
[6] Goodman, N. (1961). About. Mind, 70, 124.
[7] Hájek, P. & Pudlák, P. (1993). Metamathematics of First-Order Arithmetic. Berlin: Springer.
[8] Halbach, V. (2009). Reducing compositional to disquotational truth. Review of Symbolic Logic, 2, 786798.
[9] Halbach, V. (2016). The root of evil. A self-referential play in one act. In van Eijck, J., Iemhoff, R., and Joosten, J. J., editors. Liber Amicorum Alberti. A Tribute to Albert Visser. London: College Publications, pp. 155163.
[10] Halbach, V. & Visser, A. (2014a). Self-reference in arithmetic I. Review of Symbolic Logic, 7, 671691.
[11] Halbach, V. & Visser, A. (2014b). Self-reference in arithmetic II. Review of Symbolic Logic, 7, 692712.
[12] Heck, R Jr.. (2007). Self-reference and the languages of arithmetic. Philosophia Mathematica, III, 129.
[13] Henkin, L. (1952). A problem concerning provability. The Journal of Symbolic Logic, 17, 160.
[14] Henkin, L. (1954). Review of G. Kreisel: On a problem of Henkin’s. The Journal of Symbolic Logic, 19, 219220.
[15] Horwich, P. (1998). Truth (second edition). New York: Blackwell.
[16] Kaye, R. (1991). Models of Peano Arithmetic. Oxford: Clarendon Press.
[17] Kleene, S. (1938). On notation for ordinal numbers. The Journal of Symbolic Logic, 3, 150155.
[18] Kreisel, G. (1953). On a problem of Henkin’s. Indagationes Mathematicae, 15, 405406.
[19] Leitgeb, H. (2002). What is a self-referential sentence? Critical remarks on the alleged (non)-circularity of Yablo’s Paradox. Logique et Analyse, 177–178, 314.
[20] Leitgeb, H. (2005). What truth depends on. Journal of Philosphical Logic, 34, 155192.
[21] Löb, M. H. (1955). Solution of a problem of Leon Henkin. The Journal of Symbolic Logic, 20, 115118.
[22] Milne, P. (2007). On Gödel sentences and what they say. Philosophia Mathematica, III(15), 193226.
[23] Montague, R. (1962). Theories incomparable with respect to relative interpretability. The Journal of Symbolic Logic, 27, 195211.
[24] Priest, G. (1997). Yablo’s paradox. Analysis, 57, 236242.
[25] Putnam, H. (1958). Formalization of the concept of about. Philosophy of Science, 25, 125130.
[26] Ryle, G. (1933). Imaginary objects. Proceedings of the Aristotelian Society, 12(Suppl.), 1843.
[27] Smoryński, C. (1981). Fifty years of self-reference in arithmetic. Notre Dame Journal of Formal Logic, 22(4), 357374.
[28] Smoryński, C. (1991). The development of self-reference: Löb’s theorem. In Drucker, T., editor. Perspectives on the History of Mathematical Logic. Boston: Birkhäuser, pp. 110133.
[29] Sorensen, R. A. (1998). Yablo’s paradox and kindred infinite liars. Mind, 107, 137155.
[30] Tarski, A. (1935). Der Wahrheitsbegriff in den formalisierten Sprachen. Studia Philosophica Commentarii Societatis Philosophicae Polonorum, 1, 261405, reprinted as The Concept of Truth in Formalized Languages in Logic, Semantics and Metamathematics, pp. 152–278 .
[31] Tarski, A. (1944). The semantic conception of truth: And the foundations of semantics. Philosophy and Phenomenological Research, 4, 341376.
[31] Urbaniak, R. (2009). Leitgeb, “About,” Yablo. Logique et Analyse, 207, 239254.
[33] Visser, A. (1989). Semantics and the liar paradox. In Gabbay, D. M. and Günthner, F., editors. Handbook of Philosophical Logic, Vol. 4. Dordrecht: Reidel, pp. 617706.
[34] Yablo, S. (1985). Truth and reflexion. Journal of Philosphical Logic, 14, 297349.
[35] Yablo, S. (1993). Paradox without self-reference. Analysis, 53, 251252.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 39
Total number of PDF views: 69 *
Loading metrics...

Abstract views

Total abstract views: 330 *
Loading metrics...

* Views captured on Cambridge Core between 14th January 2018 - 20th August 2018. This data will be updated every 24 hours.