Statement:
Finite \(\Leftrightarrow\) Dedekind finite: \(W_{\aleph_{0}}\) Jech [1973b]: \(E(I,IV)\) Howard/Yorke [1989]): Every Dedekind finite set is finite.
Howard_Rubin_Number: 9
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:
Howard-Yorke-1989: Definitions of finite
Book references
The Axiom of Choice, Jech, T., 1973b
Was sind und was sollen die Zollen?, Dedekind, R., 1888
Note connections:
Note 94
Relationships between the different definitions of finite
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 9 A | For all infinite \(x\), if \(x\) is Dedekind finite,then \(\Cal P(x)\) is is Dedekind finite). Notes 8 and 94. |
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| 9 B | \(E(II,IV)\) (Howard/Yorke [1989]): \((\forall x)(x\) is \(T\)-finite if and only if \(x\) is Dedekind finite). Howard/Yorke [1989] and Note 94. |
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| 9 C | The image of a Dedekind finite set is Dedekindfinite. Jech [1973b] p 161 prob 11.2, notes 11 and 94. |
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| 9 D | The union of a Dedekind finite family of finite setsis Dedekind finite. Jech [1973b] p 161 prob 11.4, notes12 and 94. |
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| 9 E | No Dedekind finite set can be mapped onto \(\aleph_0\).(SeeForm 160.) Monro [1975] and Note 94. |
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| 9 F | There is no subset \(A\) of the class of Dedekindfinite cardinals such that \(A\) under the usual cardinalordering is order isomorphic to \(\langle{\Bbb R},\le \rangle\).Tarski [1965], Jech [1973b] p 161 prob 6, and Note 94. |
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| 9 G | Every infinite set has a denumerable subset.Cantor [1895] and G\. Moore [1982] p 9. |
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| 9 H | If \(A\) is Dedekind finite and \(B\) is Dedekind infinite,then \(A\precsim B\). G\. Moore [1982] p 28 and Note 94. |
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| 9 I | If \(\emptyset\in K\) and for all \(A\in K\) and for all\(b\not\in A\), \(A\cup \{b\}\in K\), then every Dedekind finiteset is in \(K\). G\. Moore [1982] p 28 and Note 94. |
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| 9 J | The union of a Dedekind infinite family of non-emptysets is Dedekind infinite. G\. Moore [1982] p 130 and Note 94. |
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| 9 K | If \(A\) is uncountable and \(B\) is countable, then \(|A-B|=|A|\). G\. Moore [1982] p 201. |
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| 9 L | If \(A\) is uncountable and \(B\) is countable, then\(|A\cup B| = |A|\). G\. Moore [1982] p 202. |
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| 9 M | If \(\aleph_0 < |A|\) and \(\aleph_0 = |B|\), then \(\aleph_0< |A - B|\). G\. Moore [1982] p 202. |
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| 9 N | If \(\aleph_0\le^* m\), then \(\aleph_0\le m\).Lindenbaum/Tarski [1926], G\. Moore [1982] p 216,and Note 69. |
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| 9 O | \(m\le \aleph_0\) or \(\aleph_0\le m\).Lindenbaum/Tarski [1926] and G\. Moore [1982] p 216. |
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| 9 P | If \(m + \aleph_0 < n + \aleph_0\), then \(m < n\).Lindenbaum/Tarski [1926] and G\. Moore [1982] p 216. |
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| 9 Q | If \(m + \aleph_0 = n + \aleph_0\), then \(m = n\) or\(m\), \(n \le\aleph_0\). Lindenbaum/Tarski [1926] and G\. Moore [1982] p 216. |
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| 9 R | \(E(III,IV)\): Every Dedekind finite set has a Dedekindfinite power set. Howard/Yorke [1989] and Note 94. |
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| 9 S | \(E(Ia,IV)\): Every Dedekind finite set is amorphous.Howard/Yorke [1989], notes 57 and 94. |
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| 9 T | Restricted Choice: For every infinite set \(X\) thereis an infinite \(Y\subseteq X\) such that the collection of non-emptysubsets of \(Y\) has a choice function. De la Cruz/Di Prisco [1998a]. |
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| 9 U | There exists an A-U compact Hausdorff topology onevery set. Herrlich [1996a] and Note 6. |
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| 9 V | There exist an A-U compact \(T_1\) topology on everyset. Herrlich [1996a] and Note 6. |
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| 9 W | The Alexandroff compactification of a discrete spaceis A-U compact. Herrlich [1996a] and Note 6. |
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| 9 X | In every sequentially compact pseudometric space,every infinite set has an accumulation point. Bentley/Herrlich [1998] and Note 10. |
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| 9 Y | Every infinite tree has a countably infinite chainor a countably infinite antichain. Keremedis [1999a] andNote 21. |
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