# Is there exists a set which contains as elements all its finite subsets?

Is there exists a set $$A$$ such that if $$F$$ is a finite subset of A, then $$F\in A$$? Under ZFC axiom system. I guess the answer may not, but I’m not sure yet. Any help or hint will be appreciated, thanks!

• Your question's title and your actual question seem to be different. In your question you ask whether there exists a set $\;A\;$ which contains as elements all its finite subsets. That's different from what is written in the title, so: which one do you want and what have you done so far about that? – DonAntonio Mar 26 at 15:03
• @DonAntonio Yes, Antonio, I mean it contains as elements all its finite subsets. I edit title right now. Thanks! – Landau Mar 26 at 15:08

There are such sets, an example is $$V_\omega$$, the set of all sets which are hereditarily finite, meaning that $$x\in V_\omega$$ if and only if $$x$$ is finite, all elements of $$x$$ are finite, all elements of all elements of $$x$$ are finite, and so on.
If $$y\subseteq V_\omega$$ is finite, this means that for every $$x\in y$$ we have $$x\in V_\omega$$, that is elements of $$y$$ are hereditarily finite sets. Since $$y$$ is finite this means that $$y$$ itself is an hereditarily finite set, hence $$y\in V_\omega$$.
More generally let $$\mathcal P_{\mathrm{fin}}(X)$$ denote the set of all finite subsets of $$X$$ and let $$\mathcal P^n_{\mathrm{fin}}(X)=\mathcal P_{\mathrm{fin}}(\mathcal P^{n-1}_{\mathrm{fin}}(X))$$, where $$\mathcal P^0_{\mathrm{fin}}(X)=X$$ by convention. Then $$\bigcup_{n\in\Bbb N}\mathcal P^n_{\mathrm{fin}}(X)$$ is a set with the desired property regardless of $$X$$.
• Thanks, Codenotti. By the last paragraph, if we replace “finite subsets” with “subsets whose cardinal are $\kappa$, the same claim is also true. Am I right? – Landau Mar 26 at 15:29