Abstract
This is the text of the hitherto unpublished article written at Stanford University and circulated in typescript in 1968 with the title page given below. “Recent” being no longer a true description of its contents, I publish it now under a timeless title by which it has been cited by other writers. In doing so, I have corrected the typographical errors in the original text but not the mathematical ones, to which attention is drawn instead in the notes on subsequent developments which will be found together with a supplementary bibliography at the end. The sign + in the text marks points on which those notes comment.
I record here my gratitude toDana Scott, then Professor of Logic at Stanford, who enabled me to spend eight stimulating months in California and whose brainchild the survey was.
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References
I. Basic references
[B1]R. H. Bing, A translation of the normal Moore space conjecture,Proc. Amer. Math. Soc. 16 (1965), 612–619.MR 31 # 6201
[B2]E. Ellentuck, The universal properties of Dedekind finite cardinals,Ann. of Math. 82 (1965), 225–248.MR 31 # 4729
[B3]W. Hanf, Incompactness in languages with infinitely long expressions,Fund. Math. 53 (1963), pp. 309–324.MR 28 # 3943
[B4]H. J. Keisler andA. Tarski, From accessible to inaccessible cardinals,Fund. Math. 53 (1963), pp. 225–308.MR 29 # 3385
[B5]A.Lévy, The Fraenkel—Mostowski method for independence proofs in set theory,The Theory of Models, North Holland, Amsterdam, (1965), 225–248.
[B6]A Lévy,A hierarchy of formulae of set theory, Memoirs of the Amer. Math. Soc., 1966.
[B7]A. Lévy andR. M. Solovay, Measurable cardinals and the continuum hypothesis,Israel J. of Math. 5 (1967), 234–248.
[B8]M. E. Rudin,Countable paracompactness and Souslin's problem, Canad. J. Math7 (1955), 543–547.MR 17-391
[B9]W. R. Scott,Group Theory, Prentice-Hall, Englewood Cliffs, 1964,MR 29 # 4785
[B10]R. M. Solovay, The measure problem I: A model of set theory in which all sets of reals are Lebesgue measurable. (In preparation)
[B11]R. M. Solovay, On the cardinality of∑ 12 sets of reals,Foundations of Math. (Symp.,1966,Columbus), Springer, New York, 1969, 58–73.MR 43 # 3115
[B12]R. M. Solovay, A non-constructibleΔ 13 set of integers,Trans. Amer. Math. Soc. 127 (1967), 50–75.MR 35 # 2748
[B13]H. Tanaka, Some properties of∑ 11 - andΠ 11 -sets inN n,Proc. Japan Acad. 42 (1966), 304–307.MR 34 # 5683
[B14]R. L. Vaught, The Löwenheim—Skolem theorem,Logic, Methodology and Philos. Sci. Proc. Congr. 1964), North-Holland, Amsterdam, 1965, 81–89.MR 35 # 1461
[B15]W. Reinhardt andR. M. Solovay, Strong axioms of infinity and elementary embeddings. (In preparation)
[B16]R. B. Jensen,Large cardinals. An English translation of lectures given at Oberwolfach in March, 1967. (To appear)
II. Supplementary references
[S1]C. J. Ash, A consequence of the axiom of choice,J. Austral. Math. Soc. 19 (1975), 306–308.MR 54 # 92
[S2]B. Balcar, A theorem on supports in the theory of semisets,Comment. Math. Univ. Carolinae 14 (1973), 1–6.MR 49 # 4772
[S3]J. E. Baumgartner, Ineffability properties of cardinals, I,Infinite and Finite Sets (Proc. Colloq. Keszthely,1973), North-Holland, Amsterdam, 1975, 109–130.MR 50 # 5427
[S4]J. E. Baumgartner, A. Hajnal andA. Mété, Weak saturation properties of ideals,Infinite and Finite Sets (Proc. Colloq. Keszthely,1973) North-Holland, Amsterdam, 1975, 137–158.MR 51 # 5317
[S5]J. E. Baumgartner andK. Příkry, On a theorem of Silver,Discrete Math. 14 (1976), 17–21.MR 52 # 7908
[S6]J. L. Bell, On the relationship between weak compactness inL ω1ω ,L ω1ω1 and restricted second-order languages,Arch. Math. Logik Grundlagenforsch. 15 (1972), 74–78.MR 48 # 1878
[S7]J. L. Bell andD. H. Fremlin, A geometric form of the axiom of choice,Fund. Math. 77 (1972), 167–170.MR 48 # 5865
[S8]M. Benda andJ. Ketonen, Regularity of ultrafilters,Israel J. Math. 17 (1974), 231–240.MR 53 # 132
[S9]M. Boffa, The consistency problem for NF.
