Statement:
\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\) is a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\) then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\). See Tarski [1948], p 96, Levy [1964], p. 136.
Howard_Rubin_Number: 43
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:
Tarski-1948: Algebraic and axiomatic aspects of two theorems on sums of cardinals
Levy-1964: The interdependence of certain consequences of the axiom of choice
Book references
Note connections:
Note 54
Implications involving Form 43
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 43 A | Tarski's Lemma: Let \(\{x(n): n\in\omega\}\) be adenumerable collection of subsets of a Boolean algebra \(B\) each ofwhich has a meet in B. Then for any \(a \neq 0\) in \(B\) there is aproper filter \(F\) of \(B\) that decides \(\bigwedge x(n)\) for all \(n\) andhas \(a\in F\). (\(F\) decides \(\bigwedge x(n)\) if either \(\bigwedgex(n)\in F\) or \(t\not\in F\) for some \(t\in x(n)\).) Goldblatt [1985]. |
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| 43 B | Countable Henkin Principle: Let \(\Phi\) be theclass of formulas in a language that includes the language of thesentential calculus. Let \(I\) be a countable set of inferences, and\(\Gamma\) a consistent subset of \(\Phi\) such that \(\Gamma\cup\{\psi\}\)respects \(I\) for each \(\psi\in\Phi\). Then for any \(\phi\in\Phi\), if\(\phi\) is consistent with \(\Gamma\) then \(\Gamma\) has a finitelyconsistent extension that contains \(\phi\) and decides \(I\). Goldblatt [1985] and Note 126. |
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| 43 C | Existence of a generic set: If \({\cal D}\) is adenumerable collection of dense sets in a partial order \((P,\le)\) thenfor all \(p\in P\) there is a \({\cal D}\) generic filter \(G\) with \(p\in G\).Goldblatt [1985] and Note 47. |
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| 43 D | If \({\cal D}\) is a denumerable collection of densesubsets of a partial order \((P,\le)\) then for all \(p\in P\) there isa sequence \(f: \omega\rightarrow P\) with \(f(0) = p\) and \((\foralln\in\omega)(f(n)\ge f(n+1))\) and range(\(f)\cap D\neq\emptyset\) for all \(D \in {\cal D}\). Goldblatt [1985]. |
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| 43 E | The Baire Category Theorem for \v Cech Complete-ISpaces: Every \v Cech complete space is Baire.Goldblatt [1985] and Note 28. |
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| 43 F | Baire Category Theorem for Sequentially CompleteMetric Spaces: Every sequentially complete metric space is Baire.Blair [1977], Brunner [1983c], Goldblatt [1985], and Note 28. |
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| 43 G | Form 106 +Form 154. Brunner [1983c]\cite{1984b}. |
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| 43 H | Every tree in which every element has at leastone successor has a branch. Blass [1979] and Note 21. |
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| 43 I | Form [106 A] +Form 10. Blass [1979]. |
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| 43 J | Baire Category Theorem for Cantor Complete MetricSpaces: Every Cantor complete metric space is Baire.Brunner [1983c] and Note 28. |
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| 43 K | Let \(M\) be a non-empty set partially orderedby \(<\) without \(<\)-minimal elements. Then there is a subset \(C\)of \(M\) which is linearly ordered by \(<\) such that \(C\) has no\(<\)-minimal element. Note 146. |
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| 43 L | \(Z_{\omega}\): If every chain in a partially orderedset \(P\) is finite, then \(P\) contains a maximal element.Tarski [1948], Wolk [1983], and Note 54. |
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| 43 M | If \((X,\rho)\) is a complete metric space and \(\phi:X\rightarrow {\Bbb R}\) is bounded above and upper semi-continuous then inthe Br\o ndsted ordering, (\(x\le y\) iff \(\rho(x,y)\le\phi(y)-\phi(x)\)),there is a maximal element. Brunner [1987a] and Note 38. |
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| 43 N | \(RM1,\aleph_{0}\): The representation theorem formulti-algebras with \(\aleph_{0}\) unary operations: Assume \((A,F)\) is amulti-algebra with \(\aleph_{0}\) unary operations (and no other operations),then there is an algebra \((B,G)\) with \(\aleph_{0}\) unary operations andan equivalence relation \(E\) on \(B\) such that \((B/E,G/E)\) and \((A,F)\) areisomorphic multi-algebras. Note 50 and H\"oft/Howard [1981]. |
