What is the theory without power set?

Mathematical Logic Quarterly 62 (4-5):391-406 (2016)
  Copy   BIBTEX


We show that the theory, consisting of the usual axioms of but with the power set axiom removed—specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well‐ordered—is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of in which ω1 is singular, in which every set of reals is countable, yet ω1 exists, in which there are sets of reals of every size, but none of size, and therefore, in which the collection axiom fails; there are models of for which the Łoś theorem fails, even when the ultrapower is well‐founded and the measure exists inside the model; there are models of for which the Gaifman theorem fails, in that there is an embedding of models that is Σ1‐elementary and cofinal, but not elementary; there are elementary embeddings of models whose cofinal restriction is not elementary. Moreover, the collection of formulas that are provably equivalent in to a Σ1‐formula or a Π1‐formula is not closed under bounded quantification. Nevertheless, these deficits of are completely repaired by strengthening it to the theory, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach.



    Upload a copy of this work     Papers currently archived: 91,252

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

A Natural Model of the Multiverse Axioms.Victoria Gitman & Joel David Hamkins - 2010 - Notre Dame Journal of Formal Logic 51 (4):475-484.
Does Pragmatism Have A Theory of Power?Joel Wolfe - 2012 - European Journal of Pragmatism and American Philosophy 4 (1):120-137.
The power of power—questions to Michel Foucault.Norbert Ricken - 2006 - Educational Philosophy and Theory 38 (4):541–560.
The Power of Power—Questions to Michel Foucault.Norbert Ricken - 2006 - Educational Philosophy and Theory 38 (4):541-560.
Power and values in corporate life.David R. Hiley - 1987 - Journal of Business Ethics 6 (5):343 - 353.
The technological construction of social power.Philip Brey - 2008 - Social Epistemology 22 (1):71 – 95.
Confirmation, explanation, and logical strength.David E. Nelson - 1996 - British Journal for the Philosophy of Science 47 (3):399-413.
Power Over People.Dennis Dalton - 1996 - Teaching Co..
Resurrection axioms and uplifting cardinals.Joel David Hamkins & Thomas A. Johnstone - 2014 - Archive for Mathematical Logic 53 (3-4):463-485.


Added to PP

33 (#462,035)

6 months
13 (#169,369)

Historical graph of downloads
How can I increase my downloads?

Author Profiles

References found in this work

[Omnibus Review].Thomas Jech - 1992 - Journal of Symbolic Logic 57 (1):261-262.
[Omnibus Review].Akihiro Kanamori - 1981 - Journal of Symbolic Logic 46 (4):864-866.
Ramsey-like cardinals.Victoria Gitman - 2011 - Journal of Symbolic Logic 76 (2):519 - 540.
Indestructible Strong Unfoldability.Joel David Hamkins & Thomas A. Johnstone - 2010 - Notre Dame Journal of Formal Logic 51 (3):291-321.

View all 6 references / Add more references