What is absolute undecidability? (English) Zbl 1328.03004
Summary: It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention – one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH.
MSC:
| 03A05 | Philosophical and critical aspects of logic and foundations |
| 03B25 | Decidability of theories and sets of sentences |
| 03E50 | Continuum hypothesis and Martin’s axiom |
References:
| [1] | Arrigoni, Tatiana. [2011] “V=L and Intuitive Plausibility in Set Theory. A Case Study.” Review of Symbolic Logic. Vol. 17. 337-359. · Zbl 1258.03070 |
| [2] | Azcel, Peter. [1988] Non‐Well‐Founded Sets. SLI Lecture Notes. Vol. 14. Stanford, CA: Stanford University, Center for the Study of Language and Information. · Zbl 0668.04001 |
| [3] | Balaguer, Mark. [2001] Platonism and Anti‐Platonism in Mathematics. New York: Oxford University Press. |
| [4] | Bell, John and GeoffreyHellman. [2006] “Pluralism and the Foundations of Mathematics.” in Kellert (ed.) and Waters (ed.). |
| [5] | Benacerraf, Paul and HilaryPutnam. [1983] Philosophy of Mathematics: Selected Readings (Second Edition). Cambridge: Cambridge University Press. · Zbl 0548.03002 |
| [6] | Boolos, George. [1999] “Must We Believe in Set Theory?” in Sher and Tieszen. |
| [7] | Browder, Felix (ed.) [1976] Mathematical Developments Arising from Hilbert Problems (Proceedings of Symposia in Pure Mathematics. Vol. 28). Providence, Rhode Island: American Mathematical Society. · Zbl 0326.00002 |
| [8] | Buss, Samuel. [1996] “Nelson”s Work on Logic and Foundations and Other Reflections on Foundations of Mathematics.” in Farris. |
| [9] | Devlin, Keith. [1973] Aspects of Constructibility. (Lecture Notes in Mathematics, Vol. 354). Berlin: Springer‐Verlag. · Zbl 0312.02054 |
| [10] | Devlin, Keith. [1977] The Axiom of Constructibility: A Guide for the Mathematician (Lecture Notes in Mathematics, Vol. 617). Berlin: Springer‐Verlag. · Zbl 0369.02043 |
| [11] | Devlin, Keith. [1984] Constructibility. Berlin: Springer‐Verlag. · Zbl 0542.03029 |
| [12] | Dummett, Michael. [1991] The Logical Basis of Metaphysics. Cambridge: Harvard University Press. |
| [13] | Esenin‐Volpin, Alexander. [1970] “The Ultra‐Intuitionistic Criticism and the Anti‐Traditional Program for Foundations of Mathematics”. in Kino, Myhill, and Vesley. |
| [14] | Farris, William (Ed.). Diffusion, Quantum Theory, and Radically Elementary Mathematics. Princeton University Press, 2006, pp. 183-208 · Zbl 1104.81003 |
| [15] | Feferman, Solomon. [2000] “Does Mathematics Need New Axioms?” Bulletin of Symbolic Logic. Vol. 6. 401-446. · Zbl 0977.03002 |
| [16] | Field, Hartry. [1980] Science Without Numbers. Princeton: Princeton University Press. · Zbl 0463.68065 |
| [17] | Field, Hartry. [1989] Realism, Mathematics, and Modality. Oxford: Blackwell . · Zbl 1098.00500 |
| [18] | Field, Hartry. [1998] “Which Undecidable Mathematical Sentences have Determinate Truth‐Values?” in Laurence and Macdonald. |
| [19] | Foreman, Mathew. [1998] “Generic Large Cardinals: New Axioms for Mathematics?”Proceedings of the International Congress of Mathematicians. Vol. II. Berlin. · Zbl 0917.03022 |
| [20] | Forster, Thomas. [2008] “The Iterative Conception of Set.’Review of Symbolic Logic. Vol. 1. 97-l10. · Zbl 1204.03008 |
| [21] | Forster, Thomas. [1973] Foundations of Set Theory (Studies in Logic and the Foundations of Mathematics, Volume 67). New York: Elsevier Science Publishers. · Zbl 0248.02071 |
| [22] | Friedman, H. [2002] Philosophical Problems in Logic (Lecture Notes). Princeton University. |
| [23] | Godel, Kurt. [1951] “Some Basic Theorems on the Foundations of Mathematics and their Philosophical Implications.” in Rodriguez‐Consuegra. |
| [24] | Godel, Kurt. [1964] “What is Cantor”s Continuum Problem?” (Revised Version) in Benacerraf and Putnam. First appeared in [1947] American Mathematical Monthly. Vol. 54. 515-525. · Zbl 0038.03003 |
| [25] | Jensen, Ronald. [1995] “Inner Models and Large Cardinals.”Bulletin of Symbolic Logic. Vol. 1. 393-407. · Zbl 0843.03029 |
