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Lawrence C. Paulson - One of the best experts on this subject based on the ideXlab platform.
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1.3 Arity of a Formula: Maximum Free de Bruijn Index..... 13
2012Co-Authors: Lawrence C. PaulsonAbstract:Gödel’s proof of the relative consistency of the axiom of choice [1] is one of the most important results in the foundations of mathematics. It bears on Hilbert’s first problem, namely the Continuum Hypothesis, and indeed Gödel also proved the relative consistency of the Continuum Hypothesis. Just as important, Gödel’s proof introduced the inner model method of proving relative consistency, and it introduced the concept of constructible set. Kunen [2] gives an excellent description of this body of work. This Isabelle/ZF formalization demonstrates Gödel’s claim that his proof can be undertaken without using metamathematical arguments, for example arguments based on the general syntactic structure of a formula. Isabelle’s automation replaces the metamathematics, although it does not eliminate the requirement at least to state many tedious results that would otherwise be unnecessary
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1.3 Arity of a Formula: Maximum Free de Bruijn Index..... 14
2011Co-Authors: Lawrence C. PaulsonAbstract:Gödel’s proof of the relative consistency of the axiom of choice [1] is one of the most important results in the foundations of mathematics. It bears on Hilbert’s first problem, namely the Continuum Hypothesis, and indeed Gödel also proved the relative consistency of the Continuum Hypothesis. Just as important, Gödel’s proof introduced the inner model method of proving relative consistency, and it introduced the concept of constructible set. Kunen [2] gives an excellent description of this body of work. This Isabelle/ZF formalization demonstrates Gödel’s claim that his proof can be undertaken without using metamathematical arguments, for example arguments based on the general syntactic structure of a formula. Isabelle’s automation replaces the metamathematics, although it does not eliminate the requirement at least to state many tedious results that would otherwise be unnecessary
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the relative consistency of the axiom of choice mechanized using isabelle zf
Conference on Computability in Europe, 2008Co-Authors: Lawrence C. PaulsonAbstract:Godel [3] published a monograph in 1940 proving a highly significant theorem, namely that the axiom of choice (AC) and the generalized Continuum Hypothesis (GCH) are consistent with respect to the other axioms of set theory. This theorem addresses the first of Hilbert's famous list of unsolved problems in mathematics. I have mechanized this work [8] using Isabelle/ZF [5,6]. Obviously, the theorem's significance makes it a tempting challenge; the proof also has numerous interesting features. It is not a single formal assertion, as most theorems are. Godel [3, p. 33] states it as follows, using Σto denote the axioms for set theory: What we shall prove is that, if a contradiction from the axiom of choice and the generalized Continuum Hypothesis were derived in Σ, it could be transformed into a contradiction obtained from the axioms of Σalone. Godel presents no other statement of this theorem. Neither does he introduce a theory of syntax suitable for reasoning about transformations on proofs, surely because he considers it to be unnecessary
Jose Rodriguez - One of the best experts on this subject based on the ideXlab platform.
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weak sequential properties of johnson lindenstrauss spaces
Journal of Functional Analysis, 2019Co-Authors: Antonio Aviles, Gonzalo Martinezcervantes, Jose RodriguezAbstract:Abstract A Banach space X is said to have Efremov's property ( E ) if every element of the weak⁎-closure of a convex bounded set C ⊆ X ⁎ is the weak⁎-limit of a sequence in C. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of N for which the corresponding Johnson–Lindenstrauss spaces enjoy (resp. fail) property ( E ). This is related to a gap in Plichko (2015) [12] and allows to answer (consistently) questions of Plichko and Yost.
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weak sequential properties of johnson lindenstrauss spaces
arXiv: Functional Analysis, 2018Co-Authors: Antonio Aviles, Gonzalo Martinezcervantes, Jose RodriguezAbstract:A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak$^*$-closure of a convex bounded set $C \subseteq X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of $\mathbb{N}$ for which the corresponding Johnson-Lindenstrauss spaces enjoy (resp. fail) property ($\mathcal{E}$). This is related to a gap in [A. Plichko, Three sequential properties of dual Banach spaces in the weak$^*$ topology, Topology Appl. 190 (2015), 93--98] and allows to answer (consistently) questions of Plichko and Yost.
Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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a polarized partition relation and failure of gch at singular strong limit
Fundamenta Mathematicae, 1998Co-Authors: Saharon ShelahAbstract:The main result is that for λ strong limit singular failing the Continuum Hypothesis (i.e. 2 > λ+), a polarized partition theorem holds.
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a polarized partition relation and failure of gch
arXiv: Logic, 1997Co-Authors: Saharon ShelahAbstract:The main result is that for lambda strong limit singular failing the Continuum Hypothesis (i.e. 2^lambda > lambda^+), a polarized partition theorem holds.
