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\makeatother
\title				{Variations on $\Delta^1_1$ Determinacy and $\aleph_{\omega_1}$}
\thanks				{Presented at the\,12th Panhellenic Logic Symposium -- Crete, June 2019.}
\author				{Ramez L. Sami}
\subjclass[2010]	{Primary: 03E60;\enskip{}Secondary: 03E15, 03E10.}
\address			{Department of Mathematics. Université Paris-Diderot, 75205 Paris, Cedex 13, France.}
\email				{sami@univ-paris-diderot.fr}
\begin{document}

\newgeometry{left=26mm, right=31mm
\begin{abstract}
    \setlength\parindent{0pt}
    We consider a weaker form of $\Delta^1_1$ Turing determinacy.

    Let $2 \leqslant \rho < \omega_{1}^{\scriptscriptstyle{\mathsf{CK}}}$, $\textrm{Weak-Turing-Det}_{\rho}(\Delta^1_1)$ is the statement:

    \hspace*{1.2em}\emph{Every $\Delta^1_1$ set of reals cofinal in the Turing degrees contains two Turing distinct $\Delta^0_\rho$-equivalent reals.}

    We show in $\mathsf{ZF}^-$:

        \hspace*{1.2em}$\textrm{Weak-Turing-Det}_{\rho}(\Delta^1_1)$ implies for every $\nu < \omega_{1}^{\scriptscriptstyle{\mathsf{CK}}}$ there is a transitive model: $M \models \mathsf{ZF}^- + \textup{"}\aleph_\nu \textup{ exists"}$.

    As a corollary:
    \begin{itemize}[leftmargin=1.2em]
        \item[]If every cofinal $\Delta^1_1$ set of Turing degrees contains both a degree and its jump, then for every $\nu < \omega_{1}^{\scriptscriptstyle{\mathsf{CK}}}$, there is a transitive model: $M \models \mathsf{ZF}^- + \textup{"}\aleph_\nu \textup{ exists"}$.
    \end{itemize}

    \begin{itemize}[leftmargin=1em, topsep=2pt, label={$\scriptscriptstyle\bullet$}]
        \item{}With a simple proof, this improves upon a well-known result of Harvey Friedman on the strength of Borel determinacy (though not assessed level-by-level).

        \item{}Invoking Tony Martin's proof of Borel determinacy, $\textrm{Weak-Turing-Det}_{\rho}(\Delta^1_1)$ implies $\Delta^1_1$ determinacy.

        \item{}We show further that, assuming $\Delta^1_1$ Turing determinacy, or Borel Turing determinacy, as needed:\\
            --\enskip{}Every cofinal $\Sigma^1_1$ set of Turing degrees contains a ``hyp-Turing cone'': $\{x \in \mathcal{D} \mid d_0 \leqslant_T x \leqslant_h d_0 \}$.\\
            --\enskip{}For a sequence $(A_{k})_{k < \omega}$ of analytic sets of Turing degrees, cofinal in $\mathcal{D}$, $\bigcap_{k} A_{k}$ is cofinal in $\mathcal{D}$.
    \end{itemize}
\end{abstract}
\restoregeometry
\vspace*{-10mm}
\iffalse
\begin{flushright}
 \vspace*{-2.5\baselineskip}
  	[Presented at the \textbf{12th Panhellenic Logic Symposium} -- June 2019]
	\bigskip{}
\end{flushright}
\fi
\pdfbookmark[1]{Variations on ∆₁¹ Determinacy and ℵ𝜔₁}{}
\maketitle

\section*{Introduction}

A most important result in the study of infinite games is Harvey Friedman's~\cite{Friedman:Higher_ST}, where it is shown that a proof of determinacy, for Borel games, would require \Aleph1 iterations of the power set operation \textemdash\xspace and this is precisely what Tony Martin used in his landmark proof~\cite{Martin:Borel_Det}.

Our focus here is on the Turing determinacy results of \cite{Friedman:Higher_ST}, concentrating on the theory \ensuremath{\mathsf{ZF}^-}\xspace rather than Zermelo's \ensuremath{\mathsf{Z}}\xspace. In the \DDelta11 realm, Friedman essentially shows that the determinacy of Turing closed \DDelta11 games [henceforth \TDet{\DDelta11}] implies the consistency of the theories \ZFmAleph{\nu}, for all recursive ordinals $\nu$. Friedman does produce a level-by-level analysis entailing, e.g., that the determinacy of Turing closed \SSigma{0}{n+\smash6} games implies the consistency of \ZFmAleph{n}.%
    \footnote{\,Improved by Martin to \SSigma{0}{n+\smash5}.}%
\textsuperscript{,}%
    \footnote{\,In \cite{Montalban-Shore} Montalbán and Shore considerably refine the analysis of the proof theoretic strength of \Det{\Gamma}, for~classes $\Gamma$, where $\PPi03 \subseteq \Gamma \subseteq \DDelta04$.}

Importantly, it was further observed by Friedman (unpublished) that these results extend to produce transitive models, rather than just consistency results. See Martin's forthcoming \cite{Martin:Det_Book} for details, see also Van~Wesep's~\cite{Van_Wesep}.
\smallskip{}

We forego in this paper the level-by-level analysis to provide, in §\ref{sec:Transitive-models}, a simple proof of the existence of transitive models of \ensuremath{\mathsf{ZF}^-}\xspace with uncountable cardinals, from \TDet{\DDelta11}. In so doing, we show that the full force of Turing determinacy isn't needed. The main result is Theorem\:\ref{thm:Models-with-Alephs}, with a simply stated corollary.
\smallskip{}

For context, by \textbf{Martin's Lemma} (see \ref{subsec:Turing-determinacy}), \TDet{\DDelta11} is equivalent to:\\
$\scriptstyle\bullet$\:\:\emph{Every cofinal \DDelta11 set of Turing degrees contains a cone of degrees} -- i.e., a set \Set{x \in \ensuremath{\mathcal{D}}\xspace}{d_0 \mathrel{\leqslant_{T}} x}.

\begin{theorem*}[\ref{thm:Models-with-Alephs}]
    Let $2 \le \rho < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, and assume every \DDelta11 set of reals, cofinal in the Turing degrees, contains two Turing distinct, \DDelta{0}{\rho}-equivalent reals. For every $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, there is a transitive model: \ModelZFmAleph{M}{\nu}.
\end{theorem*}

\begin{corollary*}[\ref{cor:Models-with-Alephs}]
    If every cofinal \DDelta11 set of Turing degrees contains both a degree and its jump, then for every $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, there is a transitive model: \ModelZFmAleph{M}{\nu}.
\end{corollary*}

