Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-24T17:37:43.472Z Has data issue: false hasContentIssue false

Believing the axioms. I

Published online by Cambridge University Press:  12 March 2014

Penelope Maddy*
Affiliation:
Department of Philosophy, University of California at Irvine, Irvine, California 92717

Extract

§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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bar-Hillel, Y., editor [1970] Mathematical logic and foundations of set theory, North-Holland, Amsterdam, 1970.Google Scholar
Barwise, J., editor [1977] Handbook of mathematical logic, North-Holland, Amsterdam, 1977.Google Scholar
Benacerraf, P. and Putnam, H., editors [1983] Philosophy of mathematics, 2nd ed., Cambridge University Press, Cambridge, 1983.Google Scholar
Boolos, G. [1971] The iterative conception of set, in Benacerraf and Putnam [1983], pp. 486502.CrossRefGoogle Scholar
Brouwer, L. E. J. [1912] Intuitionism and formalism, in Benacerraf and Putnam [1983], pp. 7789 CrossRefGoogle Scholar
Cantor, G. [1883] Über unendliche, lineare Punktmannigfaltigkeiten. V, Mathematische Annalen, vol. 21 (1883), pp. 545591.CrossRefGoogle Scholar
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.Google Scholar
Cantor, G. [1899] Letter to Dedekind, in van Heijenoort [1967], pp. 113117.Google Scholar
Cohen, P. [1966] Set theory and the continuum hypothesis, Benjamin, Reading, Massachusetts, 1966.Google Scholar
Cohen, P. [1971] Comments on the foundations of set theory, in Scott [1971], pp. 915.Google Scholar
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.Google Scholar
Devlin, K. J. [1977] The axiom of constructibility, Springer-Verlag, Berlin, 1977.Google Scholar
Dodd, T. and Jensen, R. B. [1977] The core model, Annals of Mathematical Logic, vol. 20 (1981), pp. 4375.CrossRefGoogle Scholar
Drake, F. [1974] Set theory, North-Holland, Amsterdam, 1974.Google Scholar
Ellentuck, E. [1975] Gödel's square axioms for the continuum, Mathematische Annalen, vol. 216 (1975), pp. 2933.CrossRefGoogle Scholar
Fraenkel, A. A., Bar-Hillel, Y., and Levy, A. [1973] Foundations of set theory, 2nd ed., North-Holland, Amsterdam, 1973.Google Scholar
Freiling, C. [1986] Axioms of symmetry: throwing darts at the real number line, this Journal, vol. 51 (1986), pp. 190200.Google Scholar
Friedman, H. M. [1971] Higher set theory and mathematical practice, Annals of Mathematical Logic, vol. 2 (1971), pp. 325357.CrossRefGoogle Scholar
Friedman, J. I. [1971] The generalized continuum hypothesis is equivalent to the generalized maximization principle, this Journal, vol. 36 (1971), pp. 3954.Google Scholar
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.CrossRefGoogle ScholarPubMed
Gödel, K. [1944] Russell's mathematical logic, in Benacerraf and Putnam [1983], pp. 447469.CrossRefGoogle Scholar
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.Google Scholar
Gödel, K. [1947/64] What is Cantor's continuum problem?, in Benacerraf and Putnam [1983], pp. 470485.CrossRefGoogle Scholar
Hallet, H. [1984] Cantorian set theory and limitation of size, Oxford University Press, Oxford, 1984.Google Scholar
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.Google Scholar
Jensen, R. B. [1970] Definable sets of minimal degree, in Bar-Hillel [1970], pp. 122128.CrossRefGoogle Scholar
Jensen, R. B. and Solovay, R. M. [1970] Some applications of almost disjoint sets, in BarHillel [1970], pp. 84104.CrossRefGoogle Scholar
Jourdain, P. [1904] On the transfinite cardinal numbers of well-ordered aggregates, Philosophical Magazine, vol. 7 (1904), pp. 6175.Google Scholar
Jourdain, P. [1905] On transfinite numbers of the exponential form, Philosophical Magazine, vol. 9 (1905), pp. 4256.Google Scholar
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.CrossRefGoogle Scholar
Kreisel, G. [1980] Kurt Godel, Biographical memoirs of fellows of the Royal Society, vol. 26 (1980), pp. 176.Google Scholar
Kunen, K. [1970] Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.Google Scholar
Levy, A. [1960] Axiom schemata of strong infinity in axiomatic set theory, Pacific Journal of Mathematics, vol. 10 (1960), pp. 223238.CrossRefGoogle Scholar
Levy, A. and Solovay, R. M. [1967] “Measurable cardinals and the continuum hypothesis,” Israel Journal of Mathematics, vol. 5 (1967), pp. 234248.CrossRefGoogle Scholar