[S10]W. Boos, Infinitary compactness without strong inaccessibility,J. Symbolic Logic 41 (1976), 33–38.MR 53 # 12946
[S11]D. Booth, A Boolean view of sequential compactness,Fund. Math. 85 (1974), 99–102.MR 51 # 4168
[S12]L. Bukovský, Characterization of generic extensions of models of set theory,Fund. Math. 83 (1973), 35–46.MR 48 # 10804
[S13]G. V. Choodnovski (Čudnovskiî), Sequentially continuous mappings and real valued measurable cardinals,Infinite and Finite Sets, Proc. Colloq. Keszthely, 1973), North-Holland, Amsterdam, 1975, 275–288.
[S14]G. V. Choodnovski (Čudnovskiî), Combinatorial properties of compact cardinals,Infinite and Finite Sets (Proc. Colloq. Keszthely, 1973), North-Holland, Amsterdam, 1975, 289–306.MR 51 # 7873
[S15]J. H. Conway, Effective implications between the “finite” choice axioms.Cambridge Summer School in Mathematical Logic (Cambridge, England, 1971) Springer, Berlin, 1973, 439–458.MR 50 # 12725
[S16]P. Dehornoy, Solution d'une conjecture de Bukovsky,C. R. Acad. Sci. Paris Sér. A-B 281 (1975), A821-A824.MR 53 # 12947
[S17]K. J. Devlin,Aspects of constructibility, Springer, Berlin 1973.MR 51 # 12527
[S18]K. J. Devlin, Some weak versions of large cardinal axioms.Ann. Math. Logic 5 (1973), 291–325.MR 51 # 161
[S19]K. J. Devlin, A note on a problem os Erdős and Hajnal,Discrete Math. 11 (1975), 9–22.MR 51 # 6842
[S20]K. J. Devlin andH. Johnsbråten,The Souslin problem, Springer, Berlin, 1974.MR 52 #5416
[S21]T. Dodd andR. B. Jensen,The core Model.
[S22]F. R. Drake,Set theory an introduction to large cardinals, North-Holland Amsterdam, 1974.
[S23]D. A. Edwards, Two theorems of functional analysis effectively equivalent to choice axioms,Fund. Math. 88 (1975), 95–101.MR 51 # 10094
[S24]P. C. Eklof, Whitehead's problem is undecidable.
[S25]P. Erdős, A. Hajnal, A. Máté andR. Rado,Partition calculus.
[S26]U. Felgner,Models of ZF-set theory, Springer, Berlin, 1971.MR 50 # 4298
[S27]U. Felgner, The independence of the Boolean prime ideal theorem from the order extension principle.
