Every real number is $Δ^1_2$ in some $ω$-model of ZFC\P. I am looking for analogous statements for $Δ^1_3$ definability and transitive models, where the situation is more subtle.
What is the consistency strength of ZFC + there is a real $r$ such that $r$ is not $Δ^1_3$ in any transitive model of ZFC\P ?
Note: A reasonable guess is that this is equivalent to $Δ^1_2$ determinacy and thus equiconsistent with a Woodin cardinal; see below.
If every real has a sharp, is every real $r$ with $0^\# ∉ L[r]$ $Δ^1_3$ (and even computable from $0^\#$) in some transitive model of ZFC + $V=L[0^\#]$?
Note: One can also generalize this question to larger canonical inner models. Also, $0^\#$ is recognized as such in every transitive model of ZFC\P containing it.
Assume $\mathbf{Δ^1_2}$ determinacy. If $T$ is a c.e. subtheory of $M_1$ (the minimal inner model with a Woodin cardinal), and $r$ is a real such that $M_1(r) ⊨ M_1^\# \text{ does not exist}$, is $r$ $Δ^1_3$ in some countably-coded inner model of $T$?
Note: Here (with all reals having sharps), an inner model is countably-coded iff for some real $x$, it is definable from $x$ in $L[x]$; this is used to avoid potential undefinability of truth in inner models.
An example I can prove is that every constructible real number is $Δ^1_3$ in some transitive model of $\text{ZFC\P} + V = L$. (Proof sketch: The $α$th real in $L$ is $Δ^1_3$ in the minimal transitive model of ZFC\P + $V=L$ + there is a c.t.m. of ZFC\P containing $α$.)
Also, for the $L[0^\#]$ question, the answer is yes if $r$ is generic over $L$: Using $Σ^1_2$ correctness of $L[r]$, existence of a counterexample $r$ would be a $Σ_2^L$ statement, and thus some such $r$ would exist in every $\operatorname{Col}(ω, κ)$ generic extension of $L$ with $L_κ ≺_{Σ_2} L$, and hence some such $r$ would be computable from $0^\#$, which is a contradiction.
There is a series of analogous results:
- Every real number is computable in some nonstandard model in the following sense: For every consistent c.e. theory $T⊢\text{EFA}$, there are $Σ^0_1$ $φ$ and $ψ$ such that $∀S⊂ω \, ∃M⊨(T ∧ ∀n (φ(n)⇔ ¬ψ(n))) \, S =\{n∈ℕ: M⊨φ(n)\}$.
- Provably in $Π^1_1\text{-CA}_0$, every real number is $Δ^1_2$ in some $ω$-model of $T$ for every c.e. $T⊢\text{ATR}_0$ such that every real is in some $ω$-model of $T$. (For the proof, note that some counterexample would be $Δ^1_2$, as true $Σ^1_2$ statements have $Δ^1_2$ witnesses.)
- Every real number is $Δ^1_4$ in some closed-under-sharps transitive model of $T$ for every c.e. $T⊢\text{ZFC\P}$ such that every real is in some transitive model of $T$ closed under $M_1^\#$. (If I recall correctly, $\mathbf{Δ^1_2}$ determinacy suffices to ensure that true $Σ^1_4$ statements have $Δ^1_4$ witnesses.) For example, using this, assuming AD in $L(ℝ)$, every real number is $Δ^1_4$ in some inner model of ZFC + projective determinacy.
- (conjectured) Assuming a measurable Woodin cardinal $κ$ and the Continuum Hypothesis, every $κ$-universally Baire set $A$ is $Δ^2_1$ (lightface) in some transitive model of $\text{ZF} + \text{AD}^+$ of containing all the reals (and similarly for stronger c.e theories such that every $κ$-uB set $A$ is in some transitive model of the theory).
The common theme is that a model may think that a real $r$ is canonically definable, but if the closure level for the model is sufficiently below the definition level, then $r$ can be arbitrary.
For $Δ^1_1$, $Δ^1_3$, and other $Δ^1_{2n+1}$ levels, I think the right form is to limit the permitted reals. For example, one can ask whether for a real $x$ being $Δ^1_1$ definable in some $ω$-model of ZFC\P is equivalent to $\operatorname{HYP}^x ⊨ \text{“} 0^\text{HJ} \text{ does not exist”}$ ($\operatorname{HYP}^x$ means hyperarithmetic in $x$; $0^\text{HJ}$ is (or corresponds to) Kleene's $\mathcal{O}$).
Regarding $Δ^1_3$, assuming $\mathbf{Δ^1_2}$ determinacy, there is a real $r$ ($M_1^\#$ works) such that a transitive model of ZFC\P containing $r$ cannot falsely claim that a real is $Δ^1_3$ in a countable ordinal. (And thus $M_1^\#$ cannot be $Δ^1_3$ in a countable ordinal in a transitive model of ZFC\P.) The reason is that:
(1) a transitive model cannot overestimate $\mathrm{HOD}_t^{L(t)}$ where $t$ is a Turing degree (the HOD gets $t$ as a parameter, and $L(t)$ gets a real of degree $t$; the HOD can be defined using definability in $L_α(t)$)
(2) under ZFC\P, every $Δ^1_3$ in a countable ordinal real is in $\mathrm{HOD}_t^{L(t)}$ for a cone of Turing degrees, and
(3) under $\mathbf{Δ^1_2}$ determinacy, for a cone of Turing degrees $t$, $HOD_t^{L(t)}$ contains precisely those reals that are $Δ^1_3$ in a countable ordinal.
