7 found
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  1. Ultrapowers Without the Axiom of Choice.Mitchell Spector - 1988 - Journal of Symbolic Logic 53 (4):1208-1219.
    A new method is presented for constructing models of set theory, using a technique of forming pseudo-ultrapowers. In the presence of the axiom of choice, the traditional ultrapower construction has proven to be extremely powerful in set theory and model theory; if the axiom of choice is not assumed, the fundamental theorem of ultrapowers may fail, causing the ultrapower to lose almost all of its utility. The pseudo-ultrapower is designed so that the fundamental theorem holds even if choice fails; this (...)
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  2.  29
    Extended Ultrapowers and the Vopěnka-Hrbáček Theorem Without Choice.Mitchell Spector - 1991 - Journal of Symbolic Logic 56 (2):592-607.
    We generalize the ultrapower in a way suitable for choiceless set theory. Given an ultrafilter, forcing is used to construct an extended ultrapower of the universe, designed so that the fundamental theorem of ultrapowers holds even in the absence of the axiom of choice. If, in addition, we assume DC, then an extended ultrapower of the universe by a countably complete ultrafilter must be well-founded. As an application, we prove the Vopěnka-Hrbáček theorem from ZF + DC only (the proof of (...)
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  3.  17
    A Measurable Cardinal with a Nonwellfounded Ultrapower.Mitchell Spector - 1980 - Journal of Symbolic Logic 45 (3):623-628.
  4.  22
    Weak Partition Relations and Measurability.Mitchell Spector - 1986 - Journal of Symbolic Logic 51 (1):33-38.
  5.  14
    Iterated Extended Ultrapowers and Supercompactness Without Choice.Mitchell Spector - 1991 - Annals of Pure and Applied Logic 54 (2):179-194.
    Working in ZF + DC with no additional use of the axiom of choice, we show how to iterate the extended ultrapower construction of Spector . This generalizes the technique of iterated ultrapowers to choiceless set theory. As an application, we prove the following theorem: Assume V = LU[κ] + “κ is λ-supercompact with normal ultrafilter U” + DC. Then for every sufficiently large regular cardinal ρ, there exists a set-generic extension V[G] of the universe in which there exists for (...)
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  6.  26
    Model Theory Under the Axiom of Determinateness.Mitchell Spector - 1985 - Journal of Symbolic Logic 50 (3):773-780.
    We initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, L ω 1 ω is no more powerful than first-order logic. The emphasis then turns to upward Lowenhein-Skolem theorems; ℵ 1 is the Hanf number of first-order logic, of L ω 1 ω , and of a strong fragment of L ω 1 ω . The main technical innovation is (...)
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  7.  12
    The Κ-Closed Unbounded Filter and Supercompact Cardinals.Mitchell Spector - 1981 - Journal of Symbolic Logic 46 (1):31-40.