1
$\begingroup$

For every set $X$ there cannot be a partition on $X$ of a larger size than the set $\iota``X$ of all singleton subsets of $X$. Formally: $$\begin{align} \forall X \forall P: P \text { is a partition on } X \to \neg [&\exists f (f: \iota``X \rightarrowtail P ) \land \\ & \not \exists g (g: P \rightarrowtail \iota ``X)] \end{align} $$

Where:

$$\begin{align} P \text { is a partition on } X \iff &\forall k \in P (k \neq \emptyset) \land \bigcup P = X \ \land \\ & \forall a ,b \in P (a \neq b \to a \cap b = \emptyset) \end{align}$$

Where $`` \rightarrowtail "$ stand for injections.

What forms of Choice this principle with the other axioms of $\sf ZF$ can prove?

Same question but with regards Stratified $\sf ZF$?

Stratified $\sf ZF$ is the theory axiomatized by the stratified axioms of $\sf ZF$, where stratified is defined after Quine as in $\sf NF$.

Improve this question
$\endgroup$
10
  • $\begingroup$ So, every partition of $X$ is either incomparable or injects to $X$? $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 10, 2023 at 17:59
  • $\begingroup$ @AsafKaragila, Yes! In the context of ZF. $\endgroup$
    – Zuhair Al-Johar
    Commented Feb 10, 2023 at 18:31
  • $\begingroup$ This is known as the Weak Partition Principle. $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 10, 2023 at 21:41
  • $\begingroup$ @AsafKaragila, what kind of known forms of choice this entails? Especially from the stratified axioms of $\sf ZF$? $\endgroup$
    – Zuhair Al-Johar
    Commented Feb 10, 2023 at 22:10
  • $\begingroup$ @ZuhairAl-Johar IIRC, it is open whether it implies the dual CB theorem (which in turns it is open whether the dual CB theorem implies PP and it is open whether PP implies AC) in ZF $\endgroup$
    – Holo
    Commented Feb 10, 2023 at 23:40

0

Reset to default

You must log in to answer this question.