## Works by Guozhen Shen

8 found
Order:
1. Factorials of infinite cardinals in zf part I: Zf results.Guozhen Shen & Jiachen Yuan - 2020 - Journal of Symbolic Logic 85 (1):224-243.
For a set x, let ${\cal S}\left$ be the set of all permutations of x. We prove in ZF several results concerning this notion, among which are the following: For all sets x such that ${\cal S}\left$ is Dedekind infinite, $\left| {{{\cal S}_{{\rm{fin}}}}\left} \right| < \left| {{\cal S}\left} \right|$ and there are no finite-to-one functions from ${\cal S}\left$ into ${{\cal S}_{{\rm{fin}}}}\left$, where ${{\cal S}_{{\rm{fin}}}}\left$ denotes the set of all permutations of x which move only finitely many elements. For all sets (...)

Export citation

Bookmark   8 citations
2. Factorials of infinite cardinals in zf part II: Consistency results.Guozhen Shen & Jiachen Yuan - 2020 - Journal of Symbolic Logic 85 (1):244-270.
For a set x, let S(x) be the set of all permutations of x. We prove by the method of permutation models that the following statements are consistent with ZF: (1) There is an infinite set x such that |p(x)|<|S(x)|<|seq^1-1(x)|<|seq(x)|, where p(x) is the powerset of x, seq(x) is the set of all finite sequences of elements of x, and seq^1-1(x) is the set of all finite sequences of elements of x without repetition. (2) There is a Dedekind infinite set (...)

Export citation

Bookmark   7 citations
3. The power set and the set of permutations with finitely many non‐fixed points of a set.Guozhen Shen - 2023 - Mathematical Logic Quarterly 69 (1):40-45.
For a cardinal, we write for the cardinality of the set of permutations with finitely many non‐fixed points of a set which is of cardinality. We investigate the relationships between and for an arbitrary infinite cardinal in (without the axiom of choice). It is proved in that for all infinite cardinals, and we show that this is the best possible result.
No categories

Export citation

Bookmark   1 citation
4. On a cardinal inequality in ZF$\mathsf {ZF}$.Guozhen Shen - forthcoming - Mathematical Logic Quarterly.
It is proved in (without the axiom of choice) that for all infinite cardinals and all natural numbers, where is the cardinality of the set of permutations with exactly non‐fixed points of a set which is of cardinality.
No categories

Export citation

Bookmark   1 citation
5. A Generalized Cantor Theorem In.Yinhe Peng & Guozhen Shen - 2024 - Journal of Symbolic Logic 89 (1):204-210.
It is proved in $\mathsf {ZF}$ (without the axiom of choice) that, for all infinite sets M, there are no surjections from $\omega \times M$ onto $\operatorname {\mathrm {\mathscr {P}}}(M)$.

Export citation

Bookmark   1 citation
6. A Note on Strongly Almost Disjoint Families.Guozhen Shen - 2020 - Notre Dame Journal of Formal Logic 61 (2):227-231.
For a set M, let |M| denote the cardinality of M. A family F is called strongly almost disjoint if there is an n∈ω such that |A∩B|<n for any two distinct elements A, B of F. It is shown in ZF (without the axiom of choice) that, for all infinite sets M and all strongly almost disjoint families F⊆P(M), |F|<|P(M)| and there are no finite-to-one functions from P(M) into F, where P(M) denotes the power set of M.

Export citation

Bookmark   1 citation
7. Cantor’s Theorem May Fail for Finitary Partitions.Guozhen Shen - forthcoming - Journal of Symbolic Logic:1-18.
A partition is finitary if all its members are finite. For a set A, $\mathscr {B}(A)$ denotes the set of all finitary partitions of A. It is shown consistent with $\mathsf {ZF}$ (without the axiom of choice) that there exist an infinite set A and a surjection from A onto $\mathscr {B}(A)$. On the other hand, we prove in $\mathsf {ZF}$ some theorems concerning $\mathscr {B}(A)$ for infinite sets A, among which are the following: (1) If there is a finitary (...)
8. (1 other version)Remarks on infinite factorials and cardinal subtraction in ZF$\mathsf{ZF}$.Guozhen Shen - 2022 - Mathematical Logic Quarterly 68 (1):67-73.