Definition. A hypno scene is "transfinite" if it contains transgender themes and is definable in the first order language of set theory. A set is a "victim" if there is a transfinite hypno scene in which the set gains a new thought. A girl is "inductively closed" if she is not a victim.
The conventional intuition about inductively closed girls is that they "have had all interesting hypnotic inductions performed on them." It is unclear a priori whether every girl is inductively closed, or even if it is consistent relative to ZFC that there is an inductively closed girl. Our main results are a new forcefem notion that adds a girl who is not inductively closed (that is, she is a victim), as well as a forcefem iteration after which a victim becomes inductively closed.
Definition. A condition in the trance forcefemming T is a pair (t, s), where t is an element of the Cohen forcefem Add(ω, 1) adding a single countable girl and t forces that s is an initial segment of the decision tree on any generic girl containing t. We say (t', s') < (t, s) if t' < t and t forces s' to end extend s.
Theorem. If G is generic for T, then V[G] models "there is a girl and she is a victim." Moreover, in any forcefem extension of V[G] that is (ω_1+1)-strategically closed, the generic victim remains a girl.
Proof. We note that the set {(t, s) : t forces s to exclude some decision} is dense, so the generic girl will have a valid target for future hypnotic induction. For the technical part of the theorem, we direct the reader to a note of Gitik on a note of Steel on a note of Shelah on indestructible femininity in projections of the Cohen forcefem. QED
Now we will forcefem a victim such that in the outer model she is inductively closed. Since our generic girl is countable, she only has countably many thoughts (and therefore can only be transfinitely inducted upon countably many times). Enumerate all her vulnerable thoughts by {t_i : i < ω}.
Definition. For each i < ω, let \odot{S_i} be the forcefemming which decides t_i relative to all previously forced thoughts. Then the surfacing forcefem, S, is the finite-support iteration of the \odot{S_i}.
The authors direct readers to [Cummings09] for further information on forcefemming a thought on a victim.
Lemma. The surfacing forcefem is (ω_1+1)-strategically closed and ccc.
Theorem. After surfacing a victim, she becomes inductively closed.
Proof. By the lemma and previous theorem, the victim remains a girl. Trivially, the surfacing is proper. The reader can verify by density arguments that each thought in the ground model is decided. Since the forcefemming is proper, no new thoughts are added in any intermediate extension, and therefore none are present in the final model either. QED.
Corollary. Since being inductively closed is upwards absolute, the forcing T*S adds a generically indestructible girl.
For further areas of research, the authors note that not much is known about inductive closure in the context of ZFC + ANBG (the axiom of nonbinary genders). The first author would like to acknowledge organizers and attendees of the 2026 Berkley Involuntary Feminization Conference for their valuable input.