In §\ref{sec:Delta-Det-Properties-Sigma} several results are derived, showing that \TDet{\DDelta11} imparts weak determinacy properties to the class \SSigma11, such as [\ref{thm:Hyp-Turing-Det-Sigma}]\,:\\
$\scriptscriptstyle\bullet$\;\:\emph{Every cofinal \SSigma11 set of degrees includes a set \Set{x \in \ensuremath{\mathcal{D}}\xspace}{d_0 \mathrel{\leqslant_{T}} x \And x \mathrel{\leqslant_{h}} d_0}, for some~$d_0 \in \ensuremath{\mathcal{D}}\xspace$}.\\
Or, from Borel Turing determinacy, [\ref{thm:Intersection of Sigma cofinal}]\,:\\
$\scriptscriptstyle\bullet$\;\emph{If $(A_{k})_{k < \omega}$ is a sequence of analytic sets of degrees each cofinal in \ensuremath{\mathcal{D}}\xspace, then \Medcap{k}{A} is cofinal in \ensuremath{\mathcal{D}}\xspace}.\medskip{}

I wish to thank Tony Martin for inspiring conversations on the present results. He provided the argument for Remark\:\ref{rem:Too-Weak}, below, and observed that my first proof of Theorem\:\ref{thm:Hyp-Det-Sigma} (used for an early version of the main result) was needlessly complex. Parts of §\ref{sec:Delta-Det-Properties-Sigma} go back to the author's dissertation \cite{Sami:Dissertation}, it is a pleasure to acknowledge Robert Solovay's direction.

\section{Preliminaries and Notation}

The effective descriptive set theory we shall need, as well as basic hyperarithmetic theory, is from Moschovakis' \cite{Moschovakis:DST}, whose terminology and notation we follow. For the theory of admissible sets, we refer to Barwise's~\cite{Barwise:Admissible_Sets}. Standard facts about the \ensuremath{\mathbb{L}}\xspace\nobreakdash-hierarchy are used without explicit mention: see Devlin's \cite{Devlin:Constructibility}, or Van~Wesep's~\cite{Van_Wesep}.
\smallskip{}

$\ensuremath{\mathcal{N}}\xspace = \omega^\omega = \ensuremath{\mathbb{N}}\xspace^\ensuremath{\mathbb{N}}\xspace$ denotes Baire's space (the set of \emph{reals}), and \ensuremath{\mathcal{D}}\xspace the set of Turing degrees. Subsets of \ensuremath{\mathcal{D}}\xspace shall be identified with the corresponding (Turing closed) sets of reals. $\mathrel{\leqslant_{T}}$, $\mathrel{\leqslant_{h}}$, and $\equiv_T$, $\equiv_h$ denote, respectively, Turing and hyperarithmetic (or \DDelta11) reducibility, and equivalence.

\subsection{Turing determinacy.}
\label{subsec:Turing-determinacy}
A set of reals $A \subseteq \ensuremath{\mathcal{N}}\xspace$ is said to be \emph{cofinal in the \textup[Turing\textup] degrees} if for~all $x \in \ensuremath{\mathcal{N}}\xspace$ there is $y \in A$, such that $x \mathrel{\leqslant_{T}} y$. For $c \in \ensuremath{\mathcal{N}}\xspace$, the \emph{Turing cone} with vertex $c$ is the set $\Cone(c) = \Set{x \in \ensuremath{\mathcal{N}}\xspace}{c \mathrel{\leqslant_{T}} x}$.
For a class of sets of reals $\Gamma$, \Det{\Gamma} is the statement that infinite games $G_\omega(A)$ where $A \in \Gamma$ are determined, whereas \TDet{\Gamma} stands for the determinacy of games $G_\omega(A)$ restricted to Turing closed sets $A \in \Gamma$. Recall the following easy yet central:

\begin{namedthm*}{Martin's Lemma \cite{Martin:Reduction_Princ}}
For a Turing closed set $A \subseteq \ensuremath{\mathcal{N}}\xspace$, the infinite game $G_\omega(A)$ is determined if, and only if, $A$ or its complement contains a cone of Turing degrees.
\end{namedthm*}

\subsection{The ambient theories.}
Our base theory is \ensuremath{\mathsf{ZF}^-}\xspace, \Ax{Zermelo-Fraenkel} set theory stripped of the Power Set axiom.%
    \footnote{\,All implicit instances of Choice, here, are \ensuremath{\mathsf{ZF}^-}\xspace-provable.}
\ensuremath{\mathcal{N}}\xspace or \ensuremath{\mathcal{D}}\xspace may be proper classes in this context, yet speaking of their ``subsets'' (\DDelta11, \SSigma11, Borel or analytic) can be handled as usual, as these sets are codable by integers, or reals. Amenities such as \Aleph1 or \L{\omega_1} aren't available but, since our results here are global (i.e.,~\DDelta11) rather than local, the reader may use instead the more comfortable $\ensuremath{\mathsf{ZF}^-}\xspace + "\ensuremath{\mathcal{P}}\xspace^2(\omega) \textup{ exists}"$.

\ensuremath{\mathsf{KP}_{\infty}}\xspace denotes the theory \Ax{Kripke-Platek + Infinity}. Much of the argumentation below takes place inside $\omega$-model\xspace{s} of \ensuremath{\mathsf{KP}_{\infty}}\xspace \textemdash\xspace familiarity with their properties is assumed.

\subsection{Constructibility and condensation.}
\label{subsec:Constructibility}
For an ordinal $\lambda > 0$, and $X \subseteq \L{\lambda}$, $\hull{\lambda}(X)$ denotes the set of elements of \L{\lambda} definable from parameters in $X$, and $\thull{\lambda}(X)$ its transitive collapse. For $X = \O$, one simply writes \hull{\lambda} and \thull{\lambda}. Gödel's Condensation Lemma is the relevant tool here.
Note that, since $\L{\lambda} = \thull{\lambda}(\lambda) = \hull{\lambda}(\lambda)$, all elements of \L{\lambda} are definable in \L{\lambda} from ordinal parameters.

