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There are (at least) two variants of this question.

Is it possible to modify the axioms of set theory, without arriving at obvious contradiction, in such a way that in a model of the theory there is a set $X$ with empty powerset? (Note that this requires quite serious changes, e. g. $X\subseteq X$ shall not hold)

Category-theoretic version: is it possible to adjoin to an elementary topos an object $X$ with an isomorphism $\Omega^X\cong0$ without collapsing the topos? (The analog of the above parenthetic remark here is that one obviously has to seriously change the structure, maybe even work in categories-without-identity-morphisms)

In response to a comment below I must admit such sort of thing requires at least some motivation. I'll try to provide some, risking even more downvotes - it is less-than-half-serious; but then it is a soft question anyway.

Recently I encountered an inductive construction which goes like this (I omit details). One constructs $X_0\subseteq X_1\subseteq X_2\subseteq\cdots$; one starts with $X_0=\varnothing$; having constructed $X_n$, elements of $X_{n+1}$ are certain pairs $(\chi,x)$ with $\chi$ something that is not relevant here, and $x\subseteq X_n$, which for $n>0$ must satisfy $x\nsubseteq X_{n-1}$.

So out of vanity urge I thought - would not it be great if one would have some such $X_{-1}\subseteq X_0$ which would enable one to remove the above "for $n>0$"? That is, $X_{-1}$ should be such that $X_{-1}\subseteq X_0$ and moreover $x\subseteq\varnothing$ with $x\nsubseteq X_{-1}$ would not exclude $x=\varnothing$, and moreover $\{x\subseteq X_{-1}|\text{whatever}\}$ should be $\varnothing$.

As you see initially I did not dare to add one more requirement that this $X$ should satisfy: it must be subset of the empty set :D

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    $\begingroup$ Sorry, but I am going to say it: Why?? $\endgroup$
    – Matthias Ludewig
    Commented Mar 15, 2017 at 9:03
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    $\begingroup$ how do you even make sense of $\Omega^X$ in a category without identities ? you will also need to deal with the problem of the empty subobject and the singleton map... In the end the object you'll produce will have nothing to do with a topos anyway. so unless you do something like adding a $X \rightarrow 0$ with no maps to $X$ I don't really see a way out.. $\endgroup$
    – Simon Henry
    Commented Mar 15, 2017 at 9:29
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    $\begingroup$ I wouldn't be surprised if you could do something of the sort in a category or set theory whose internal logic, instead of being classical or intuitionist, is given by (Girard's) linear logic. I think in a $*$-autonomous category, e.g., given by Chu's construction, you can get an object $X=\top$ that satisfies $Z^X=0$ for every $Z$ (because this is just $Z\otimes 0 =0$). (contd.) $\endgroup$
    – Gro-Tsen
    Commented Mar 15, 2017 at 10:52
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    $\begingroup$ I don't think getting rid of identities in the category will help you; not only the identity $X\to X$ but also the unique morphism $0\to X$ from the initial object will give a subobject of $X$ that ought to be classified by a morphism $1\to\Omega^X$. And if you get rid of the initial object 0, then you need to rephrase the desired isomorphismn $\Omega^X\cong0$. $\endgroup$
    – Andreas Blass
    Commented Mar 15, 2017 at 16:00
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    $\begingroup$ I feel obliged to post this here $\endgroup$
    – fosco
    Commented Mar 17, 2017 at 13:56

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