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One hundred and two problems in mathematical logic

Published online by Cambridge University Press:  12 March 2014

Harvey Friedman
Harvey Friedman*
Affiliation:
State University of New York at Buffalo, Amherst, New York 14226
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Extract

This expository paper contains a list of 102 problems which, at the time of publication, are unsolved. These problems are distributed in four subdivisions of logic: model theory, proof theory and intuitionism, recursion theory, and set theory. They are written in the form of statements which we believe to be at least as likely as their negations. These should not be viewed as conjectures since, in some cases, we had no opinion as to which way the problem would go.

In each case where we believe a problem did not originate with us, we made an effort to pinpoint a source. Often this was a difficult matter, based on subjective judgments. When we were unable to pinpoint a source, we left a question mark. No inference should be drawn concerning the beliefs of the originator of a problem as to which way it will go (lest the originator be us).

The choice of these problems was based on five criteria. Firstly, we are only including problems which call for the truth value of a particular mathematical statement. A second criterion is the extent to which the concepts involved in the statements are concepts that are well known, well denned, and well understood, as well as having been extensively considered in the literature. A third criterion is the extent to which these problems have natural, simple and attractive formulations. A fourth criterion is the extent to which there is evidence that a real difficulty exists in finding a solution. Lastly and unavoidably, the extent to which these problems are connected with the author's research interests in mathematical logic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 40 , Issue 2 , June 1975 , pp. 113 - 129
Copyright
Copyright © Association for Symbolic Logic 1975