\subsection{Reflection.}
The following reflection principle will be used a few times, to make for shorter proofs.%
    \footnote{\,Longer ones can always be produced using \DDelta11 selection + \SSigma11 separation.}
A property $\Phi(X)$ of subsets $X \subseteq \ensuremath{\mathcal{N}}\xspace$ is said to be ``\PPi11 \emph{on} \SSigma11'' if, for any \SSigma11 relation $U \subseteq \ensuremath{\mathcal{N}}\xspace \times \ensuremath{\mathcal{N}}\xspace$, the set \Set{x \in \ensuremath{\mathcal{N}}\xspace}{\Phi(U_x)} is \PPi11.

\begin{theorem*}
    Let $\Phi(X)$ be a \PPi11 on \SSigma11 property. For any \SSigma11 set $S \subseteq \ensuremath{\mathcal{N}}\xspace$ such that $\Phi(S)$ there is a \DDelta11 set $D \supseteq S$ such that $\Phi(D)$.
\end{theorem*}

\begin{proof}
See Kechris'~\cite[§35.10]{Kechris:Classical_DST} for a boldface version, easily transcribed to lightface.
\end{proof}

\section{Weak Turing Determinacy}
\label{sec:Weak-Turing-determinacy}

Examining what's needed to derive the existence of transitive models from Turing determinacy hypotheses, it is possible to isolate a seemingly weaker statement. For $1 \le \rho < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, let $x \equiv_\rho y$ denote \DDelta{0}{\rho}-equivalence on \ensuremath{\mathcal{N}}\xspace, that is: $x \in \DDelta{0}{\rho}(y) \;\&\; y \in \DDelta{0}{\rho}(x)$. \,$\equiv_1$ is just Turing equivalence.

\begin{definition}
    For a class $\Gamma$, and $2 \le \rho < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, \WTDet{\rho}{\Gamma} is the statement:
     \begin{itemize}[leftmargin=\parindent, topsep=0pt]
     		\item[]\emph{For every set of reals $A \in \Gamma$ cofinal in the degrees, there are two Turing distinct $x, y \in A$ such that $x \equiv_\rho y$.}
     \end{itemize}

\end{definition}

For any recursive $\rho \ge 2$, \WTDet{\rho}{\DDelta11} will suffice to derive the existence of transitive models. The property lifts from \DDelta11 to \SSigma11 \textemdash\xspace note that it is, \emph{a priori}, asymmetric.

\begin{theorem}
\label{thm:WTuringDelta->WTuringSigma}
    Let $2 \le \rho < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$. \WTDet{\rho}{\DDelta11} implies \WTDet{\rho}{\SSigma11}.
\end{theorem}

\begin{proof}
Assume \WTDet{\rho}{\DDelta11}. Let $S \in \SSigma11$ and suppose there are no Turing distinct $x, y \in S$ such that $x \equiv_\rho y$, that is
\[
        \All{x,y} (x, y \in S \And x \equiv_\rho y \Implies x \equiv_T y).
\]
This is a statement $\Phi(S)$, where $\Phi(X)$ is a \PPi11 on \SSigma11 property. Reflection yields a \DDelta11 set $D \supseteq S$ such that $\Phi(D)$. By \WTDet{\rho}{\DDelta11}, $D$ is not cofinal in the degrees; \emph{a fortiori}, $S$ isn't either.\qedhere
\end{proof}

\begin{remark}
\label{rem:Too-Weak}

One may be tempted to substitute for \WTDet{\rho}{\DDelta11} a simpler hypothesis:
\\
\emph{Every \DDelta11 set of reals cofinal in the degrees contains two Turing distinct reals $x, y$ such that $x \equiv_h y$}.
\\
It turns out to be too weak and, indeed, provable in Analysis. (Tony Martin, private communication: Building on his \cite{Martin:Friedman_Conj}, he shows that every uncountable \DDelta11 set of reals contains two Turing distinct reals, in every hyperdegree $\ge$ Kleene's \ensuremath{\mathcal{O}}\xspace.) The weaker statement does suffice however, when asserted about the class \SSigma11, see Theorem \ref{thm:Models-with-Alephs-from-Sigma}, below.
\end{remark}

\section{Transitive Models from Weak Turing Determinacy}
\label{sec:Transitive-models}

We now state the main result, and a simple special case. The proof is postponed towards the end of the present section.

\begin{theorem}
\label{thm:Models-with-Alephs}
    Let $2 \le \rho < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, and assume \WTDet{\rho}{\DDelta11}. For every $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, there is a transitive model: \ModelZFmAleph{M}{\nu}.
\end{theorem}

\begin{corollary}
\label{cor:Models-with-Alephs}
    If every cofinal \DDelta11 set of Turing degrees contains both a degree and its jump, then for every $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, there is a transitive model: \ModelZFmAleph{M}{\nu}.\qed
\end{corollary}

\SideTitle{Term models.}

Given a complete%
    \footnote{\,Complete extensions are always meant to be consistent.}
theory $U \supseteq \ensuremath{\mathsf{KP}_{\infty}~+~(\mathbb{V}=\mathbb{L})}\xspace$, one constructs its term model. To be specific, owing to the presence of the axiom $\ensuremath{\mathbb{V}}\xspace = \ensuremath{\mathbb{L}}\xspace$, to every formula $\psi(v)$ is associated $\xbar\psi(v)$ such that ${U \mathrel{{|}\mkern-3mu{\relbar}} \Exists{v}\psi(v) \SEquiv \Exists{!v}\xbar\psi(v)}$. Just take for $\xbar\psi(v)$ the formula $\psi(v) \land (\All{w <_\ensuremath{\mathbb{L}}\xspace v}) \neg \psi(w)$.

Let $(\phi_n(v))_{n < \omega}$ be a recursive in $U$ enumeration of the formulas $\phi(v)$, in the single free variable~$v$, having $U \mathrel{{|}\mkern-3mu{\relbar}} \Exists{!v} \phi(v)$. Using, as metalinguistic device, $(\iota v) \phi(v)$ for ``\emph{the unique $v$ such that $\phi(v)$}'', set:
\[
        M_U = \Set{n \in \omega}{\All{i < n,}~U \mathrel{{|}\mkern-3mu{\relbar}} (\iota v)\phi_n \neq (\iota v)\phi_i},
\]
and define on $M_U$ the relation $\in_U$\,:
\[
        {\hspace{-7.7eM}} m \in_U n \Equiv U \mathrel{{|}\mkern-3mu{\relbar}} (\iota v)\phi_m \in (\iota v)\phi_n.
\]
$(M_U, \in_U)$ is a prime model of $U$ and, $U$ being complete, $(M_U, \in_U)$ is recursive in $U$. Using the canonical enumeration $\omega \to M_U$, substitute $\omega$ for $M_U$ and remap $\in_U$ accordingly. The resulting model $\M{U} = (\omega, \in^\M{U})$ shall be called the \emph{term model of} $U$. The function $U \mapsto \M{U}$ is recursive.

