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  1. Is there any infinite set which is Dedekind finite and weakly Dedekind finite?

  2. If $X$ is a weakly Dedekind finite amorphous set, can we show that $\mathcal P(X)$, the power set of $X$, is also weakly Dedekind finite?

If someone know, please tell (or hint) them.

Thank you.

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    $\begingroup$ Could you please add the definitions of these concepts? Also, there seem to be some spelling errors $\endgroup$
    – Yemon Choi
    Commented May 26, 2016 at 13:26

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Recall Kuratowski's theorem:

$X$ can be mapped onto $\omega$ if and only if there is an injection from $\omega$ into $\mathcal P(X)$.

And also, recall that:

$X$ is infinite if and only if $\mathcal P(X)$ can be mapped onto $\omega$.

So to answer the first question, in Cohen's first model every infinite set can be mapped onto $\omega$, so there are no infinite weakly Dedekind finite sets; on the other hand, if $A$ is amorphous, then $A$ cannot be mapped onto $\omega$, therefore its power set is Dedekind finite.

The second question follows easily from the second quoted statement.

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  • $\begingroup$ Oh, you made my day. Thank you so much :) $\endgroup$
    – Jaruwat Rodbanjong
    Commented May 26, 2016 at 14:08
  • $\begingroup$ Sorry, but i want to know whether P(X) is "weakly" Dedekind finite or not, not Dedekind finite. $\endgroup$
    – Jaruwat Rodbanjong
    Commented May 26, 2016 at 16:44
  • $\begingroup$ Weakly Dedekind finite sets, if my memory serves me right (and you provided no definition) are Dedekind finite sets that cannot be mapped onto $\omega$. $\endgroup$
    – Asaf Karagila ♦
    Commented May 26, 2016 at 16:46
  • $\begingroup$ In my class, a set X is weakly Dedekind finite if there is no surjection from X onto $/omega$. $\endgroup$
    – Jaruwat Rodbanjong
    Commented May 26, 2016 at 16:51
  • $\begingroup$ Yes, as I wrote. And as I wrote in my answer, if $X$ is infinite (Dedekind finite or not), then $\mathcal P(X)$ can be mapped onto $\omega$. $\endgroup$
    – Asaf Karagila ♦
    Commented May 26, 2016 at 16:54

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