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    2014. The Theory of Models.


    Kanamori, Akihiro 2012. Sets and Extensions in the Twentieth Century.


    Larson, Jean A. 2012. Sets and Extensions in the Twentieth Century.


    Kanamori, Akihiro 2009. Philosophy of Mathematics.


    Kirby, Laurence 2008. A hierarchy of hereditarily finite sets. Archive for Mathematical Logic, Vol. 47, Issue. 2, p. 143.


    Бунина, Елена Игоревна Bunina, Elena Igorevna Бунина, Елена Игоревна Bunina, Elena Igorevna Захаров, Валерий Константинович Zakharov, Valeriy Konstantinovich Захаров, Валерий Константинович and Zakharov, Valeriy Konstantinovich 2007. Формульно-недостижимые кардиналы и характеризация всех натуральных моделей теории множеств Цермело - Френкеля. Известия Российской академии наук. Серия математическая, Vol. 71, Issue. 2, p. 3.


    Bunina, E. I. and Zakharov, V. K. 2006. Characterization of model Mirimanov-von Neumann cumulative sets. Journal of Mathematical Sciences, Vol. 138, Issue. 4, p. 5830.


    Бунина, Елена Игоревна Bunina, Elena Igorevna Захаров, Валерий Константинович and Zakharov, Valeriy Konstantinovich 2005. Канонический вид множеств Тарского в теории множеств Цермело - Френкеля. Математические заметки, Vol. 77, Issue. 3, p. 323.


    Бунина, Елена Игоревна Bunina, Elena Igorevna Захаров, Валерий Константинович and Zakharov, Valeriy Konstantinovich 2003. Канонический вид супертранзитивных стандартных моделей в теории множеств Цермело - Френкеля. Успехи математических наук, Vol. 58, Issue. 4, p. 143.


    Augenstein, Bruno W. 1996. Links between physics and set theory. Chaos, Solitons & Fractals, Vol. 7, Issue. 11, p. 1761.


    1992.


    Hellman, Geoffrey 1990. Toward a modal-structural interpretation of set theory. Synthese, Vol. 84, Issue. 3, p. 409.


    Perlis, Donald 1988. Autocircumscription. Artificial Intelligence, Vol. 36, Issue. 2, p. 223.


    Moore, Gregory H. 1987. Logic Colloquium '86, Proceedings of the Colloquium held in Hull.


    1986. Theory of Relations.


    1980. Descriptive Set Theory.


    1978.


    1974. Set Theory - An Introduction to Large Cardinals.


    Gloede, K. 1973. Cambridge Summer School in Mathematical Logic.


    1973. The Axiom of Choice.


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Inner models for set theory—Part I

  • J. C. Shepherdson (a1)
  • DOI: http://dx.doi.org/10.2307/2266389
  • Published online: 01 March 2014
Abstract

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.

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(1)K. Gödel . The consistency of the continuum hypothesis. Princeton1940.

(2)J. v. Neumann . Die Axiomatisierung der Mengenlehre. Mathematische Zeitschrift, vol. 27 (1928), pp. 669752.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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