Whenever \M{U} is an $\omega$-model\xspace, we say that $a \subseteq \omega$ is realized in \M{U} if there is $\mathring{a} \in \omega$ such that $a = \Set{k \in \omega}{k^\M{U} \in^\M{U} \mathring{a}}$. We state, for later reference, a couple of standard facts.

\begin{proposition}
\label{prop:Realized-reals}
    Let $U$ be as above. If \M{U} is an $\omega$-model\xspace, and $a \subseteq \omega$ realized in \M{U}, then:
    \begin{enumerate}
        \item{}For all $x \mathrel{\leqslant_{h}} a$\,\emph: $x$ is realized in \M{U}.
        \item{}$a \mathrel{\leqslant_{T}} U$. Hence $U$ is not realized in \M{U}, lest the jump $U'$ be realized in \M{U}, and ${U' \mathrel{\leqslant_{T}} U}$.\qed
    \end{enumerate}
\end{proposition}

Note that if $U = \Th(\L{\alpha})$, where $\alpha$ is admissible, then \M{U} is a copy of \hull{\alpha}. Hence $\M{U} \cong \L{\beta}$, for some $\beta \le \alpha$. The following easy proposition is quite familiar.

\begin{proposition}
\label{prop:Countable-admissible-term-model}
    Assume $\ensuremath{\mathbb{V}}\xspace = \ensuremath{\mathbb{L}}\xspace$. For cofinally many countable admissible $\alpha$'s, $\L{\alpha} = \hull{\alpha}$, equivalently, $\M{\Th(\L{\alpha})} \cong \L{\alpha}$.
\end{proposition}

\begin{proof}
Suppose not. Let $\lambda$ be the sup of the admissible $\alpha$'s having $\L{\alpha} = \hull{\alpha}$, and let $\kappa > \lambda$ be the first admissible such that $\lambda$ is countable in \L{\kappa}. Since $\lambda$ is definable and countable in \L{\kappa}, $\lambda \cup \{\lambda\} \subseteq \hull{\kappa}$. It follows readily that $\L{\kappa} = \thull{\kappa} = \hull{\kappa}$\,: a contradiction.
\end{proof}

\SideTitle{Cardinality in the constructible levels.}

Set theory within the confines of \L{\lambda}, $\lambda$ an arbitrary limit ordinal, imposes some contortions. For technical convenience, the notion of cardinal needs to be slightly twisted \textemdash\xspace for a time only.

\begin{definition}\ \par
    \begin{enumerate}
        \item{}For an ordinal $\alpha$, $\Card(\alpha) = \min_{\xi \le \alpha}(\textit{there is a surjection \,} \xi \to \alpha)$.
        \item{}$\alpha$ is a cardinal if $\alpha = \Card(\alpha)$.
        \item{}$\Card_\lambda \subseteq \L{\lambda}$ is the class of infinite cardinals as computed in \L{\lambda}.
    \end{enumerate}
\end{definition}

\subsection{\kern-.6em}
\label{subsec:Biject}
Note that, for any limit $\lambda$, from a surjection $g \colon \gamma \to \alpha$ in \L{\lambda}, one can extract $a \subseteq \gamma$ and $r \subseteq \gamma \times \gamma$ such that ${g \mathord{\upharpoonright} a} \colon (a,r) \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} (\alpha, \in)$, and both $(a,r)$, $g \mathord{\upharpoonright} a$ are in \L{\lambda}. Further, if $\lambda$ is admissible, in \L{\lambda} the altered notion of cardinality coincides with the standard one.

\begin{convention}
\label{label:Convention}
    For simplicity's sake, the assertion ``\emph{\Aleph{\nu} exists in \L{\lambda}}'' should be understood as:

    \emph{There is an isomorphism $\nu+1 \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} I$, where $I$ is an initial segment of\/ $\Card_\lambda$}.

    \noindent{}Its negation is equivalent in \ensuremath{\mathsf{KP}_{\infty}}\xspace to: \emph{There is $\kappa \le \nu$ such that $\Card_\lambda \cong \kappa$}. The notation \AlephL{\nu}{\lambda} carries the obvious meaning.
\end{convention}

We need the following, presumably ``folklore'', result. A proof is provided in the Appendix, for lack of a convenient reference.

\begin{proposition}
\label{prop:Folklore}
    For $\lambda$ a limit ordinal, $\L{\lambda} \models "\*\mu > \omega \textup{ is a successor cardinal}"$ implies $\L{\mu} \models \ensuremath{\mathsf{ZF}^-}\xspace$.
\end{proposition}

\SideTitle{The theories \texorpdfstring{\T{\nu}}{T\_ν}.}

For $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, pick $e_\nu \in \omega$ a recursive index for a wellordering $<_{e_\nu}$ of a subset of $\omega$, of length~$\nu$. Using $e_\nu$, statements about $\nu$ can tentatively be expressed in \ensuremath{\mathsf{KP}_{\infty}}\xspace. In an $\omega$-model\xspace \ensuremath{\mathcal{M}}\xspace of \ensuremath{\mathsf{KP}_{\infty}}\xspace, the~truth of such statements is independent of the choice of $e_\nu$. Indeed, $<_{e_\nu}$ is realized in \ensuremath{\mathcal{M}}\xspace, and its realization is isomorphic in \ensuremath{\mathcal{M}}\xspace to the \ensuremath{\mathcal{M}}\xspace\nobreakdash-ordinal of order-type $\nu$, to be denoted $\nu^\ensuremath{\mathcal{M}}\xspace$.

\begin{definition}
    For $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, \T{\nu} is the theory
    \[
        \ensuremath{\mathsf{KP}_{\infty}~+~(\mathbb{V}=\mathbb{L})}\xspace + "\text{For all limit $\lambda$,\;\Aleph{\nu+1} doesn't exist in \L{\lambda}}".
    \]
\end{definition}

This definition is clearly lacking: a recursive index $e_\nu$ coding the ordinal $\nu$ is not made explicit. This is immaterial, as we shall be interested only in $\omega$-model\xspace{s} of \T{\nu}. They possess the following rigidity property.

\keep{4}
\begin{lemma}
\label{lem:Rigidity-Property}
    Let $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, and $\M1,\,\M2$ be $\omega$-model\xspace{s} of \T{\nu}. Let $u \in \mathbb{O}\mathrm{n}^\M1$, and $w,\,w^\low* \in \mathbb{O}\mathrm{n}^\M2$, for any two isomorphisms $f \colon \LL{u}{\M1} \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} \LL{w}{\M2}$ and $f{^\ast\mkern-2mu} \colon \LL{u}{\M1} \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} \LL{w^\low*}{\M2}$, $f = f{^\ast\mkern-2mu}$.
\end{lemma}

\begin{proof}
By an easy reduction, it suffices to prove this for $u$, a limit \M1-ordinal.

