Question: Is the following theory conservative towards ZFC? (If not, what is your strength?)
Language: $ ∈ $. $ j $ (unary function symbol)
axioms:
1. ZFC (without separation and replacement for formulas using $ j $).
2. (scheme) $ j $ is a non-trivial elementary embedding $ (V, ∈) → (V, ∈) $,
3. (scheme) $ S∩φ ≡ {s∈S: φ (s) } $ exists whenever $ φ $ is a formula with parameters and a free variable and quantity $ S $ can be defined from a quantity $ T $ With $ T = j (T) $,
Note that (3) implies the existence of $ S & # 39; ∩φ $ for all $ S & # 39; $ With $ | S & # 39; | ≤ | S | $ for some $ S $ as above. Thus, (3) simply confirms the complete separation scheme for quantities that are not too large, including all definable (allowable) $ j $) puts.
background
An important phenomenon in mathematical logic is that if we expand a theory with new symbols (and sometimes new types) and add meaningful new axioms, the new theory will sometimes be conservative of the original; and such correspondences often provide important structural insights into both theories. A relevant special case here is that the new theory is apparently stronger and covers part of the structure of a stronger theory, but a key ingredient is missing.
Taking various sensible formalizations (which imply substitutes for $ j $Formulas), existence of a non-trivial elementary embedding $ V → M $ ($ M $ transitive) is synonymous with the existence of a measurable cardinal, and $ V = M $ is not compatible with ZFC.
However, (1) + (2) is conservative towards ZFC: add Skolem functions for $ V $ and add $ ω $ Constant symbols for imperceptible atomic numbers, use the compactness to find a model, take the Skolem torso of the imperceptible elements and add a non-trivial order-preserving injection between the imperceptible elements, which then relates to the desired one elementary embedding stretches.
Such a model (see above) may be unfounded, but we can try to do well in small quantities. Any countable ZFC model $ M $ has a nontrivial elementary end extension $ N $; out of elementary, $ N $ is also a top extension that is for everyone $ α∈ mathrm {Ord} ^ M $. $ V_α ^ M $ and $ V_α ^ N $ have the same elements. And maybe a way to integrate $ j $ There will be a positive answer to the question.
Also under (1) – (3), $ {s: s = j (s) } $ (under & # 39; ∈ & # 39;) is an actual elementary substructure of $ (V, ∈) $, but it doesn't exist as a set unless its cardinality is $ n $– great for everyone $ n $,
That being said, there is a rich hierarchy theory based on how & # 39; close & # 39; $ j $ is to $ V $, For example (see this question) if (3) is replaced by the axiom of the critical point (i.e. the smallest atomic number that has been shifted) $ j $ exists), the resulting theory is conservative towards ZFC + {there is $ n $-inffable cardinal}$ _ {n∈ℕ} $,