Just random thinking.
ω = ∃ x ( ∅ ∈ x ∧ ∀ y ∈ x ( y ∪ { y } ∈ x ) ) {\displaystyle \omega =\exists x(\emptyset \in x\land \forall y\in x(y\cup \{y\}\in x))}
ω + 1 = ∃ x ( ∃ y ( ∅ ∈ y ∧ ∀ z ∈ y ( z ∪ { z } ∈ y ) ) ∧ y ∪ { y } ∈ x ) {\displaystyle \omega +1=\exists x(\exists y(\emptyset \in y\land \forall z\in y(z\cup \{z\}\in y))\land y\cup \{y\}\in x)}
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