[S28]J. E. Fenstad, The axiom of determinateness,Proc. Second Scandinavian Logic Sympos. (Oslo, 1970), North-Holland, Amsterdam, 1971, 41–61.MR 48 # 10806
[S29]W. Fleissner, Normal Moore spaces in the constructible universe,Proc. Amer. Math. Soc. 46 (1974), 294–298.MR 50 # 14682
[S30]H. M. Friedman, Higher set theory and mathematical practice,Ann. Math. Logic 2 (1971), 325–357.MR 44 # 1556
[S31]H. Friedman, One hundred and two problems in mathematical logic,J. of Symbolic Legic 40 (1975), 113–129.MR 51 # 5254
[S32]F. Galvin andA. Hajnal, Inequalities for cardinal powers,Ann. Math. 101 (1975), 491–498.MR 51 # 12535
[S33]G. Gardiner, The equivalence of Boolean prime ideal theorem and a theorem of functional analysis,Fund. Math. 84 (1974), 81–86.MR 50 # 10749
[S34]S. Grigorieff, Combinatorics on ideals and forcing,Ann. Math. Logic 3 (1971), 363–394.MR 45 # 6614
[S35]S. Grigorieff, Intermediate submodels and generic extensions in set theory,Ann. Math. 101 (1975), 447–490.MR 51 # 10089
[S36]D. Guaspari,Thin and wellordered analytical sets, University of Cambridge, 1973. (Ph. D. Thesis)
[S37]D. Guaspari, The largest countableΠ 11 .
[S38]L. Harrington,Π 21 sets andΠ 21 singletons,Proc. Amer. Math. Soc. 52 (1975), 356–360.MR 51 # 10096
[S39]L. Harrington, Long projective wellorderings.
[S40]L. Harrington, The constructable reals can be (almost) anything.
[S41]L. Harrington andT. Jech, OnΣ 1 well orderings of the universe,J. Symbolic Logic 41 (1976), 167–170.MR 53 # 7780
[S42]J. Harrison, Recursive pseudo-well-orderings,Trans. Amer. Math. Soc. 131 (1968), 526–543.MR 39 # 5366
[S43]P. E. Howard, Limitations on the Fraenkel-Mostowski method of independence proofs,J. Symbolic Logic 38 (1973), 416–422.MR 52 # 2878
[S44]P. E. Howard, Los's theorem plus the Boolean prime ideal theorem imply the axiom of choice,Proc. Amer. Math. Soc. 49 (1975), 426–428.MR 52 # 5422
[S45]T. J. Jech,Lectures in set theory with particular emphasis on the method of forcing, Springer, Berlin, 1971.MR 48 # 105
[S46]T. J. Jech, Forcing with trees and ordinal definability,Ann. Math. Logic 7 (1975), 387–409.MR 51 # 142
[S47]T. J. Jech,The axiom of choice, North-Holland, Amsterdam, 1973.MR 53 # 139
[S48]T. J. Jech, Trees,J. Symbolic Logic 36 (1971), 1–14.MR 44 # 1560
[S49]T. J. Jech, Some combinatorial problemsn concernig uncountable cardinals,Ann. Math. Logic 5 (1973), 165–198.Zbl.262. 02062
[S50]T. Jech andK. Příkry, On ideals of sets and the power set operation.
[S51]R. Jensen, Definable sets of minimal degree, in:Mathematical Logic and Foundations of Set Theory (ed. Y. Bar-Hillel), North-Holland, Amsterdam, 1970.
[S52]R. B. Jensen, Coding the universe by a real.
[S53]R. B. Jensen, Marginalia to a theorem of Silver (with two subsequent corrections: More Marginalia; Marginalia III). See the paper byDevlin andJensen,Proc. Logic Colloq. (Kiel, 1974), Springer, Berlin, 1975.
