Language: Logic of single-sorted first-order predicates

Primitives: $ =; in, emptyset, $ 0 where the last two are constants.

To define: $ set (x) equiv_ {df} exists y (x in y) $

Axioms: ID +

**1. Low extensionality:**

$ non-empty (x) land forall z (z in x leftrightarrow z in y) to x = y $

**2. Foundation:** $ exists y (y in x) to exists y in x ( no exists c (c in y earth c in x)) $

**3. Class comprehension:** if $ phi $ is a formula in which $ x $ is not free, so all closures of: $$ exists x forall y (y in x leftrightarrow set (y) land phi) $$; are axioms.

To define: $ x = {y | phi } equiv_ {df} forall y (y in x leftrightarrow set (y) land phi) $

**4. Empty sets:** $ 0 neq emptyset land forall x (empty (x) leftrightarrow x = 0 lor x = emptyset) $

**5. Twinning:** $ forall sets x, y (set ( {x, y })) $

**6. Sub-assemblies:** $ set (x) land y subseteq x to set (y) $

**7. Power:** $ set (x) to set ( mathcal P (x)) $

**8. Union:** $ set (x) to set ( bigcup (x)) $

To define: $ x = y ^ o equiv_ {df} x = o lor (not empty (x) land forall z in TC (x) (empty (z) to z = o)) $

Or $ TC $ means transitive closure.

**9. Modified size limit:** $ | x ^ o | <| V ^ o | set (x ^ o) $

Or $ V ^ o = {x ^ o | set (x ^ o) } $

To define:

$ x ^ 0 cong y emptyset equiv_ {df} TC (x ^ 0) is $$ in $–$ isomorphic to TC (y ^ emptyset) $

Or $ TC $ means transitive closure.

**10. Describability:** if $ phi ^ 0 $ is a parameterless formula in which all variables are set to 0, and if $ phi ^ emptyset $ is the formula obtained from $ phi ^ 0 $ simply replacing each occurrence of the symbol 0 with the symbol $ emptyset $then:

$ {x ^ 0 | phi ^ 0 } cong {x ^ emptyset | phi ^ emptyset } to {x ^ 0 | phi ^ 0 }, {x ^ emptyset | phi ^ emptyset } text {are sets} $

, is an axiom.

Question: would this theory prove the coherence of "ZFC + there is an inaccessible set"?