Skip to main content Accessibility help

Believing the axioms. I

  • Penelope Maddy (a1)


§0. Introduction. Ask a beginning philosophy of mathematics student why we believe the theorems of mathematics and you are likely to hear, “because we have proofs!” The more sophisticated might add that those proofs are based on true axioms, and that our rules of inference preserve truth. The next question, naturally, is why we believe the axioms, and here the response will usually be that they are “obvious”, or “self-evident”, that to deny them is “to contradict oneself” or “to commit a crime against the intellect”. Again, the more sophisticated might prefer to say that the axioms are “laws of logic” or “implicit definitions” or “conceptual truths” or some such thing.

Unfortunately, heartwarming answers along these lines are no longer tenable (if they ever were). On the one hand, assumptions once thought to be self-evident have turned out to be debatable, like the law of the excluded middle, or outright false, like the idea that every property determines a set. Conversely, the axiomatization of set theory has led to the consideration of axiom candidates that no one finds obvious, not even their staunchest supporters. In such cases, we find the methodology has more in common with the natural scientist's hypotheses formation and testing than the caricature of the mathematician writing down a few obvious truths and preceeding to draw logical consequences.

The central problem in the philosophy of natural science is when and why the sorts of facts scientists cite as evidence really are evidence. The same is true in the case of mathematics. Historically, philosophers have given considerable attention to the question of when and why various forms of logical inference are truth-preserving. The companion question of when and why the assumption of various axioms is justified has received less attention, perhaps because versions of the “self-evidence” view live on, and perhaps because of a complacent if-thenism.