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References

REFERENCES

[Ba]Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226–252.Google Scholar
[Bau]Baumgartner, J., All ω1 dense sets of reals can be isomorphic, Fundamenta Mathematicae, vol. 79 (1973), pp. 101–106.CrossRefGoogle Scholar
[Co]Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, New York, 1966.Google Scholar
[Eh]Ehrenfeucht, A., Elementary theories with models without automorphisms, The theory of models, North-Holland, Amsterdam, 1965, pp. 70–76.Google Scholar
[Fe 1]Feferman, S., Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae, vol. 56 (1965), pp. 325–345.CrossRefGoogle Scholar
[Fe 2]Feferman, S., Formal theories for transfinite iterations of generalized inductive definitions and some subsystems of analysis, Intuitionism and proof theory, North-Holland, Amsterdam, 1970, pp. 303–326.Google Scholar
[Fr 1]Friedman, H., Subsystems of set theory and analysis, Chapter 2, Thesis, M.I.T., Cambridge, Mass., 1967.Google Scholar
[Fr 2]Friedman, H., Uncountable prepositional logic, Stanford University, 1968 (unpublished).Google Scholar
[Fr 3]Friedman, H., Iterated inductive definitions and Σ21-AC, Intuitionism and proof theory, North-Holland, Amsterdam, 1970, pp. 435–442.Google Scholar
[Fr 4]Friedman, H., Why first order logic, Madison, 11 1970 (unpublished).Google Scholar
[Fr 5]Friedman, H., Determinateness in the low projective hierarchy, Fundamenta Mathematicae, vol. 72 (1971), pp. 79–84.CrossRefGoogle Scholar
[Fr 6]Friedman, H., Higher set theory and mathematical practice, Annals of Mathematical Logic, vol. 2 (1971), pp. 326–357.CrossRefGoogle Scholar
[Fr 7]Friedman, H., On the Δ operation on semantic systems, Stanford University, 05 1972 (unpublished).Google Scholar
[Fr 8]Friedman, H., The consistency of classical set theory relative to a set theory with intuitionistic logic, this Journal, vol. 38 (1973), pp. 315–319.Google Scholar
[Fr 9]Friedman, H., Countable models of set theories, section 5, Springer Lecture Notes in Mathematics, vol. 337 (1973), pp. 539–573.CrossRefGoogle Scholar
[Fr 10]Friedman, H., Borel sets and hyperdegrees, this Journal, vol. 38 (1973), pp. 405–409.Google Scholar
[Fr 11]Friedman, H., Beth's theorem in cardinality logics, Israel Journal of Mathematics, vol. 14 (1973), pp. 205–212.CrossRefGoogle Scholar
[Fr 12]Friedman, H., On closed sets of ordinals, Proceedings of the American Mathematical Society, vol. 43 (1974), pp. 190–192.CrossRefGoogle Scholar
[Fr 13]Friedman, H., Minimality in the Δ21-degree, Fundamenta Mathematicae, (1974).Google Scholar
[Fr 14]Friedman, H., Large models of countable height, Transactions of the American Mathematical Society, (to appear).Google Scholar
[Fr 15]Friedman, H., Recursiveness in Π11 paths through , Proceedings of the American Mathematical Society, (to appear).Google Scholar
[FJ]Friedman, H. and Jensen, R., Note on admissible ordinals, Springer Lecture Notes in Mathematics, vol. 72 (1968), pp. 77–79.CrossRefGoogle Scholar
[Go]Gödel, K., On formally undecidable propositions of Principia Mathematica and related systems. I, The undecidable (Davis, Martin, Editor), Raven Press, New York, 1965, pp. 5–38.Google Scholar
[Goo 1]Goodman, N., Intuitionistic arithmetic as a theory of constructions, Thesis, Stanford University, 1968.Google Scholar
[Goo 2]Goodman, N., A realizability proof of my theorem, 02 1973 (unpublished).Google Scholar
[Gr]Gregory, J., Beth definability in infinitary languages, this Journal, vol. 39 (1974), pp. 22–26.Google Scholar
[Ha]Hanf, W., Model-theoretic methods in the study of elementary logic, The theory of models, North-Holland, Amsterdam, 1965, pp. 132–145.Google Scholar
[Har]Harrington, L., A basis theorem for Borel sets, Recursive function theory newsletter, no. 3, 10 1972, p. 10.Google Scholar
[HK]Harrington, L. and Kechris, A., Π21 singletons and 0#, Fundamenta Mathematicae (to appear).Google Scholar
[Harr]Harrison, J., Recursive pseudo-well-orderings, Transactions of the American Mathematical Society, vol. 131 (1968), pp. 539–543.CrossRefGoogle Scholar