Maddy, P. [BAII] Believing the axioms. II, this Journal (to appear).Google Scholar
Martin, D. A. [PSCN] Projective sets and cardinal numbers, circulated photocopy.Google Scholar
Martin, D. A. [SAC] Sets versus classes, circulated photocopy.Google Scholar
Martin, D. A. [1970] Measurable cardinals and analytic games, Fundamenta Mathematicae, vol. 66 (1970), pp. 287291.Google Scholar
Martin, D. A. [1975] Borel determinacy, Annals of Mathematics, ser. 2, vol. 102 (1975), pp. 363371.Google Scholar
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.Google Scholar
Martin, D. A. and Solovay, R. M. [1970] Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143178.CrossRefGoogle Scholar
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.Google Scholar
Mirimanoff, D. [1917] “Remarques sur la théorie des ensembles et les antinomies Cantoriennes. I, L'Enseignement Mathématique, vol. 19 (1917), pp. 208217.Google Scholar
Moore, G. H. [1982] Zermelo's axioms of choice, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
Moore, G. H. [198?] Introductory note to 1947 and 1964, The collected works of Kurt Gödel. Vol. II, Oxford University Press, Oxford (to appear).Google Scholar
Moschovakis, Y. N. [1980] Descriptive set theory, North-Holland, Amsterdam, 1980.Google Scholar
Parsons, C. [1974] Sets and classes, Noûs, vol. 8 (1974), pp. 112.CrossRefGoogle Scholar
Parsons, C. [1977] What is the iterative conception of set?, in Benacerraf and Putnam [1983], pp. 503529.CrossRefGoogle Scholar
Reinhardt, W. N. [1974] Remarks on reflection principles, large cardinals and elementary embeddings, in Jech [1974], pp. 189205.CrossRefGoogle Scholar
Reinhardt, W. N. [1974] Set existence principles of Shoenfield, Ackermann, and Powell, Fundamenta Mathematicae, vol. 84 (1974), pp. 534.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
Scott, D. S. [1977] Foreword to Bell, J. L., Boolean-valued Models and independence proofs in set theory, Clarendon Press, Oxford, 1977.Google Scholar
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.CrossRefGoogle Scholar
Shoenfield, J. R. [1977] Axioms of set theory, in Barwise [1977], pp. 321344.Google Scholar
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.Google Scholar
Silver, J. [1971] The consistency of the GCH with the existence of a measurable cardinal, in Scott [1971], pp. 391396.Google Scholar
Solovay, R. M. [1967] A nonconstructible Δ⅓ set of integers, Transactions of the American Mathematical Society, vol. 127 (1967), pp. 5075.Google Scholar
Solovay, R. M. [1969] The cardinality of Σ½ sets, Foundations of Mathematics (Bulloff, J. J. et al., editors), Symposium papers commemorating the sixtieth birthday of Kurt Godel, Springer-Verlag, Berlin, 1969, pp. 5873.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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. A. et al., editors), North-Holland, Amsterdam, 1985, pp. 1186.CrossRefGoogle Scholar
Tarski, A. [1938] Über unerreichbare Kardinalzahlen, Fundamenta Mathematicae, vol. 30 (1938), pp. 6889.CrossRefGoogle Scholar
Ulam, S. [1930] Zur Masstheorie in der allgemeinen Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 140150.CrossRefGoogle Scholar
van Heuenoort, J., editor [1967] From Frege to Gödel. A source-book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, Massachusetts, 1967.Google Scholar
von Neumann, J. [1923] On the introduction of transfinite numbers, in van Heijenoort [1967], pp. 346354.Google Scholar
von Neumann, J. [1925] An axiomatization of set theory, in von Heijenoort [1967], pp. 393413.Google Scholar
Wang, H. [1974] The concept of set, in Benacerraf and Putnam [1983], pp. 530570.Google Scholar
Wang, H. [1981] Some facts about Kurt Gödel, this Journal, vol. 46 (1981), pp. 653659.Google Scholar
Zermelo, E. [1904] Proof that every set can be well-ordered, in van Heijenoort [1967], pp. 139141.Google Scholar
Zermelo, E. [1908] A new proof of the possibility of a well-ordering, in van Heijenoort [1967], pp. 183198.Google Scholar
Zermelo, E. [1908] Investigations in the foundations of set theory. I, in van Heijenoort [1967], pp. 199215.Google Scholar
Zermelo, E.[1930] Über Grenzzahlen und Mengenbereiche, Fundamenta Mathematicae, vol. 14 (1930), pp. 2947.Google Scholar