[S54]R. B. Jensen, The fine structure of the constructible hierarchy,Ann. Math. Logic 4 (1972), 229–308 and 443.MR 46 # 8834
[S55]R. B. Jensen, On the consistency of a slight (?) modification of Quine's New Foundations,Synthese 19 (1968), 250–263.Zbl.202. 10
[S56]R. B. Jensen andH. Johnsbråten, A new construction of a non-constructibleΔ 13 subset of ω,Fund. Math. 81 (1974), 279–290.MR 52 # 13382
[S57]R. R. Jensen andB. Koppelberg, A note on ultrafilters (and an Addendum)
[S58]R. B. Jensen andK. Kunen,Some combinatorial properties of L and V. (Manuscript)
[S59]R. B. Jensen andR. M. Solovay, Some applications of almost disjoint sets,Mathematical Logic and Foundations of Set Theory (Proc. Colloq., Jerusalem, 1968), North-Holland, Amsterdam, 1970, 84–104.MR 44 # 6482
[S60]A. Kanamori,Ultrafilters over uncountable cardinals, University of Cambridge, 1975. (Ph. D. Thesis)
[S61]A. Kanamori, Wenkly normal filters and irregular ultrafilters.
[S62]A. Kanamori, W. N. Reinhardt andR. Solovay, Strong axioms of infinity and elementary embeddings,Ann. Math. Logic. (To appear)
[S63]A. S. Kechris, The theory of countable analytical sets,Trans. Amer. Math. Soc. 202 (1975), 259–297.MR 54 # 7259
[S64]A. S. Kechris, Measure and category in effective descriptive set theory,Ann. Math. Logic 5 (1973), 337–384.MR 51 # 5308
[S65]A. S. Kechris, On projective ordinals,J. Symbolic Logic 39 (1974), 269–282MR 54 # 2684
[S66]A. S. Kechris andY. N. Moschovakis,Notes on the theory of scales. (Typescript)
[S67]J. Ketonen, Strong compactness and other cardinal sins.Ann. Math. Logic 5 (1972), 47–76.Zbl 257. 02055
[S68]J. Ketonen, Non-regular ultrafilters and large cardinals,Trans. Amer. Math. Soc. 224 (1976), 61–73.MR 54 # 7260
[S69]E. M. Kleinberg, Rowbottom cardinals and Jónsson cardinals are almost the same,J. Symbolic Logic 38 (1973), 423–427.MR 43 # 2385
[S70]E. M. Kleinberg, AD The ℵ Π are Jónsson cardinals and ℵ ω is a Rowbottom cardinal.
[S71]K. Kunen, Some applications of iterated ultrapowers in set theory,Ann. Math. Logic 1 (1970), 179–227.MR 43 # 3080
[S72]K. Kunen, On the GCH at Measurable Cardinals,Proc. Colloq (Manchester 1969), North-Holland, Amsterdam, 1971, 107–110.MR 43 # 3081
[S73]K. Kunen, A model for the negation of the axiom of choice,Cambridge Summer School in Mathematical Logic (Cambridge, England, 1971), Springer, Berlin, 1973, 489–494.MR 49 # 2372
[S74]K. Kunen, Saturated ideals.
[S75]K. Kunen andJ. B. Paris, Boolean extension and measurable cardinals,Ann. Math. Logic 2 (1971), 359–377.MR 43 # 3114
[S76]J. Lake, Natural models and Ackermann-type set theories,J. Symbolic Logic 40 (1975), 151–158.MR 51 # 5307
[S77]R. Laver, on the consistency of Borel's conjecture.
[S78]A. Lévy andR. M. Solovay, On the decomposition of sets of reals to Borel sets,Ann. Math. Logic 5 (1972), 1–20.MR 47 # 38
[S79]M. Magidor, On the singular cardinals problem, I–II.
[S80]M. Magidor, How large is the first strongly compact cardinal? or Study in Identity Crises,Ann. Math. Logic 10 (1976), 33–59
[S81]R. Mansfield, The non-existence ofΣ 12 well-orderings of the Cantor set,Fund. Math. 86 (1976), 279–282.MR 52 # 7903
[S82]R. Mansfield, Omitting types: application to descriptive set theory,Proc. Amer. Math. Soc. 47 (1975), 198–200.MR 50 # 6851
[S83]W. Marek andM. Srebrny, Gaps in the constructible universe,Ann. Math. Logic 6 (1974), 359–394.MR 51 # 10102
[S84]D. A. Martin, Measurable cardinals and analytic games,Fund. Math. 66 (1960), 281–291.MR 41 # 3283
[S85]D. A. Martin, Borel determinacy,Ann. of Math. 102 (1975), 363–371.MR 53 # 7785
[S86]D. A. Martin, Projective sets and cardinal numbers.