[Hu]Hugill, D. F., Initial segments of Turing degrees, Proceedings of the London Mathematical Society, vol. 19 (1969), pp. 1–16.CrossRefGoogle Scholar
[Hut]Hutchinson, J., Models of set theories and their applications, Thesis, Stanford University, 1974.Google Scholar
[Je]Jensen, R., On the consistency of a slight (?) modification of Quine's New Foundations, Synthese, vol. 19 (1968), pp. 250–263.CrossRefGoogle Scholar
[JS]Jensen, R. and Solovay, R., Some applications of almost disjoint sets, Mathematical logic and foundations of set theory (Bar-Hillel, Y., Editor) North-Holland, Amsterdam, 1966, pp. 87–91.Google Scholar
[Jo]Jockusch, C., An application of Σ40 determinacy to the degrees of unsolvability, this Journal, vol. 38 (1973), pp. 293–294.Google Scholar
[Jo S 1]Jockusch, C. Jr., and Soare, R., Minimal covers and arithmetical sets, Proceedings of the American Mathematical Society, vol. 25 (1970), pp. 856–859.CrossRefGoogle Scholar
[Jo S 2]Jockusch, C., A minimal pair of Π10 classes, this Journal, vol. 36 (1971), pp. 66–78.Google Scholar
[Jon S]Jones, N. and Selman, A., Turing machines and the spectra of first-order formulas with equality, Proceedings of the 4th ACM Symposium on Theory of Computing, 1972, pp. 157–167.Google Scholar
[Kl 1]Kleene, S. C., Introduction to metamathematics, Van Nostrand, Princeton, N.J., 1950, p. 82.Google Scholar
[Kl 2]Kleene, S. C., Quantification of number-theoretic functions, Compositie, vol. 14 (1959), pp. 23–40.Google Scholar
[Kr]Kreisel, G., The axiom of choice and the class of hyperarithmetic functions, Dutch Academy A, vol. 65 (1962), pp. 307–319.Google Scholar
[Ku]Kunen, K., Inaccessibility properties of cardinals, Thesis, Stanford University, 1968, pp. 51–55.Google Scholar
[La 1]Lachlan, A., The elementary theory of recursively enumerable sets, Duke Mathematical Journal, vol. 35 (1968), pp. 123–146.CrossRefGoogle Scholar
[La 2]Lachlan, A., Distributive initial segments of the degrees of unsolvability, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 457–472.CrossRefGoogle Scholar
[LS]Lerman, M. and Sacks, G., Some minimal pairs of α-r.e. degrees, Annals of Mathematical Logic, vol. 4 (1972), pp. 415–442.CrossRefGoogle Scholar
[Li]Lindstrom, P., On extensions of elementary logic, Theoria, vol. 35 (1969), pp. 1–11.CrossRefGoogle Scholar
[Lo]Lopez-Escobar, E. G. K., An interpolation theorem for denumerably long formulas, Fundamenta Mathematicae, vol. 57 (1965), pp. 253–272.CrossRefGoogle Scholar
[Ma]MacIntyre, J., Minimal α-recursion theoretic degrees, this Journal, vol. 38 (1973), pp. 18–28.Google Scholar
[Mag 1]Magidor, M., The first measurable cardinal can be strongly compact, 11 1972 (unpublished).Google Scholar
[Mag 2]Magidor, M., Changing cofinality of cardinals (to appear).Google Scholar
[Mal]Malitz, J., Infinitary analogs of theorems from first order model theory, this Journal, vol. 36 (1971), pp. 216–228.Google Scholar
[Mar 1]Martin, D. A., The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689.CrossRefGoogle Scholar
[Mar 2]Martin, D. A., Measurable cardinals and analytic games, Fundamenta Mathematicae, vol. 66 (1970), pp. 287–291.Google Scholar
[Mar 3]Martin, D. A., A form of determinateness equivalent to the existence of 0#, Rockefeller University, 1971 (unpublished).Google Scholar
[Mar 4]Martin, D. A., Projective sets and cardinal numbers: some questions related to the continuum problem (to appear).Google Scholar
[Mar 5]Martin, D. A., Analysis and Σ40 games, Rockefeller University, 03 1974 (unpublished).Google Scholar
[Mat]Matijasevic, Yu. V., Diophantine representation of recursively enumerable predicates, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J., Editor), North-Holland, Amsterdam, 1971, pp. 171–177.CrossRefGoogle Scholar
[Mc]McKay, C., The decidability of certain intermediate prepositional logics, this Journal, vol. 33 (1968), pp. 258–264.Google Scholar
[Mi]Mitchell, W., Aronszajn trees and the independence of the transfer property, Annals of Mathematical Logic, vol. 4 (1972), pp. 21–46.CrossRefGoogle Scholar