| [26] | Kellert, Stephen Helen Longino, and C.Kenneth Waters (eds.) [2006] Scientific Pluralism: Minnesota Studies in the Philosophy of Science, XIX. Minneapolis, MN: University of Minnesota Press. |
| [27] | Kilmister, Clive W. [1980] “Zeno, Aristotle, Weyl and Shuard: Two‐and‐a‐Half Millenia of Worries over Number.” Mathematical Gazette. Vol. 64. 149-158. |
| [28] | Kino, Akino, JohnMyhill and RichardVesley (eds.) [1970] Intuitionism and Proof Theory. Amsterdam and London: North‐Holland Publishing. · Zbl 0195.01201 |
| [29] | Koellner, Peter. [2006] “On the Question of Absolute Undecidability.” (Revised Version) in Parsons, Feferman, and Simpson. First appeared as [2006] Philosophia Mathematica. Vol. 14. 153-188. · Zbl 1113.03011 |
| [30] | Kracht, Marcus (ed.), MaartendeRijke (ed.), HeinrichWansing (ed.), and MichaelZakharyaschev (ed.) (eds.) [1998] Advances in Modal Logic. Vol. 1. (Berlin, 1996). Volume 87 of CSLI Lecture Notes. 307-359. Stanford, CA: CSLI. · Zbl 0897.00022 |
| [31] | Kreisel, George. [1967] “Observations on Popular Discussions of the Foundations of Mathematics”. in Scott. |
| [32] | Laurence, Stephen and CynthiaMacdonald. [1998] Contemporary Readings in the Foundations of Metaphysics. Oxford: Basil Blackwell. |
| [33] | Maddy, Penelope. [2011] Defending the Axioms. Oxford: Oxford University Press. · Zbl 1219.00014 |
| [34] | Martin, Donald. [1976] “Hilbert”s First Problem: The Continuum Hypothesis.” in Browder. |
| [35] | Nelson, Edward. [1986] Predicative Arithmetic (Mathematical Notes. No. 32). Princeton, NJ: Princeton University Press. · Zbl 0617.03002 |
| [36] | Parsons, Charles, SolomonFeferman, and Stephen G.Simpson. [2009] Kurt Gödel: Essays for his Centennial (Lecture Notes in Logic. Vol. 33). Association of Symbolic Logic. |
| [37] | Potter, Michael. [2004] Set Theory and Its Philosophy: A Critical Introduction. Oxford: Oxford University Press. · Zbl 1066.03002 |
| [38] | Quine, W. [1937] “New Foundations for Mathematical Logic.” American Mathematical Monthly. Vol. 44. 70-80. · JFM 63.0022.02 |
| [39] | Quine, W. [1969] Set Theory and Its Logic. (Revised Edition) Cambridge: Harvard University Press. · Zbl 0193.30402 |
| [40] | Rodriguez‐Consuegra, Francisco. [1995] Kurt Gödel: Unpublished Philosophical Essays. Berlin: Birkhauser Verlag. · Zbl 0854.01046 |
| [41] | Schiffer, Stephen. [2002] “Moral Realism and Indeterminacy.” in Sosa and Villanueva. |
| [42] | Schroeter, Laura, and FrancoisSchroeter. [2012] “Do Moral Realists Need Convergence?” Unpublished Manupscript. |
| [43] | Scott, Dana (ed.) (ed.) [1967] Axiomatic Set Theory (Proceedings of Symposia in Pure Mathematics, V. XIII, Part I). Providence, RI: American Mathematical Association. |
| [44] | Shapiro, Stewart. [2000] Philosophy of Mathematics: Structure and Ontology. Oxford: Oxford University Press. · Zbl 0990.00005 |
| [45] | Sher, Gila and RichardTieszen. [1999] Between Logic and Intuition: Essays in Honor of Charles Parsons. Cambridge: Cambridge University Press. |
| [46] | Sosa, Ernest and EnriqueVillanueva (eds.) [2002] Realism and Relativism (Philosophical Issues, Vol. 12). Oxford: Blackwell. |
| [47] | Visser, Albert. [1998] “An Overview of Interpretability Logic.” in Kracht, de Rijke, Wansing, and Zakharyaschev. |
| [48] | Waters, Kenneth, HelenLongino, and StephenKellert (eds.) [2006] Scientific Pluralism (Minnesota Studies in Philosophy of Science, Volume 19). Minneapolis: University of Minnesota Press. |
| [49] | Williamson, Timothy. [2004] Vagueness. New York: Routledge. |
| [50] | Woodin, Hugh. [2001a] “The Continuum Hypothesis I.” Notices of the American Mathematical Society. Vol. 48. 567-576. · Zbl 0992.03063 |
| [51] | Woodin, Hugh. [2001b] “The Continuum Hypothesis II.” Notices of the American Mathematical Society. Vol. 48. 681-690. · Zbl 1047.03041 |
| [52] | Wright, Crispin. [1994] Truth and Objectivity. Cambridge: Harvard University Press. |
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