Potent axioms

Author:
Matthew Foreman

Journal:
Trans. Amer. Math. Soc. **294** (1986), 1-28

MSC:
Primary 03E65; Secondary 03E15, 03E50, 03E55, 04A30

DOI:
https://doi.org/10.1090/S0002-9947-1986-0819932-2

MathSciNet review:
819932

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper suggests alternatives to the ordinary large cardinal axioms of set theory. These axioms can be viewed as generalizations of large cardinals and exhibit many of the same phenomena. They are shown to imply the G.C.H., every set of reals in is Lebesgue measurable, and various results in combinatorics, algebra and model theory.

**[B]**J. E. Baumgartner,*A generic graph construction*, J. Symbolic Logic (to appear). MR**736618 (85j:03082)****[B-T]**J. E. Baumgartner and A. D. Taylor,*Saturation of ideals in generic extensions*. II, Trans. Amer. Math. Soc.**271**(1982), 587-609. MR**654852 (83k:03040b)****[F1]**M. Foreman,*More saturated ideals*, Cabal Seminar 1979-1981. Lecture Notes in Math., vol. 1019, Springer-Verlag, 1983. MR**730584****[F2]**-,*Large cardinals and strong model-theoretic transfer properties*, Trans. Amer. Math. Soc.**272**(1982), 427-463. MR**662045 (84d:03038)****[F-L]**M. Foreman and R. Laver,*A graph transfer property*(to appear).**[F-M1]**M. Foreman and M. Magidor,*is consistent with a*-*saturated ideal on*(to appear).**[F-M2]**-,*An*-*saturated ideal on*.**[G]**H. Gaifman,*Elementary embeddings of models of set-theory and certain subtheories*, Axiomatic Set Theory, Proc. Sympos. Pure Math., Vol. 13, Part II (T. Jech, ed.). Amer. Math. Soc., Providence, R.I., 1971, pp. 33-101. MR**0376347 (51:12523)****[Ga]**F. Galvin,*Chromatic numbers of subgraphs*, Period. Mat. Hungar.**4**(1973), 117-119. MR**0345859 (49:10589)****[J1]**T. Jech,*Set theory*, Academic Press, New York, 1978. MR**506523 (80a:03062)****[JMMP]**T. Jech, M. Magidor, W. Mitchell and K. Prikry,*Precipitous ideals*, J. Symbolic Logic**45**(1980), 1-8. MR**560220 (81h:03097)****[J-P]**T. Jech and K. Prikry,*On ideals of sets and the power set operation*, Bull. Amer. Math. Soc.**82**(1976), 593-595. MR**0505504 (58:21618)****[K1]**K. Kunen,*Some applications of iterated ultrapowers in set theory*, Ann. of Math. Logic**1**(1970). 179-227. MR**0277346 (43:3080)****[K2]**-,*Saturated ideals*, J. Symbolic Logic**43**(1978), 65-76. MR**495118 (80a:03068)****[L]**R. Laver,*Saturated ideals and non-regular ultrafilters*. Proc. Bernays Conference (Patras, Greece, 1980), North-Holland, 1982.**[M1]**M. Magidor,*Precipitous ideals and*-*sets*.**[Ma]**A. R. D. Mathias,*Happy families*, Ann. of Math. Logic**12**(1977), 59-111. MR**0491197 (58:10462)****[M-S]**J. Mycielski and S. Swierczkowski,*On Lebesgue measureability and the axiom of determinateness*, Fund. Math.**54**(1964). MR**0161788 (28:4992)****[S]**J. Silver,*The consistency of the**with the existence of a measurable cardinal*, Axiomatic Set Theory, Proc. Sympos. Pure Math., Vol. 13, Part I. Amer. Math. Soc., Providence. R.I., 1971, pp. 391-395. MR**0278937 (43:4663)****[Sh]**S. Shelah. unpublished.**[Sh1]**-,*Proper forcing*, Lecture Notes in Math., vol. 940, Springer-Verlag, 1982. MR**675955 (84h:03002)****[Sh2]**-,*Infinite Abelian groups Whitehead problem and some constructions*, Israel J. Math.**13**(1977).**[So1]**R. M. Solovay,*A model of set theory in which every set of reals is Lebesgue measurable*, Ann. of Math (2)**92**(1970), 1-56. MR**0265151 (42:64)****[So2]**-,*Real-valued measurable cardinals*, Axiomatic Set Theory, Proc. Sympos. Pure Math., Vol. 13, Part I, Amer. Math. Soc., Providence, R. I., 1971, pp. 397-428. MR**0290961 (45:55)****[SRK]**R. M. Solovay, N. N. Reinhardt and A. Kanamori,*Strong axioms of infinity and elementary embeddings*, Ann. of Math. Logic**13**(1978), 73-116. MR**482431 (80h:03072)****[W1]**W. H. Woodin,*Discontinuous homomorphisms of**and set theory*(to appear).**[W2]**-,*An*-*dense ideal on*(to appear).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
03E65,
03E15,
03E50,
03E55,
04A30

Retrieve articles in all journals with MSC: 03E65, 03E15, 03E50, 03E55, 04A30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1986-0819932-2

Article copyright:
© Copyright 1986
American Mathematical Society