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| 43 O | (Depends on \(n\in \omega\), \(n\ge 1\)) \(RM1,n\): Therepresentation theorem for multi-algebras with \(n\) unary operations:Assume \((A,F)\) is a multi-algebra with \(n\) unary operations (and noother operations), then there is an algebra \((B,G)\) with \(n\) unaryoperations and an equivalence relation \(E\) on \(B\) such that \((B/E,G/E)\)and \((A,F)\) are isomorphic multi-algebras. H\"oft/Howard [1981] and Note 50. |
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| 43 P | The ultraproduct of a well founded structure by acountably complete ultrafilter is well founded. Spector [1980]and Note 54. |
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| 43 Q | Matrix\((\omega)\): For every atomless poset \((P,\le)\)and every family \(D=\{D_i: i\in\omega\}\) of dense open subsets of \(P\),there exists a matrix \(C=\{C_i: i\in\omega\}\), \(|\bigcup C|=\omega\) of \(P\)for \(D\). Keremedis [1996b] and Note 132. |
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| 43 R | Any partial order \((A,\le)\) with no infinitedescending chains is well founded, (that is, for all \(X\) such that\(\emptyset\neq X\subseteq A\), \(X\) has a minimal element.)Spector [1980] and Note 54. |
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| 43 S | If \(S\) is a relation on \(A\) such that\((\forall x \in A)( \exists y \in A)(x \mathrel S y)\) and \(x_0\) is anyelement of A, then there is a sequence \(a(1),a(2),\ldots \) ofelements of \(A\) such that \(a(n) \mathrel S a(n+1)\) for all\(n \in \omega \) and such that \(a(0) = x_0\). Notes 101 and 54. |
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| 43 T | A sequentially complete metric space is not meager.Note 28. |
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| 43 U | Baire Category Theorem for Filter Complete MetricSpaces: Every filter complete metric space is Baire, Morillon [1988] and Note 28. |
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| 43 V | Every convex-compact set in a Hausdorff topologicalvector space with a basis of convex open sets is Baire.Fossy/Morillon [1998] and Note 23. |
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| 43 W | Countable products of compact Hausdorff spaces areBaire. Brunner [1983c] and Note 28. |
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| 43 X | Products of compact Hausdorff spaces are Baire.Herrlich/Keremedis [1999] and Note 28. |
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| 43 Y | Products of pseudo-compact spaces are Baire.Herrlich/Keremedis [1999] and Note 28. |
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| 43 Z | Products of countably compact, regular spaces areBaire. Herrlich/Keremedis [1999] and Note 28. |
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| 43 AA | Products of regular-closed spaces are Baire.Herrlich/Keremedis [1999] and Note 28. |
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| 43 AB | Products of \v Cech-complete spaces are Baire.Herrlich/Keremedis [1999] and Note 28. |
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| 43 AC | Products of pseudo-complete spaces are Baire.Herrlich/Keremedis [1999] and Note 28. |
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| 43 AD | Every Frechet complete (pseudo)metric space is Baire.Herrlich/Keremedis [1999a], notes 10 and 28. |
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| 43 AE | Countable products of compact Hausdorf spaces areBaire. Herrlich/Keremedis [1999a] and Note 28. |
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| 43 AF | Every infinite branching poset (a partially orderedset in which each element has at least two lower bounds) has either acountably infinite chain or a countably infinite antichain. Keremedis [1999a]. |
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| 43 AG | Let \(B\) be a Boolean algebra, \(b\) a non-zero elementof \(B\) and \(\{A_i: i\in\omega\}\) a sequence of subsets of \(B\) such thatfor each \(i\in\omega\), \(A_i\) has a supremum \(a_i\). Then there existsa filter \(D\) in \(B\) such that \(b\in D\) and, for each \(i\in\omega\),if \(a_i\in D\), then \(D\cap\ A_i\neq\emptyset\). Bacsich [1972b]. |
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| 43 AH | Ekeland's Variational Principle: If \((E,d)\) isa non-empty complete metric space, \(f: E\to\Bbb R\) is lowersemi-continuous and bounded from below, and \(\epsilon\) is a positivereal number, then there exists \(a\in E\) such that for all \(x\in E\),\(f(a)\le f(x)+\epsilon d(x,a)\). Dodu/Morillon [1999] andNote 28. |
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