Set $\ensuremath{\mathcal{C}}\xspace_u = \Set{c <^\M1 \! u}{\M1 \models \*c \in \Card_\*u}$. The relevant point here is that $\ensuremath{\mathcal{C}}\xspace_u$ is wellordered by $<^\M1$. Indeed, since $\M1 \models "\Aleph{\nu+1} \text{ doesn't exist in } \L{\*u}"$, there is $\kappa \le \nu+1$ such that $\M1 \models {\Card_\*u \cong \kappa^\M1}$ (see~\ref{label:Convention}), and consequently $(\ensuremath{\mathcal{C}}\xspace_u, <^\M1) \cong (\kappa, \in)$.

We check first that $f$ and $f{^\ast\mkern-2mu}$ agree on the \M1-ordinals $o <^\M1\! u$. Set $\kappa_u(o) = \Card(o)$, evaluated in \LL{u}{\M1}. Evidently $f(o)=f{^\ast\mkern-2mu}(o)$ for $o \le \omega$ in \M1, next show by transfinite induction on $c \in \ensuremath{\mathcal{C}}\xspace_u$\,:
\[
        \text{for all \:} o <^\M1 \! u : \enskip{} \kappa_u(o) \le^\M1 \! c \Implies f(o) = f{^\ast\mkern-2mu}(o).
\]
The inductive hypothesis yields, for all $o <^\M1 \! c$, $f(o) = f{^\ast\mkern-2mu}(o)$, hence $f(c) = f{^\ast\mkern-2mu}(c)$. Let now $o <^\M1 \! u$ have $\kappa_u(o) = c$. Inside \LL{u}{\M1}, $(o, \in)$ is isomorphic to an ordering $s = (a, r)$, where $a \subseteq c$ and $r \subseteq c \times c$, (see\:\ref{subsec:Biject}). Since $f$ and $f{^\ast\mkern-2mu}$ agree on the \M1-ordinals up to $c$, one readily gets $f(s) = f{^\ast\mkern-2mu}(s)$. In \M2, the ordering $f(s)$ is isomorphic to both the ordinals $f(o)$ and $f{^\ast\mkern-2mu}(o)$, hence $f(o) = f{^\ast\mkern-2mu}(o)$.

This entails $w = w^\low*$. Now for $x \in \LL{u}{\M1}$, $x$ is definable in \LL{u}{\M1} from \M1-ordinals (see \ref{subsec:Constructibility}), thus $f(x)$ and $f{^\ast\mkern-2mu}(x)$ satisfy in \LL{w}{\M2} the same definition from equal parameters, hence ${f(x) = f{^\ast\mkern-2mu}(x)}$.
\end{proof}

\SideTitle{Pseudo-wellfounded models.}

A relation $\mathord{\vartriangleleft} \subseteq \omega \times \omega$ is said to be \emph{pseudo-wellfounded} if every nonempty $\DDelta11(\vartriangleleft)$ subset of $\omega$ has a $\vartriangleleft$-minimal element. By the usual computation, this is a \SSigma11 property.

\begin{definition}
    For $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, \S{\nu} is the set of theories:
    \[
        \S{\nu} = \Set{U}{\text{$U$ is a complete extension of \T{\nu}, and \M{U} is pseudo-wellfounded} }.
    \]
\end{definition}

The first clause in the definition of \S{\nu} is arithmetical, while the ``pseudo-wellfounded'' clause can be written as:
\[
        \All{x \mathrel{\leqslant_{h}} \M{U}}(x \neq \O \Implies \Exists{k \,{\in}\, x}\All{m \,{\in}\, x}\neg(m \in^\M{U}\! k)).
\]
Since \M{U} is uniformly $\DDelta11(U)$, \S{\nu} is \SSigma11. Further, for $U \in \S{\nu}$, \M{U} is an $\omega$-model\xspace. The sets \S{\nu} play the central role in the proof. They are sparse, in the following sense.

\begin{proposition}
    For $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, no two distinct members of \S{\nu} have the same hyperdegree.
\end{proposition}

\begin{proof}
Let $U_1,\,U_2 \in \S{\nu}$ have $U_1 \equiv_h U_2$, and let $\M1,\,\M2$ stand for $\M{U_1},\,\M{U_2}$. We will obtain $U_1 = U_2$ by showing $\M1 \cong \M2$. Define a relation between `ordinals' $u \in \M1$ and $w \in \M2$.%
\begin{align*}
        u \simeq w 		& \Equiv \LL{u}{\M1} \cong \LL{w}{\M2}.\\
\intertext{%
Set $I_1 = \Dom(\simeq)$ and $I_2 = \Img(\simeq)$. $I_1$ and $I_2$ are initial segments of $\mathbb{O}\mathrm{n}^\M1$ and $\mathbb{O}\mathrm{n}^\M2$, respectively. Using Lemma \ref{lem:Rigidity-Property}, the relation ``$u \simeq w$'' defines a bijection $I_1 \to I_2$ which is, indeed, the restriction of an isomorphism $F \colon \raisebox{-.15ex}{\scalebox{1.5}{\ensuremath\cup}}_{u \in I_1}\LL{u}{\M1} \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} \raisebox{-.15ex}{\scalebox{1.5}{\ensuremath\cup}}_{w \in I_2} \LL{w}{\M2}$. By the same lemma, it can be expressed as:
}%
        u \simeq w		& \Equiv \Exists{f} ( f \colon \LL{u}{\M1} \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} \LL{w}{\M2} )\\
						& \Equiv \Exists{!f}( f \colon \LL{u}{\M1} \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} \LL{w}{\M2} ).
\intertext{%
The expression on the last RHS reads $\Exists{!f} \ensuremath{\mathcal{I}}\xspace(f, U_1, u, U_2, w)$, where \ensuremath{\mathcal{I}}\xspace is a \DDelta11 predicate, thus
}%
        u \simeq w		& \Equiv \Exists{f \mathrel{\leqslant_{h}} U_1 \oplus U_2}(f \colon \LL{u}{\M1} \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} \LL{w}{\M2}).
\end{align*}
By the usual computation, the relation ``$u \simeq w$'' is $\DDelta11(U_1 \oplus U_2)$ $[\:= \DDelta11(U_1) = \DDelta11(U_2)]$. Consequently, $I_1$ and $I_2$ are also $\DDelta11(U_1)$ $[\:= \DDelta11(U_2)]$. $\M1,\,\M2$ being pseudo-wellfounded, $\mathbb{O}\mathrm{n}^\M1 - I_1$ and $\mathbb{O}\mathrm{n}^\M2 - I_2$ each, if nonempty, has a minimum. Call $m_1,\,m_2$ the respective potential minima, and consider the~cases:

\keep{2}
\begin{itemize}[leftmargin=16pt, itemsep=3pt,label={--}]
    \item{}$\mathbb{O}\mathrm{n}^\M1 - I_1$ and $\mathbb{O}\mathrm{n}^\M2 - I_2$ both are nonempty. This isn't possible, as there would be an isomorphism $\LL{m_1}{\M1} \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} \LL{m_2}{\M2}$, entailing $m_1 \in I_1$ and $m_2 \in I_2$.

