We know the Shroeder-Bernstein (SB) theorem can be proved in ZF, while the Dual Schroeder-Bernstein (DSB) can be proved in ZF+AC but not in ZF. Define as ISB the property that whenever there are both an injection and a surjection $X \to Y$, then there must be a bijection too. Trivially ZF+DSB $\implies$ ZF+ISB (because an injection $X\to Y$ allows to define a surjection $Y\to X$, without using AC). Is the converse true? I'm assuming here that ZF $\nRightarrow$ ISB, but I don't know a proof of that either (UPDATE: this is the case, as pointed out by Asaf Karagila in a comment).
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1What do you mean "the converse"? – Asaf Karagila Jan 14 '15 at 21:44
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2As for your last remark, this thread shows that $\sf ISB$ is not provable in $\sf ZF$. – Asaf Karagila Jan 14 '15 at 21:45
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1Does ZF+ISB $\implies$ DSB? Should I edit? – Yaakov Baruch Jan 14 '15 at 21:46
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3This is a difficult question, since we don't know any explicit models where $\sf ZF+\lnot AC+DSB$ hold, or $\sf ISB$ for that matter. At best we have "local examples" (where the general principle fails, but nontrivially holds for some set). My guess is that the implication is false. – Asaf Karagila Jan 14 '15 at 21:51
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What you call $\sf ISB$, is better known as $\sf WPP$ (Weak Partition Principle) and can be formulated as "If there is a surjection from $X$ onto $Y$, then $X$ cannot have a strictly smaller cardinality than $Y$" (alternatively, $|X|\leq^*|Y|\leq|X|\rightarrow |X|=|Y|$ or $|X|\leq^*|X|\rightarrow |X|\nless|Y|$).
In their paper Banaschewski and Moore write that this is still open, and Higasikawa makes no mention of a possible answer in his paper from 1995. I am unaware of any new papers since then that have dealt with these topics, meaning that this problem is probably still open.
Bibliography.
Bernhard Banaschewski, Gregory H. Moore, The dual Cantor-Bernstein theorem and the partition principle, Notre Dame J. Formal Logic 31 (3), (1990), 375–381.
Masasi Higasikawa, Partition principles and infinite sums of cardinal numbers. Notre Dame J. Formal Logic 36 (1995), no. 3, 425–434.
Additional reading:
Asaf Karagila
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