Random gaps under CH
Author:
James Hirschorn
Journal:
Trans. Amer. Math. Soc. 356 (2004), 1281-1290
MSC (2000):
Primary 03E05; Secondary 03E40, 03E50, 28E15
DOI:
https://doi.org/10.1090/S0002-9947-03-03380-4
Published electronically:
November 25, 2003
MathSciNet review:
2034309
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: It is proved that if the Continuum Hypothesis is true, then one random real always produces a destructible gap.
- [AT97] Uri Abraham and Stevo Todorčević, Partition properties of 𝜔₁ compatible with CH, Fund. Math. 152 (1997), no. 2, 165–181. MR 1441232
- [Dow95] Alan Dow, More set-theory for topologists, Topology Appl. 64 (1995), no. 3, 243–300. MR 1342520, https://doi.org/10.1016/0166-8641(95)00034-E
- [Hau36]
Felix Hausdorff, Summen
von Mengen, Fund. Math. 26 (1936), 241-255.
- [Hir00a] James Hirschorn, Random trees under CH, preprint, 2000.
- [Hir00b] James Hirschorn, Towers of measurable functions, Fund. Math. 164 (2000), no. 2, 165–192. MR 1784706
- [Hir01] James Hirschorn, Summable gaps, Ann. Pure Appl. Logic 120 (2003), no. 1-3, 1-63.
- [Hir03] James Hirschorn, Random gaps, preprint, October 2003.
- [Jec97] Thomas Jech, Set theory, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1997. MR 1492987
- [Kan94] Akihiro Kanamori, The higher infinite, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1994. Large cardinals in set theory from their beginnings. MR 1321144
- [Kun76a]
Kenneth Kunen,
gaps under MA, handwritten note, August 1976.
- [Kun76b] Kenneth Kunen, Some points in 𝛽𝑁, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 3, 385–398. MR 427070, https://doi.org/10.1017/S0305004100053032
- [Lav79] Richard Laver, Linear orders in (𝜔)^{𝜔} under eventual dominance, Logic Colloquium ’78 (Mons, 1978) Stud. Logic Foundations Math., vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 299–302. MR 567675
- [Sch93] Marion Scheepers, Gaps in 𝜔^{𝜔}, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 439–561. MR 1234288
- [Sol71] Robert M. Solovay, Real-valued measurable cardinals, 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. 397–428. MR 0290961
- [TF95] S. Todorchevich and I. Farah, Some applications of the method of forcing, Yenisei Series in Pure and Applied Mathematics, Yenisei, Moscow; Lycée, Troitsk, 1995. MR 1486583
- [Tod89] Stevo Todorčević, Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. MR 980949
- [Tod00] Stevo Todorčević, A dichotomy for P-ideals of countable sets, Fund. Math. 166 (2000), no. 3, 251–267. MR 1809418
- [Woo84]
W. Hugh Woodin, Discontinuous homomorphisms of
and set theory, Ph.D. thesis, University of California, Berkeley, 1984.
Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 03E05, 03E40, 03E50, 28E15
Retrieve articles in all journals with MSC (2000): 03E05, 03E40, 03E50, 28E15
Additional Information
James Hirschorn
Affiliation:
Department of Mathematics, University of Helsinki, Helsinki, Finland
Address at time of publication:
Centre de Recerca Matemàtica, Apartat 50, E-08193 Bellaterra, Spain
Email:
jhirschorn@crm.es, James.Hirschorn@logic.univie.ac.at
DOI:
https://doi.org/10.1090/S0002-9947-03-03380-4
Keywords:
Gap,
destructible gap,
random real,
Continuum Hypothesis
Received by editor(s):
October 1, 2001
Published electronically:
November 25, 2003
Article copyright:
© Copyright 2003
American Mathematical Society