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.

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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