Statement:
\(C(\aleph_{0},< \aleph_{0})\): Every denumerable family of non-empty finite sets has a choice function.
Howard_Rubin_Number: 10
Parameter(s): This form does not depend on parameters
This form's transferability is: Transferable
This form's negation transferability is: Negation Transferable
Article Citations:
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 10 A | \(UT(\aleph_{0},< \aleph_{0},\aleph_{0})\): Theunion of denumerably many pairwise disjoint finite sets is denumerable.Shannon [1988] p 569. |
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| 10 B | Tychonoff's Compactness Theorem for CountableFamilies of Finite \(T_2\) Spaces: The product of a countable family offinite \(T_2\) spaces is compact. Krom [1981]. |
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| 10 C | Every set mapping \(f: X\rightarrow [X]^{WO}\) on aDedekind finite, infinite set \(X\) has arbitrarily large, finite freesubsets. Brunner [1989], notes 22 and 94. |
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| 10 D | In every Hilbert space with a Dedekind finiteorthonormal basis, the closed unit ball is sequentiallycompact. Brunner [1983b] and Note 94. |
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| 10 E | \(PC(\aleph_{0},< \aleph_{0},\infty)\): Everydenumerable family of finite sets has an infinitesubfamily with a choice function. Brunner [1983b]. |
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| 10 F | K\"onig's Lemma: Every \(\omega\) tree has aninfinite chain. Pincus [1972c], Note 21, and Note 35. |
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| 10 G | For all metric spaces \(X\) and \(Y\) and continuousfunctions \(f\) from \(X\) onto \(Y\) such that \(f^{-1}(y)\) is finite for all\(y\in Y\), if \(X\) has a dense Dedekind finite subset, then so does \(Y\).Brunner [1982d] and Note 94. |
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| 10 H | \(\sigma\)-compact, paracompact \(T_2\) spaces areweakly Lindel\"of. Brunner [1982b] and Note 43. |
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| 10 I | Assume \((P,\le)\) is a partial order such that \(P\)is a denumerable union of finite sets and contains a non-empty subset\(Q\) relative to which depths are defined for all \(x\in \) P. If all\(Q\)-strong antichains of \(P\) are finite then for each denumerablefamily \({\Cal D}\) of dense sets there is a \({\Cal D}\) generic filter.Shannon [1990] and Note 47. |
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| 10 J | Let \(S\) be a denumerable set of propositionalsentences over an infinite set of propositional letters (in apropositional language where conjunction and disjunction are operatorson unordered finite sets of sentences). Then \(S\) has a model if andonly if every finite subset of \(S\) has a model. Lolli [1977]. |
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| 10 K | Let \(S\) be a consistent denumerable set of sentencesin a first order language (where conjunction and disjunction areoperators on unordered finite sets of sentences). Then \(S\)has a model. Note 19. |
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| 10 L | Let \(S\) be a denumerable set of sentences in a firstorder language (where conjunction and disjunction areoperators on unordered finite sets of sentences) such that everyfinite subset has a model. Then \(S\) has a model. Note 19. |
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| 10 M | \(PUT(\aleph_0,<\aleph_0,\aleph_0)\): The union ofdenumerably many pairwise disjoint finite sets has a denumerablesubset. |
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| 10 N | \(PUT(WO,<\aleph_0,WO)\): The union of an infinite wellordered set of pairwise disjoint finite sets has an infinite wellordered subset. |
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| 10 O | \(PC(WO,<\aleph_0,\infty)\): Every infinite wellordered family of finite sets has an infinite subset with achoice function. |
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| 10 P | Countable products of finite Hausdorff spacesare Baire. Herrlich\slash Keremedis [1999b] and Note 28. |
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| 10 Q | Countable products of non-empty finite sets arenon-empty. Herrlich\slash Keremedis [1999b]. |
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| 10 R | For every sequence \((X_n)_{n\in\omega}\) of non-emptyfinite sets, \(\Cal P(\bigcup_{n\in\omega}X_n)\) is linearly orderable.Herrlich\slash Keremedis [1999b]. |
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| 10 S | \(UT(\aleph_{0},<\aleph_{0},WO)\): Theunion of denumerably many pairwise disjoint finite sets can be well ordered. ([10 A]\(\to\) [10 S] \(to\) 10) |
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| 10 T | \(PUT(\aleph_0,<\aleph_0,WO)\): The union of adenumerable set of pairwise disjoint finite sets has an infinite well ordered subset.([10 A] \(\to\) [10 T] \(\to\) [10 M]) |
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| 10 U | A countable product of topological spaces all of whichhave finite topologies is second countable. Gutierres [2004]. |
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