[Mo]Morley, M., The number of countable models, this Journal, vol. 35 (1970), pp. 14–18.Google Scholar
[Mos]Mostowski, A., A generalization of the incompleteness theorem, Fundamenta Mathematicae, vol. 49 (1961), pp. 205–232.CrossRefGoogle Scholar
[MS]Martin, D. and Solovay, R., Internal Cohen extensions, Annals of Mathematical Logic, vol. 2 (1970), pp. 143–178.CrossRefGoogle Scholar
[My]Myhill, J., Some properties of intuitionistic Zermelo-Fraenkel set theory, Lecture Notes in Mathematics, Springer-Verlag, vol. 337 (1973), pp. 206–231.Google Scholar
[Pa]Parikh, R., Some results on the lengths of proofs, Transactions of the American Mathematical Society, vol. 177 (1973), pp. 29–36.CrossRefGoogle Scholar
[Par]Paris, J., ZF ⊦ Σ40 determinateness, this Journal, vol. 37 (1972), pp. 661–667.Google Scholar
[Pr 1]Prikry, K., Changing measurable into accessible cardinals, Dissertationes Mathematicae. Rozprawy Matematyczne, vol. 68 (1970), pp. 5–52.Google Scholar
[Pr 2]Prikry, K., Unpublished manuscript, 06 1973.Google Scholar
[Ro]Rosser, J. B., Extensions of some theorems of Gödel and Church, this Journal, vol. 1 (1936), pp. 87–91.Google Scholar
[Sa 1]Sacks, G., Degrees of unsolvability, Annals of Mathematical Studies, no. 55 (1963).Google Scholar
[Sa 2]Sacks, G., F-recursiveness, Logic Colloquium '69, North-Holland, Amsterdam, 1971, pp. 289–303.CrossRefGoogle Scholar
[Sa 3]Sacks, G., Forcing with perfect closed sets, Axiomatic set theory, Proceedings of Symposia in Pure Mathematics, Vol. 13, Part I, American Mathematical Society, Providence, R.I., 1971, pp. 331–343.CrossRefGoogle Scholar
[Sa 4]Sacks, G., Countable admissible ordinals and hyperdegrees (to appear).Google Scholar
[SS]Sacks, G. and Simpson, S., The α-infinite injury method, Annals of Mathematical Logic, vol. 4 (1972), pp. 343–367.CrossRefGoogle Scholar
[Sh]Shelah, S., Generalized quantifiers and compact logic (to appear).Google Scholar
[She]Shepherdson, J., The rule of induction in the free variable arithmetic based on + and ·, Proceedings of a Symposium at Clermont Ferrand, 1961, pp. 25–31.Google Scholar
[Sho]Shore, R., Minimal α-degrees, Annals of Mathematical Logic, vol. 4 (1972), pp. 393–414.CrossRefGoogle Scholar
[Si 1]Silver, J., Some applications of model theory in set theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 101–105.CrossRefGoogle Scholar
[Si 2]Silver, J., Measurable cardinals and Δ31-well-orderings, Annals of Mathematics, vol. 94 (1971), pp. 414–446.CrossRefGoogle Scholar
[Sim]Simmons, H., The solution of a decision problem for several classes of rings, Pacific Journal of Mathematics, vol. 34 (1970), pp. 547–557.CrossRefGoogle Scholar
[Simp]Simpson, S., Minimal covers and hyperdegrees (to appear).Google Scholar
[So 1]Soare, R., Automorphisms of the lattice of recursively enumerable sets, Part I: Maximal sets, Annals of Mathematics (to appear).Google Scholar
[So 2]Soare, R., Automorphisms of the lattice of recursively enumerable sets, Part II: High degrees (to appear).Google Scholar
[Sol]Solovay, R., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1–56.CrossRefGoogle Scholar
[ST]Solovay, R. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1971), pp. 201–245.CrossRefGoogle Scholar
[Sz]Szmielew, W., Elementary properties of abelian groups, Fundamenta Mathematicae, vol. 41 (1955), pp. 203–271.CrossRefGoogle Scholar
[Ta]Tanaka, H., A property of arithmetic sets, Proceedings of the American Mathematical Society, vol. 31 (1972), pp. 521–524.CrossRefGoogle Scholar
[Tar]Tarski, A., A decision method for elementary algebra and geometry, revised 2nd ed., University of California, Berkeley, 1951.CrossRefGoogle Scholar
[Ya]Yates, C. E. N.Initial segments of the degrees of unsolvability, Part I: a survey, Mathematical logic and foundations of set theory (Bar-Hillel, Y., Editor), North-Holland, Amsterdam, 1966, pp. 63–83.Google Scholar
[Zu]Zucker, J.Iterated inductive definitions, trees and ordinals, Lecture Notes in Mathematics, Springer-Verlag, vol. 344 (1973), pp. 392–453.Google Scholar