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the ax kochen isomorphism theorem
1995Co-Authors: Saharon ShelahAbstract:We show in x1 that the Ax-Kochen isomorphism theorem [AK] requires the Continuum Hypothesis. Most of the applications of this theorem are insensitive to set theoretic considerations. (A probable exception is the work of Moloney [Mo].) In x2 we give an unrelated result on cuts in models of Peano arithmetic which answers a question on the ideal structure of countable ultraproducts of Z posed in [LLS]. Inx1 we also answer a question of Keisler and Schmerl regarding Scott complete ultrapowers of R.
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bext 2 g t can be nontrivial even assuming gch
arXiv: Logic, 1994Co-Authors: Menachem Magidor, Saharon ShelahAbstract:Using the consistency of some large cardinals we produce a model of Set Theory in which the generalized Continuum Hypothesis holds and for some torsion-free abelian group G of cardinality ℵ!+1 and for some torsion group T Bext 2 (G,T) 6 0. Hence G.C.H. is not sufficient for getting the results of (10).
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vive la difference ii the ax kochen isomorphism theorem
Israel Journal of Mathematics, 1994Co-Authors: Saharon ShelahAbstract:We show in §1 that the Ax-Kochen isomorphism theorem [AK] requires the Continuum Hypothesis. Most of the applications of this theorem are insensitive to set theoretic considerations. (A probable exception is the work of Moloney [Mo].) In §2 we give an unrelated result on cuts in models of Peano arithmetic which answers a question on the ideal structure of countable ultraproducts of ℤ posed in [LLS]. In §1 we also answer a question of Keisler regarding Scott complete ultrapowers of ℝ (see 1.18).
Antonio Aviles - One of the best experts on this subject based on the ideXlab platform.
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weak sequential properties of johnson lindenstrauss spaces
Journal of Functional Analysis, 2019Co-Authors: Antonio Aviles, Gonzalo Martinezcervantes, Jose RodriguezAbstract:Abstract A Banach space X is said to have Efremov's property ( E ) if every element of the weak⁎-closure of a convex bounded set C ⊆ X ⁎ is the weak⁎-limit of a sequence in C. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of N for which the corresponding Johnson–Lindenstrauss spaces enjoy (resp. fail) property ( E ). This is related to a gap in Plichko (2015) [12] and allows to answer (consistently) questions of Plichko and Yost.
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weak sequential properties of johnson lindenstrauss spaces
arXiv: Functional Analysis, 2018Co-Authors: Antonio Aviles, Gonzalo Martinezcervantes, Jose RodriguezAbstract:A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak$^*$-closure of a convex bounded set $C \subseteq X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of $\mathbb{N}$ for which the corresponding Johnson-Lindenstrauss spaces enjoy (resp. fail) property ($\mathcal{E}$). This is related to a gap in [A. Plichko, Three sequential properties of dual Banach spaces in the weak$^*$ topology, Topology Appl. 190 (2015), 93--98] and allows to answer (consistently) questions of Plichko and Yost.
Gonzalo Martinezcervantes - One of the best experts on this subject based on the ideXlab platform.
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weak sequential properties of johnson lindenstrauss spaces
Journal of Functional Analysis, 2019Co-Authors: Antonio Aviles, Gonzalo Martinezcervantes, Jose RodriguezAbstract:Abstract A Banach space X is said to have Efremov's property ( E ) if every element of the weak⁎-closure of a convex bounded set C ⊆ X ⁎ is the weak⁎-limit of a sequence in C. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of N for which the corresponding Johnson–Lindenstrauss spaces enjoy (resp. fail) property ( E ). This is related to a gap in Plichko (2015) [12] and allows to answer (consistently) questions of Plichko and Yost.
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weak sequential properties of johnson lindenstrauss spaces
arXiv: Functional Analysis, 2018Co-Authors: Antonio Aviles, Gonzalo Martinezcervantes, Jose RodriguezAbstract:A Banach space $X$ is said to have Efremov's property ($\mathcal{E}$) if every element of the weak$^*$-closure of a convex bounded set $C \subseteq X^*$ is the weak$^*$-limit of a sequence in $C$. By assuming the Continuum Hypothesis, we prove that there exist maximal almost disjoint families of infinite subsets of $\mathbb{N}$ for which the corresponding Johnson-Lindenstrauss spaces enjoy (resp. fail) property ($\mathcal{E}$). This is related to a gap in [A. Plichko, Three sequential properties of dual Banach spaces in the weak$^*$ topology, Topology Appl. 190 (2015), 93--98] and allows to answer (consistently) questions of Plichko and Yost.