Hide All
Bar-Hillel, Y., editor [1970] Mathematical logic and foundations of set theory, North-Holland, Amsterdam, 1970.
Barwise, J., editor [1977] Handbook of mathematical logic, North-Holland, Amsterdam, 1977.
Benacerraf, P. and Putnam, H., editors [1983] Philosophy of mathematics, 2nd ed., Cambridge University Press, Cambridge, 1983.
Boolos, G. [1971] The iterative conception of set, in Benacerraf and Putnam [1983], pp. 486502.
Brouwer, L. E. J. [1912] Intuitionism and formalism, in Benacerraf and Putnam [1983], pp. 7789
Cantor, G. [1883] Über unendliche, lineare Punktmannigfaltigkeiten. V, Mathematische Annalen, vol. 21 (1883), pp. 545591.
Cantor, G. [1895] Contributions to the founding of the theory of transfinite numbers (translated and with an introduction by Jourdain, P.E.B.), Open Court Press, Chicago, Illinois, 1915; reprint, Dover, New York, 1952.
Cantor, G. [1899] Letter to Dedekind, in van Heijenoort [1967], pp. 113117.
Cohen, P. [1966] Set theory and the continuum hypothesis, Benjamin, Reading, Massachusetts, 1966.
Cohen, P. [1971] Comments on the foundations of set theory, in Scott [1971], pp. 915.
Dedekind, R. [1888] Essays on the theory of numbers (translated by Beman, W. W.), Open Court Press, Chicago, Illinois, 1901; reprint, Dover, New York, 1963.
Devlin, K. J. [1977] The axiom of constructibility, Springer-Verlag, Berlin, 1977.
Dodd, T. and Jensen, R. B. [1977] The core model, Annals of Mathematical Logic, vol. 20 (1981), pp. 4375.
Drake, F. [1974] Set theory, North-Holland, Amsterdam, 1974.
Ellentuck, E. [1975] Gödel's square axioms for the continuum, Mathematische Annalen, vol. 216 (1975), pp. 2933.
Fraenkel, A. A., Bar-Hillel, Y., and Levy, A. [1973] Foundations of set theory, 2nd ed., North-Holland, Amsterdam, 1973.
Freiling, C. [1986] Axioms of symmetry: throwing darts at the real number line, this Journal, vol. 51 (1986), pp. 190200.
Friedman, H. M. [1971] Higher set theory and mathematical practice, Annals of Mathematical Logic, vol. 2 (1971), pp. 325357.
Friedman, J. I. [1971] The generalized continuum hypothesis is equivalent to the generalized maximization principle, this Journal, vol. 36 (1971), pp. 3954.
Gödel, K. [1938] The consistency of the axiom of choice and of the generalized continuum hypothesis, Proceedings of the National Academy of Sciences of the United States of America, vol. 24 (1938), pp. 556557.
Gödel, K. [1944] Russell's mathematical logic, in Benacerraf and Putnam [1983], pp. 447469.
Gödel, K. [1946] Remarks before the Princeton bicentennial conference on problems in mathematics, The undecidable (Davis, M., editor), Raven Press, Hewlett, New York, 1965, pp. 8488.
Gödel, K. [1947/64] What is Cantor's continuum problem?, in Benacerraf and Putnam [1983], pp. 470485.
Hallet, H. [1984] Cantorian set theory and limitation of size, Oxford University Press, Oxford, 1984.
Jech, T. J., editor [1974] Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part II, American Mathematical Society, Providence, Rhode Island, 1974.
Jensen, R. B. [1970] Definable sets of minimal degree, in Bar-Hillel [1970], pp. 122128.
Jensen, R. B. and Solovay, R. M. [1970] Some applications of almost disjoint sets, in BarHillel [1970], pp. 84104.
Jourdain, P. [1904] On the transfinite cardinal numbers of well-ordered aggregates, Philosophical Magazine, vol. 7 (1904), pp. 6175.
Jourdain, P. [1905] On transfinite numbers of the exponential form, Philosophical Magazine, vol. 9 (1905), pp. 4256.
Kanamori, K. and Magidor, M. [1978] The evolution of large cardinal axioms in set theory, Higher set theory (Müller, G. H. and Scott, D. S., editors), Lecture Notes in Mathematics, vol. 669, Springer-Verlag, Berlin, 1978, pp. 99275.
Kreisel, G. [1980] Kurt Godel, Biographical memoirs of fellows of the Royal Society, vol. 26 (1980), pp. 176.
Kunen, K. [1970] Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.
Levy, A. [1960] Axiom schemata of strong infinity in axiomatic set theory, Pacific Journal of Mathematics, vol. 10 (1960), pp. 223238.
Levy, A. and Solovay, R. M. [1967] “Measurable cardinals and the continuum hypothesis,” Israel Journal of Mathematics, vol. 5 (1967), pp. 234248.
Maddy, P. [BAII] Believing the axioms. II, this Journal (to appear).
Martin, D. A. [PSCN] Projective sets and cardinal numbers, circulated photocopy.
Martin, D. A. [SAC] Sets versus classes, circulated photocopy.
Martin, D. A. [1970] Measurable cardinals and analytic games, Fundamenta Mathematicae, vol. 66 (1970), pp. 287291.
Martin, D. A. [1975] Borel determinacy, Annals of Mathematics, ser. 2, vol. 102 (1975), pp. 363371.