[S87]D. A. Martin andJ. B. Paris, AD implies that there are exactly two normal measures onω 2
[S88]D. A. Martin, andR. M. Solovay, Internal Cohen extension,Ann. Math. Logic 2 (1970), 143–178.MR 42 # 5787
[S89]A. R. D. Mathias, Happy Families,Ann. Math. Logic (To appear)
[S90]A. R. D. Mathias, On sequences generic in the sense of Příkry,J. Austral. Math. Soc. 15 (1973), 409–414.MR 48 # 10809
[S91]K. McAloon, On the sequence of models HODt,Fund. Math. 82 (1974), 85–93.MR 50 # 98
[S92]K. McAloon, Consistency results about ordinal definability,Ann. Math. Logic 2 (1971), 449–467.MR 45 # 1753
[S93]T. K. Menas, On strong compactness and super-compactness,Ann. Math. Logic 7 (1975) 327–359.MR 50 # 9589
[S94]T. K. Menas,On strong compactness and supercompactness, University of California, Berkeley, 1974. (Ph. D. Thesis)
[S95]W. J. Mitchell, Sets constructible from sequences of ultrafilters,J. Symbolic Logic 39 (1974), 57–66.MR 49 # 8863
[S96]W. Mitchell, Aronszajn trees and the independence of the transfer property,Ann. Math. Logic 5 (1972), 21–46.MR 47 # 1612
[S97]G. P. Monro, Decomposable cardinals,Fund. Math. 80 (1973), 101–104.MR 48 # 3741
[S98]C. Morgenstern, On the independence of the ordering of certain large cardinals.
[S99]D. B. Morris,Adding total indiscernibles to models of set theory, University of Wisconsin, Madison, 1970. (Thesis)
[S100]Y. N. Moschovakis, Determinancy and prewellorderings of the continuum,Mathematical Logic and Foundations of Set Theory (Proc. Colloq., Jerusalem, 1968), North-Holland, Amsterdam, 1970, 24–62.MR 43 # 6082
[S101]A. J. Ostaszewski, On countably compact, perfectly normal spaces,J. London Math. Soc. 14 (1976), 505–516.
[S102]D. Pincus, Support structures for the axiom of choice,J. Symbolic Logic 36 (1971), 28–38.MR 44 # 61
[S103]D. Pincus, Zermelo—Fraenkel consistency results by Fraenkel—Mostowski methods,J. Symbolic Logic 37 (1972), 721–743.MR 49 # 2374
[S104]D. Pincus, The strength of the Hahn—Banach theorem,Victoria Sympos. Nonstand. Analysis (1972), Springer, Berlin, 1974, 203–248.Zbl 279. 02044
[S105]K. Příkry, Changing measurable into accessible cardinals,Dissertationes Math. Rozprawy Mat. 68 (1970).MR 41 # 6685
[S106]K. Příkry, On a problem of Gillamn and Keisler,Ann. Math. Logic 2 (1970), 179–187.MR 42 # 4408
[S107]K. Příkry, Determinateness and partitions,Proc. Amer. Math. Soc. 54 (1976), 303–306.
[S108]W. N. Reinhardt, Ackermann's set theory equals ZF,Ann. Math. Logic 2 (1970), 189–249.MR 43 # 42
[S109]F. Rothberger, Eine Äquivalenz zwischen der Kontinuumhypothese und der Existenz der Lusinschen und Sierpińskischen Mengen,Fund. Math. 30 (1938), 215–217.Zbl 18, 394
[S110]F. Rowbottom, Some strong axioms of infinity incompatible with the axiom of constructibility,Ann. Math. Logic. 3 (1971), 1–44.MR 48 # 1928
[S111]M. E. Rudin,Lectures on set theoretic topology, American Mathematical Society, Regional Conference Series in Mathematics, No. 23.
