Consistency results concerning supercompactness

Author:
Telis K. Menas

Journal:
Trans. Amer. Math. Soc. **223** (1976), 61-91

MSC:
Primary 02K35; Secondary 02H13, 02K05

MathSciNet review:
0540771

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Abstract: A general framework for proving relative consistency results with regard to supercompactness is developed. Within this framework we prove the relative consistency of the assertion that every set is ordinal definable with the statement asserting the existence of a supercompact cardinal. We also generalize Easton's theorem; the new element in our result is that our forcing conditions preserve supercompactness.

**[1]**William B. Easton,*Powers of regular cardinals*, Ann. Math. Logic**1**(1970), 139–178. MR**0269497****[2]**Paul R. Halmos,*Lectures on Boolean algebras*, Van Nostrand Mathematical Studies, No. 1, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR**0167440****[3]**Thomas J. Jech,*Lectures in set theory, with particular emphasis on the method of forcing*, Lecture Notes in Mathematics, Vol. 217, Springer-Verlag, Berlin-New York, 1971. MR**0321738****[4]**Thomas J. Jech,*Trees*, J. Symbolic Logic**36**(1971), 1–14. MR**0284331****[5]**M. Magidor, Dissertation, University of Jerusalem, 1972.**[6]**John Myhill and Dana Scott,*Ordinal definability*, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., (1967), Amer. Math. Soc., Providence, R.I., 1971, pp. 271–278. MR**0281603****[7]**W. Reinhardt and R. Solovay,*Strong axioms of infinity and elementary embeddings*(to appear).**[8]**D. Scott and R. Solovay,*Boolean-valued models for set theory*, mimeographed notes.**[9]**J. R. Shoenfield,*Unramified forcing*, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 357–381. MR**0280359****[10]**Jack Silver,*The independence of Kurepa’s conjecture and two-cardinal conjectures in model theory*, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 383–390. MR**0277379****[11]**-, Forthcoming paper on large cardinals and the G.C.H.**[12]**R. M. Solovay and S. Tennenbaum,*Iterated Cohen extensions and Souslin’s problem*, Ann. of Math. (2)**94**(1971), 201–245. MR**0294139****[13]**D. H. Stewart, M. Sci. Thesis, Bristol, 1966.**[14]**F. Tall, Doctoral Dissertation, University of Wisconsin, 1969.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0540771-8

Keywords:
Large cardinals,
backward Easton forcing,
ordinal definability,
supercompactness

Article copyright:
© Copyright 1976
American Mathematical Society