    \item{}$I_1 = \mathbb{O}\mathrm{n}^\M1$ and $\mathbb{O}\mathrm{n}^\M2 - I_2 \ne \O$. Here $\M1 = \raisebox{-.15ex}{\scalebox{1.5}{\ensuremath\cup}}_{u \in I_1} \LL{u}{\M1} \cong \LL{m_2}{\M2}$, and $U_1$ being now the \nohyphens{theory} of \LL{m_2}{\M2}, is realized in \M2. By hypothesis $U_2 \equiv_h U_1$ hence, by Proposition \ref{prop:Realized-reals}(1), $U_2$ is realized in \M2 (that's \M{U_2}). This contradicts (2) of the same proposition.

    \item{}The third case, symmetric of the last one, is equally impossible.

    \item{}The remaining case: $I_1 = \mathbb{O}\mathrm{n}^\M1$ and $I_2 = \mathbb{O}\mathrm{n}^\M2$. Here $\M1 = \raisebox{-.15ex}{\scalebox{1.5}{\ensuremath\cup}}_{u \in I_1} \LL{u}{\M1}$ and ${\M2 = \raisebox{-.15ex}{\scalebox{1.5}{\ensuremath\cup}}_{w \in I_2} \LL{w}{\M2}}$, thus $F \colon \M1 \stackrel{\mkern-5mu\longsim}{\smash\rightarrow} \M2$ is the desired isomorphism.\qedhere
\end{itemize}

\end{proof}

\begin{proof}[\textbf{Proof of Theorem\:\ref{thm:Models-with-Alephs}}]
We may assume $\ensuremath{\mathbb{V}}\xspace = \ensuremath{\mathbb{L}}\xspace$. Fix $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$.

\textbf{Claim.} There is a limit ordinal $\lambda$, such that: \Aleph{\nu+1} exists in \L{\lambda}.

Suppose no such $\lambda$ exists. It follows that for all admissible $\alpha > \omega$, $\L{\alpha} \models \T{\nu}$.
This entails that the \SSigma11 set \S{\nu} is cofinal in the degrees: indeed, using Proposition~\ref{prop:Countable-admissible-term-model}, given $x \subseteq \omega$ there is an ${\alpha > \omega}$, admissible, such that ${x \in \L{\alpha}}$ and $\M{\Th(\L{\alpha})} \cong \L{\alpha}$. Thus $x \mathrel{\leqslant_{T}} \Th(\L{\alpha})$ and, \M{\Th(\L{\alpha})} being wellfounded, $\Th(\L{\alpha}) \in \S{\nu}$.

Invoking now \WTDet{\rho}{\DDelta11} and Proposition~\ref{thm:WTuringDelta->WTuringSigma}, \WTDet{\rho}{\SSigma11} holds. Hence, there are distinct $U_1,\,U_2 \in \S{\nu}$ such that $U_1 \equiv_\rho U_2$, contradicting the previous proposition.\qed\textsubscript{\textbf{Claim}}
\smallskip{}

Let $\lambda$ be as claimed, and set $\mu = \AlephL{\nu+1}{\lambda}$. In \L{\lambda}, $\mu$ is a successor cardinal hence, by Proposition~\ref{prop:Folklore}, $\L{\mu} \models \ensuremath{\mathsf{ZF}^-}\xspace$. Further for all $\xi \le \nu$, $\AlephL{\xi}{\lambda} \in \L{\mu}$, and \AlephL{\xi}{\lambda} is an \Lcard{\mu} (now in the usual sense), hence ${\L{\mu} \models \ZFmAleph{\nu}}$.
\end{proof}

Note the following byproduct of the previous proposition, and the proof just given (substituting $U_1 \equiv_h U_2$ for $U_1 \equiv_\rho U_2$, in the proof) \textemdash\xspace in contradistinction to Remark\:\ref{rem:Too-Weak}.

\begin{theorem}
\label{thm:Models-with-Alephs-from-Sigma}
    Assume every \SSigma11 set of reals, cofinal in the degrees, contains two Turing distinct reals $x, y$, such that $x \equiv_h y$. For every $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, there is a transitive model:\,$M \models \ZFmAleph{\nu}$.\qed
\end{theorem}

An easy consequence of the main result: \WTDet{\rho}{\DDelta11} implies full \DDelta11 determinacy. The proof goes through Martin's Borel determinacy theorem: no direct argument is known for this sort of implication \textemdash\xspace apparently first observed by Friedman for \TDet{\DDelta11}.

\begin{theorem}
\label{thm:Full-Det}
    For $2 \le \rho < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$, \WTDet{\rho}{\DDelta11} implies \Det{\DDelta11}.
\end{theorem}

\begin{proof}
	Assume \WTDet{\rho}{\DDelta11}. Let $A \subseteq \ensuremath{\mathcal{N}}\xspace$ be \DDelta11, say $A \in \SSigma{0}{\nu}$ where $\nu < \omega_{1}^{\scriptscriptstyle\mathsf{CK}}$. Applying Theorem\:\ref{thm:Models-with-Alephs}, there is a transitive \ModelZFmAleph{M}{\nu}. Invoking (non-optimally) Martin's main result from \cite{Martin:Borel_Det} inside $M$, \SSigma{0}{\nu} games are determined. The statement ``\emph{the game $G_\omega(A)$ is determined}\,'' is \SSigma12. By Mostowki's absoluteness theorem, being true in $M$, it holds in the universe: $G_\omega(A)$ is indeed determined.
\end{proof}

\section{\texorpdfstring{\DDelta11 Determinacy and Properties of \SSigma11 Sets}{∆₁¹ Determinacy and Properties of Σ₁¹ Sets}}
\label{sec:Delta-Det-Properties-Sigma}

We proceed now to show that \DDelta11 determinacy imparts weak determinacy properties to the class~\SSigma11. In view of Theorem\:\ref{thm:Full-Det}, there is no point, here, in working from weaker hypotheses.