[S112]G. Sageev, An independence result concerning the axiom of choice,Ann. Math. Logic 8 (1975), 1–184.MR 51 # 2915
[S113]S. Shelah, Infinite abelian groups Whitehead problem and some constructions,Israel J. Math. 18 (1974), 243–256.MR 50 # 9582
[S114]S. Shelah, A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals,Israel J. Math. 21 (1975), 319–349.MR 52 # 10410
[S115]R. A. Shore, Square bracker partition relations inL, Fund. Math. 84 (1974), 101–106.MR 51 # 7880
[S116]J. H. Silver, On the singular cardinals problem,Proc. International Congress of Mathematicians, Vancouver 1974, 265–268.
[S117]J. H. Silver, Some applications of model theory in set theory,Ann. Math. Logic 3 (1971), 45–110.MR 53 # 12950
[S118]J. H. Silver, Measurable cardinals andΔ 13 well-orderings,Ann. Math. 94 (1971), 414–446MR 45 # 8517
[S119]R. M. Solovay, Hyperarithmetically encodable sets.
[S120]R. M. Solovay, A model of set-theory in which every set of reals is Lebesgue measurable,Ann. Math. 92 (1970) 1–56.MR 42 # 64
[S121]R. M. Solovay, Strongly compact cardinals and the GCH,Proc. Tarski Symposium (Berkeley, California, 1971), Amer. Math. Soc., Providence, R. I., 1974, 365–372.MR 52 # 106
[S122]R. M. Solovay andS. Tennenbaum, Iterated Cohen extension and Souslin's problem,Ann. Math. 94 (1971), 201–245.MR 45 # 3212
[S123]F. D. Tall, The countable chain condition versus separability—applications of Martin's axiom,General Topology and Appl. 4 (1974), 315–339.MR 54 # 11264
[S124]F. Tall, An alternative to the continuum hypothesis and its uses in general topology.
[S125]A. Tarski, Ideale in vollständigen Mengenkörpern, II.Fund. Math. 33 (1945), 51–56.MR 8 # 193
[S126]J. Truss, On successors in cardinal arithmetic,Fund. Math. 78 (1973), 7–21.MR 47 # 8307
[S127]J. Truss, Finite axioms of choice,Ann. Math. Logic 6 (1973), 147–176.Zbl 273. 02049
[S128]J. Truss, Models of set theory containing many perfect sets,Ann. Math, Logic 7 (1974), 197–219.MR 51 # 5304
[S129]S. Ulam, Zur Maßtheorie in der allgemeinen Mengenlehre,Fund. Math. 16 (1930), 140–150.
[S130]P. Vopěnka andP. Hájek,The theory of semisets, North-Holland, Amsterdam, 1972
[S131]S. Wagon,Decompositions of saturated ideals, Darthmouth College, Hanover, New Hampshire, 1975. (Ph. D. Thesis)
[S132]N. H. Williams,Cardinal numbers with partition properties, Australian National University, Canberra, 1969. (Ph. D. Thesis)
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Work on the original version was supported by NSF Grant 7655.
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Mathias, A.R.D. Surrealist landscape with figures (a survey of recent results in set theory). Period Math Hung 10, 109–175 (1979). https://doi.org/10.1007/BF02025889
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DOI: https://doi.org/10.1007/BF02025889
AMS (MOS) subject classifications (1970)
- Primary 02-02
- 04-02
- 02K05
- 02K30
- 02K35
- 04A15
- Secondary 02F27
- 02K10
- 04A20
- 04A25
- 04A30
- 02H13
- 02H25
- 20K99
- 28A05
- 54H05
- 54G99
- 54F45
- 54D20
- 54A25