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Free algebras in Von Neumann-Bernays-Gӧdel set theory and positive elementary inductions in reasonable structures


Rubin, Arthur (1978) Free algebras in Von Neumann-Bernays-Gӧdel set theory and positive elementary inductions in reasonable structures. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/62t8-9b85.


This thesis consists of two independent chapters. The first chapter deals with universal algebra. It is shown, in von Neumann-Bernays-Gӧdel set theory, that free images of partial algebras exist in arbitrary varieties. It follows from this, as set-complete Boolean algebras form a variety, that there exist free set-complete Boolean algebras on any class of generators. This appears to contradict a well-known result of A. Hales and H. Gaifman, stating that there is no complete Boolean algebra on any infinite set of generators. However, it does not, as the algebras constructed in this chapter are allowed to be proper classes. The second chapter deals with positive elementary inductions. It is shown that, in any reasonable structure ᶆ, the inductive closure ordinal of ᶆ is admissible, by showing it is equal to an ordinal measuring the saturation of ᶆ. This is also used to show that non-recursively saturated models of the theories ACF, RCF, and DCF have inductive closure ordinals greater than ω.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Kechris, Alexander S.
Thesis Committee:
  • Unknown, Unknown
Defense Date:5 May 1978
Funding AgencyGrant Number
Department of Health, Education and WelfareUNSPECIFIED
Ford FoundationUNSPECIFIED
Record Number:CaltechTHESIS:05282015-144505104
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8901
Deposited On:02 Jun 2015 15:23
Last Modified:09 Nov 2022 19:20

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