\begin{definition}
    The hyp-Turing cone with vertex $d \in \ensuremath{\mathcal{D}}\xspace$ is the set of degrees
    \[
        \Cone_{h}(d) = \Set{x \in \ensuremath{\mathcal{D}}\xspace}{d \mathrel{\leqslant_{T}} x \And x \mathrel{\leqslant_{h}} d} = \Cone(d) \cap \DDelta11(d).
    \]
    \HTDet{\Gamma} is the statement: \emph{Every cofinal set of degrees $A \in \Gamma$ contains a hyp-Turing~cone}.
\end{definition}

\keep{5}
\begin{theorem}
\label{thm:Sigma-seq-degrees}
    Assume \TDet{\DDelta11}. If $(S_{k})_{k < \omega}$ is a \SSigma11 sequence of sets of Turing degrees, each cofinal in \ensuremath{\mathcal{D}}\xspace, then $\Medcap{k}{S} \neq \O$ \textemdash\xspace and, indeed, \Medcap{k}{S} contains a hyp-Turing cone.
\end{theorem}

\begin{proof}
Let the $S_{k}$'s be given as the sections of a \SSigma11 relation $S \subseteq \omega \times \ensuremath{\mathcal{N}}\xspace$ and assume \Medcap{k}{S} contains no hyp-Turing cone, that is:%
\[
        \All{x \in \ensuremath{\mathcal{N}}\xspace} \Exists{y \mathrel{\leqslant_{h}} x} (x \mathrel{\leqslant_{T}} y \And y \notin \Medcap{k}{S}).
\]
This is a statement $\Phi(S)$, where $\Phi(X)$ is a \PPi11 on \SSigma11 property. Reflection yields a \DDelta11 relation $D \supseteq S$ such that $\Phi(D)$. By shrinking $D$, if needed, we may ensure that its sections $D_{k}$ are Turing closed, preserving $\Phi(D)$. Now, $D_{k} \supseteq S_{k}$ and \Medcap{k}{D} contains no hyp-Turing cone. A contradiction ensues using \TDet{\DDelta11} and Martin's Lemma: each $D_{k}$, being cofinal in the degrees, contains a Turing cone, hence so does the intersection \Medcap{k}{D}.
\end{proof}

The converse is immediate: if \TDet{\DDelta11} fails, using Martin's Lemma there is a \DDelta11 set ${A \subseteq \ensuremath{\mathcal{D}}\xspace}$, such that $A$ and $\mathord{\sim} A$ are both cofinal in \ensuremath{\mathcal{D}}\xspace, and the \DDelta11 ``sequence'' $\langle A, \mathord{\sim} A \rangle$ has empty intersection.
Relativizing \ref{thm:Sigma-seq-degrees}, one readily gets:

\begin{corollary}
\label{thm:Intersection of Sigma cofinal}
    Assume Borel Turing determinacy. If $(A_{k})_{k < \omega}$ is a sequence of analytic sets of Turing degrees, each cofinal in \ensuremath{\mathcal{D}}\xspace, then \Medcap{k}{A} is cofinal in \ensuremath{\mathcal{D}}\xspace.\qed
\end{corollary}

An interesting degenerate case of \ref{thm:Sigma-seq-degrees}, where the sequence $(S_{k})_{k < \omega}$ consists of a single \SSigma11 term.

\begin{theorem}
\label{thm:Hyp-Turing-Det-Sigma}
    \TDet{\DDelta11} implies \HTDet{\SSigma11}.\qed
\end{theorem}

In view of Theorem\:\ref{thm:Full-Det}, the implication is, of course, an equivalence. A similar result holds for full determinacy.

\begin{definition}
    For a game $G_\omega(A)$, a strategy $\sigma$ for Player \pI is called a hyp-winning strategy if $\All{\tau \mathrel{\leqslant_{h}} \sigma}(\sigma{*}\tau \in A)$, i.e., applying $\sigma$, Player \pI wins against any $\DDelta11(\sigma)$ sequence of moves by Player~\pII.
\end{definition}

\begin{theorem}
\label{thm:Hyp-Det-Sigma}
    Assume \Det{\DDelta11}. For all \SSigma11 sets $S$, one of the following holds for the game $G_\omega(S)$.
    \begin{enumerate}
        \item{}Player \pI has a hyp-winning strategy.
        \item{}Player \pII has a winning strategy.
    \end{enumerate}
\end{theorem}

\begin{proof}
Let $S$ be \SSigma11, and assume Player \pI has no hyp-winning strategy for $G_\omega(S)$\,: $\All{\sigma} \Exists{\tau \mathrel{\leqslant_{h}} \sigma}$ $(\sigma{*}\tau \notin S)$. As above, Reflection yields a \DDelta11 set $D \supseteq S$ such that Player \pI has no hyp-winning strategy for $G_\omega(D)$, hence no winning strategy. Invoking \Det{\DDelta11}, Player \pII has a winning strategy for $G_\omega(D)$ which is, \emph{a fortiori}, winning for $G_\omega(S)$.
\end{proof}

\section*{Appendix}
\stepcounter{section}

The point of the present section is to sketch a proof of Proposition~\ref{prop:Folklore} (\ref{prop:Folklore2}, below), without dissecting the \ensuremath{\mathbb{L}}\xspace construction \textemdash\xspace albeit with a recourse to admissible sets. Finer results are certainly known. \ensuremath{\mathcal{F}}\xspace is the set of formulas, $\ensuremath{\mathcal{F}}\xspace \in \L{\omega+1}$. $\models_\L{\alpha}$ is the satisfaction relation for \L{\alpha},
\[
        {\models_\L{\alpha}} (\phi,\vec{s}) \LEquiv \phi \in \ensuremath{\mathcal{F}}\xspace \And \vec{s} \in \LL{\alpha}{<\omega} \And \L{\alpha} \models \phi[\vec{s}].
\]
Apart from the classic Condensation Lemma (see \ref{subsec:Constructibility}), we shall need the following familiar result.

