Generalizations of Cantor's theorem in

Shen, Guozhen

doi:10.1002/malq.201600039

Cited by 10 publications

(21 citation statements)

References 20 publications

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“…If f is a finite-to-one map from x into y and g is a finite-to-one map from y into z, then g • f is a finite-to-one map from x into z. Hence, if a fto b and b fto c then a fto c. The following three facts are Fact 2.8 and Corollaries 2.9 & 2.11 of [18], respectively. Fact 2.8.…”

Section: Preliminariesmentioning

confidence: 89%

“…a a and that fin(a) * seq 1-1 (a) seq(a). The next two facts are Facts 2.13 & 2.14 of [18], respectively. Proof.…”

Section: Some Special Cardinalsmentioning

confidence: 98%

“…dfto S pdfin (a). Our proof is based on ideas in [18], which are originally from [21]. The strategy is as follows:…”

Section: Permutations That Move Power Dedekind Finitely Many Elementsmentioning

confidence: 99%

“…and a, ℵ 0 · a, seq 1-1 (a), seq(a), [a] 3 , or 2 a in the following This question is known as the dual Specker problem and is asked in [24] (cf. also [8, p. 133] or [18,Problem 5.8]). Question 6.6.…”

Section: 2mentioning

confidence: 99%

“…dfto S pdfin (a). Our proof is based on ideas in [7] and [11], which are originally from [15]. The strategy is as follows: Assume towards a contradiction that there exists a set x such that there is a Dedekind finite-to-one function from S(x) into S pdfin (x) and such that there is an injection h from into S(x).…”

Section: Some Notation On Permutationsmentioning

confidence: 99%

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Factorials of Infinite Cardinals in Zf Part I: Zf Results

Shen¹,

Yuan²

2019

J. symb. log.

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For a set x, let S(x) be the set of all permutations of x.We study several aspects of this notion in ZF. The main results are as follows:(1) ZF proves that for all sets x, if S(x) is Dedekind infinite, then there are no finite-to-one maps fromis the set of all permutations of x which move only finitely many elements.(2) ZF proves that for all sets x, the cardinality of S(x) is strictly greater than that of [x] 2 . (3) It is consistent with ZF that there exists an infinite set x such that the cardinality of S(x) is strictly less than that of [x] 3 . (4) It is consistent with ZF that there exists an infinite set x such that there is a finite-to-one map from S(x) into x.2010 Mathematics Subject Classification. Primary 03E10, 03E25.

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Section: Preliminariesmentioning

confidence: 89%

“…a a and that fin(a) * seq 1-1 (a) seq(a). The next two facts are Facts 2.13 & 2.14 of [18], respectively. Proof.…”

Section: Some Special Cardinalsmentioning

confidence: 98%

“…dfto S pdfin (a). Our proof is based on ideas in [18], which are originally from [21]. The strategy is as follows:…”

Section: Permutations That Move Power Dedekind Finitely Many Elementsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Some Notation On Permutationsmentioning

confidence: 99%

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Factorials of Infinite Cardinals in Zf Part I: Zf Results

Shen¹,

Yuan²

2019

J. symb. log.

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Remarks on infinite factorials and cardinal subtraction in ZF$\mathsf{ZF}$

Shen

2021

Mathematical Logic Qtrly

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The factorial of a cardinal a$\mathfrak {a}$, denoted by fraktura!$\mathfrak {a}!$, is the cardinality of the set of all permutations of a set which is of cardinality a$\mathfrak {a}$. We give a condition that makes the cardinal equality fraktura!−fraktura=fraktura!$\mathfrak {a}!-\mathfrak {a}=\mathfrak {a}!$ provable without the axiom of choice. In fact, we prove in ZF$\mathsf {ZF}$ that, for all cardinals a$\mathfrak {a}$, if ℵ0⩽2a$\aleph _0\leqslant 2^\mathfrak {a}$ and there is a permutation without fixed points on a set which is of cardinality a$\mathfrak {a}$, then fraktura!−fraktura=fraktura!$\mathfrak {a}!-\mathfrak {a}=\mathfrak {a}!$.

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The power set and the set of permutations with finitely many non‐fixed points of a set

Shen

2023

Mathematical Logic Qtrly

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For a cardinal a$\mathfrak {a}$, we write scriptSfin(a)$\operatorname{\mathcal {S}_{\text{fin}}}(\mathfrak {a})$ for the cardinality of the set of permutations with finitely many non‐fixed points of a set which is of cardinality a$\mathfrak {a}$. We investigate the relationships between 2fraktura$2^\mathfrak {a}$ and scriptSfin(a)$\operatorname{\mathcal {S}_{\text{fin}}}(\mathfrak {a})$ for an arbitrary infinite cardinal a$\mathfrak {a}$ in ZF$\mathsf {ZF}$ (without the axiom of choice). It is proved in ZF$\mathsf {ZF}$ that 2a≠scriptSfin(a)$2^\mathfrak {a}\ne \operatorname{\mathcal {S}_{\text{fin}}}(\mathfrak {a})$ for all infinite cardinals a$\mathfrak {a}$, and we show that this is the best possible result.

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Generalizations of Cantor's theorem in

Cited by 10 publications

References 20 publications

Factorials of Infinite Cardinals in Zf Part I: Zf Results

Factorials of Infinite Cardinals in Zf Part I: Zf Results

Remarks on infinite factorials and cardinal subtraction in ZF$\mathsf{ZF}$

The power set and the set of permutations with finitely many non‐fixed points of a set

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