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Inner models for set theory—Part I

Published online by Cambridge University Press:  12 March 2014

J. C. Shepherdson
J. C. Shepherdson*
Affiliation:
University of Bristol, England
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Extract

One of the standard ways of proving the consistency of additional hypotheses with the basic axioms of an axiom system is by the construction of what may be described as ‘inner models.’ By starting with a domain of individuals assumed to satisfy the basic axioms an inner model is constructed whose domain of individuals is a certain subset of the original individual domain. If such an inner model can be constructed which satisfies not only the basic axioms but also the particular additional hypothesis under consideration, then this affords a proof that if the basic axiom system is consistent then so is the system obtained by adding to this system the new hypothesis. This method has been applied to axiom systems for set theory by many authors, including v. Neumann (4), Mostowski (5), and more recently Gödel (1), who has shown by this method that if the basic axioms of a certain axiomatic system of set theory are consistent then so is the system obtained by adding to these axioms a strong form of the axiom of choice and the generalised continuum hypothesis. Having been shown in this striking way the power of this method it is natural to inquire whether it has any limitations or whether by the construction of a sufficiently ingenious inner model one might hope to decide other outstanding consistency questions, such as the consistency of the negations of the axiom of choice and continuum hypothesis. In this and two following papers we prove some general theorems concerning inner models for a certain axiomatic system of set theory which lead to the result that as far as a fairly large family of inner models are concerned this method of proving consistency has been exhausted, that no essentially new consistency results can be obtained by the use of this kind of model.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 16 , Issue 3 , September 1951 , pp. 161 - 190
Copyright
Copyright © Association for Symbolic Logic 1951

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References

BIBLIOGRAPHY

(1)Gödel, K.. The consistency of the continuum hypothesis. Princeton 1940.CrossRefGoogle Scholar
(2)Neumann, J. v.. Die Axiomatisierung der Mengenlehre. Mathematische Zeitschrift, vol. 27 (1928), pp. 669–752.CrossRefGoogle Scholar
(3)Neumann, J. v.. Eine Axiomatisierung der Mengenlehre. Journal für die reine und angewandte Mathematik, vol. 154 (1925), pp. 219–240.CrossRefGoogle Scholar
(4)Neumann, J. v.. Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre. Journal für die reine und angewandte Mathematik, vol. 160 (1929), pp. 227–241.CrossRefGoogle Scholar
(5)Mostowski, A.. Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip. Fundamenta mathematicae, vol. 32 (1939), pp. 201–252.CrossRefGoogle Scholar
(6)Bernays, P.. A system of axiomatic set theory—part I. This Journal, vol. 2 (1937), pp. 65–77.Google Scholar
(7)Markoff, A. A.. O zavisimosti aksiomy B6 ot drugih aksiom sistémy Bernays'a-Gödel'á. Izvéstiá Akadémii Nauk SSSR, mathematical series, vol. 12 (1948), pp. 569–570.Google Scholar
(8)Zermelo, E.. Über Grenzzahlen und Mengenbereiche. Fundamenta mathematicae, vol. 16 (1930), pp. 29–47.CrossRefGoogle Scholar
(9)Bernays, P.. A system of axiomatic set theory—part II. This Journal, vol. 6 (1941), pp. 1–17.Google Scholar
(10)Tarski, A.. Unerreichbare Kardinalzahlen. Fundamenta mathematicae, vol. 30 (1938), pp. 66–89.CrossRefGoogle Scholar