\emph{For a limit ordinal $\lambda > \omega$, and $\beta < \lambda$, $\models_\L{\beta} \in \L{\lambda}$}. See \cite[§7.1]{Van_Wesep}.

\begin{notation*}
    $X \surj{\lambda} Y$ abbreviates $\Exists{f \,{\in}\, \L{\lambda}}(f \colon X \twoheadrightarrow Y)$, where `$\twoheadrightarrow$' stands for surjective map.
\end{notation*}

Recall: in the present context, ``$\mu$ is an \Lcard{\lambda}'' means ``there is no $\xi < \mu$ such that $\xi \surj{\lambda} \mu$''.

\begin{lemma}
      Let $\lambda > \omega$ be a limit ordinal. For $0 < \alpha \le \gamma < \lambda$, if $\L{\beta} =\thull{\gamma}(\alpha)$, then $\seq{\alpha} \surj{\lambda} \beta$.
\end{lemma}

\begin{proof}
Observe that $\L{\beta} = \hull{\beta}(\alpha)$, and $\beta \le \gamma < \lambda$. In \L{\beta}, every $\xi \in \beta$ is the unique solution of some formula $\phi(v, \vec{\*\eta})$, with parameters $\vec{\eta} \in \seq{\alpha}$. Thus, from the fact that $\models_\L{\beta} \in \L{\lambda}$ one readily derives $\ensuremath{\mathcal{F}}\xspace \times \seq{\alpha} \surj{\lambda} \beta$. Using an injection $\ensuremath{\mathcal{F}}\xspace \times \seq{\alpha} \to \seq{\alpha}$, in \L{\lambda}, one gets $\seq{\alpha} \surj{\lambda} \beta$.
\end{proof}

\begin{proposition}
\label{prop:Appendix-prop}
    Let $\lambda > \omega$ be a limit ordinal, and $\mu < \lambda$ an \Lcard{\lambda}, $\mu > \omega$.
    \begin{enumerate}
    \item{}$\mu$ is admissible.
    \item{}For $\alpha < \mu$ and $\alpha \le \gamma < \lambda$, if $\L{\beta} = \thull{\gamma}(\alpha)$, then $\beta < \mu$. \textup{(A downward Löwenheim-Skolem property.)}
    \end{enumerate}
\end{proposition}

\begin{proof}
We check (1) and (2) simultaneously, by induction on $\mu$.
Set $\xbar\mu = \min_{\eta \le \mu}(\seq{\eta} \surj{\lambda} \mu)$.

We claim that $\xbar\mu = \mu$. Easily, $\xbar\mu$ is an \Lcard{\lambda} and $\xbar\mu > \omega$. If $\mu$ is the first \Lcard{\lambda} $> \omega$, then $\xbar\mu = \mu$. Else, if $\xbar\mu < \mu$, then by the induction hypothesis $\xbar\mu$ is admissible, thus there is an \L{\xbar\mu}\nobreakdash-definable bijection $\xbar\mu \to \seq{\xbar\mu}$. Hence $\xbar\mu \surj{\lambda} \seq{\xbar\mu}$, and since $\seq{\xbar\mu} \surj{\lambda} \mu$, $\xbar\mu \surj{\lambda} \mu$, contradicting the fact that $\mu$ is an \Lcard{\lambda}.

Now (2) holds for $\mu$. Indeed, if $0 < \alpha < \mu$, and $\alpha \le \gamma < \lambda$, and $\L{\beta} = \thull{\gamma}(\alpha)$, then by the previous lemma, $\seq{\alpha} \surj{\lambda} \beta$. Hence, since $\alpha < \mu = \xbar\mu$, $\beta < \mu$.
\smallskip{}

To complete the proof that $\mu$ is admissible, only $\Delta_0$ \Ax{Collection} needs checking.

Say $\L{\mu} \models \All{x \,{\in}\, \*a} \Exists{y} \phi(x, y, \vec{\*p})$, where $\phi$ is a $\Delta_0$ formula, and $a, \vec{p} \in \L{\mu}$. Pick $\alpha < \mu$ with $a,\,\vec{p} \in \L{\alpha}$, and set ${\L{\beta} = \thull{\mu}(\alpha)}$. $\beta \ge \alpha$, and $\L{\beta} \models \All{x \,{\in}\, \*a} \Exists{y} \phi(x, y, \vec{\*p})$. Applying (2), $\beta < \mu$ and $b =^\text{def} \L{\beta} \in \L{\mu}$. By $\Delta_0$ absoluteness, $\L{\mu} \models \All{x \,{\in}\, \*a} \Exists{y \,{\in}\, \*b} \phi(x,y,\vec{\*p})$.
\end{proof}

\begin{proposition}[\ref{prop:Folklore}]
\label{prop:Folklore2}
    For $\lambda$ a limit ordinal, $\L{\lambda} \models "\*\mu > \omega \textup{ is a successor cardinal}" \Implies \L{\mu} \models \ensuremath{\mathsf{ZF}^-}\xspace$.
\end{proposition}

\begin{proof}
Let $\pi$ be the cardinal predecessor of $\mu$ in \L{\lambda}. We argue that $\pi$ is the largest cardinal in \L{\mu}. Indeed, for $\pi \le \eta < \mu$, pick $\gamma < \lambda$ such that $\Exists{f \in \L{\gamma}}(f \colon \pi \twoheadrightarrow \eta)$, and set $\L{\beta} = \thull{\gamma}(\eta+1)$. We~get $\Exists{f \in \L{\beta}}(f \colon \pi \twoheadrightarrow \eta)$ and, invoking Proposition~\ref{prop:Appendix-prop}(2), $\beta < \mu$.
\smallskip{}

Next, check that $\mu$ is regular inside \L{\lambda}. The standard \ensuremath{\mathsf{ZFC}}\xspace proof for the regularity of infinite successor cardinals goes through here: for each nonzero $\eta < \mu$, using the wellordering $<_\L{\mu}$, select $f_\eta \in \L{\mu}$, $f_\eta \colon \pi \twoheadrightarrow \eta$, and note that the sequence $(f_\eta)_{0 < \eta < \mu}$ is in $\L{\mu+1} \subseteq \L{\lambda}$, etc.
\smallskip{}

Finally, to show that $\L{\mu} \models \ensuremath{\mathsf{ZF}^-}\xspace$: Since by \ref{prop:Appendix-prop}(1) $\mu$ is admissible, using the definable bijection $\mu \to \L{\mu}$, it suffices to verify \Ax{Replacement} for ordinal class-functions.

Let therefore $F \colon \mu \to \mu$ be definable over \L{\mu}, from parameters $\vec{p}$. Given $a \in \L{\mu}$, $a \subseteq \mu$, by regularity of $\mu$ in \L{\lambda}, $F[a]$ is bounded in $\mu$. Pick ${\alpha < \mu}$, with $F[a] \subseteq \alpha$ and $a,\,\vec{p} \in \L{\alpha}$. Set $\L{\beta} = \thull{\mu}(\alpha)$, applying Proposition~\ref{prop:Appendix-prop}(2), $\beta < \mu$. Since $F[a] \in \L{\mu+1}$, $F[a] \in \L{\beta+1} \subseteq \L{\mu}$.
\end{proof}

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