Martin, D. A. [1976] Hilbert's first problem: the continuum hypothesis, Mathematical developments arising from Hilbert problems (Browder, F. E., editor), Proceedings of Symposia in Pure Mathematics, vol. 28, American Mathematical Society, Providence, Rhode Island, 1976, pp. 8192.
Martin, D. A. and Solovay, R. M. [1970] Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143178.
Mirimanoff, D. [1917] Les antinomies de Russell et de Burali-Forti et le problème fondamental de la théorie des ensembles, L'Enseignement Mathématique, vol. 19 (1917), pp. 3752.
Mirimanoff, D. [1917] “Remarques sur la théorie des ensembles et les antinomies Cantoriennes. I, L'Enseignement Mathématique, vol. 19 (1917), pp. 208217.
Moore, G. H. [1982] Zermelo's axioms of choice, Springer-Verlag, Berlin, 1982.
Moore, G. H. [198?] Introductory note to 1947 and 1964, The collected works of Kurt Gödel. Vol. II, Oxford University Press, Oxford (to appear).
Moschovakis, Y. N. [1980] Descriptive set theory, North-Holland, Amsterdam, 1980.
Parsons, C. [1974] Sets and classes, Noûs, vol. 8 (1974), pp. 112.
Parsons, C. [1977] What is the iterative conception of set?, in Benacerraf and Putnam [1983], pp. 503529.
Reinhardt, W. N. [1974] Remarks on reflection principles, large cardinals and elementary embeddings, in Jech [1974], pp. 189205.
Reinhardt, W. N. [1974] Set existence principles of Shoenfield, Ackermann, and Powell, Fundamenta Mathematicae, vol. 84 (1974), pp. 534.
Rowbottom, R. [1964] Some strong axioms of infinity incompatible with the axiom of constructibility, Ph.D. dissertation, University of Wisconsin, Madison, Wisconsin, 1964; reprinted in Annals of Mathematical Logic, vol. 3 (1971), pp. 1–44.
Russell, B. [1906] On some difficulties in the theory of transfinite numbers and order types, Proceedings of the London Mathematical Society, ser. 2, vol. 4 (1906), pp. 2953.
Scott, D. S. [1961] Measurable cardinals and constructible sets, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 7 (1961), pp. 145149.
Scott, D. S. [1977] Foreword to Bell, J. L., Boolean-valued Models and independence proofs in set theory, Clarendon Press, Oxford, 1977.
Scott, D. S., editor [1971] Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, part I, American Mathematical Society, Providence, Rhode Island, 1971.
Shoenfield, J. R. [1977] Axioms of set theory, in Barwise [1977], pp. 321344.
Silver, J. [1966] Some applications of model theory in set theory, Ph.D. dissertation, University of California, Berkeley, California, 1966; reprinted in Annals of Mathematical Logic, vol. 3 (1971), pp. 45–110.
Silver, J. [1971] The consistency of the GCH with the existence of a measurable cardinal, in Scott [1971], pp. 391396.
Solovay, R. M. [1967] A nonconstructible set of integers, Transactions of the American Mathematical Society, vol. 127 (1967), pp. 5075.
Solovay, R. M. [1969] The cardinality of sets, Foundations of Mathematics (Bulloff, J. al., editors), Symposium papers commemorating the sixtieth birthday of Kurt Godel, Springer-Verlag, Berlin, 1969, pp. 5873.
Solovay, R. M. [1970] A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, ser. 2, vol. 92 (1970), pp. 156.
Solovay, R. M., Reinhardt, W. N., and Kanamori, A. [1978] Strong axioms of infinity and elementary embeddings, Annals of Mathematical Logic, vol. 13 (1978), pp. 73116.
Stanley, L. J. [1985] Borel diagonalization and abstract set theory: recent results of Harvey Friedman, Harvey Friedman's research on the foundations of mathematics (Harrington, L. al., editors), North-Holland, Amsterdam, 1985, pp. 1186.
Tarski, A. [1938] Über unerreichbare Kardinalzahlen, Fundamenta Mathematicae, vol. 30 (1938), pp. 6889.
Ulam, S. [1930] Zur Masstheorie in der allgemeinen Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 140150.
van Heuenoort, J., editor [1967] From Frege to Gödel. A source-book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, Massachusetts, 1967.
von Neumann, J. [1923] On the introduction of transfinite numbers, in van Heijenoort [1967], pp. 346354.
von Neumann, J. [1925] An axiomatization of set theory, in von Heijenoort [1967], pp. 393413.
Wang, H. [1974] The concept of set, in Benacerraf and Putnam [1983], pp. 530570.
Wang, H. [1981] Some facts about Kurt Gödel, this Journal, vol. 46 (1981), pp. 653659.
Zermelo, E. [1904] Proof that every set can be well-ordered, in van Heijenoort [1967], pp. 139141.
Zermelo, E. [1908] A new proof of the possibility of a well-ordering, in van Heijenoort [1967], pp. 183198.
Zermelo, E. [1908] Investigations in the foundations of set theory. I, in van Heijenoort [1967], pp. 199215.
Zermelo, E.[1930] Über Grenzzahlen und Mengenbereiche, Fundamenta Mathematicae, vol. 14 (1930), pp. 2947.

Related content

Powered by UNSILO

Believing the axioms. I

  • Penelope Maddy (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.