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${\mathfrak t}$ is the least $\beta$ such that there is a $\gamma<\beta$ with $L_\gamma \prec L_\beta$. That ${\mathfrak t} \leq$ the least such $\beta$ is obvious. On the other hand, if $X \subset L_{\mathfrak t}$ is $\subseteq$-least with $X \prec L_{\mathfrak t}$, then $X \not= L_{\mathfrak t}$; hence if $\sigma \colon L_\gamma \cong X$, then either $\sigma$ is the identity and thus $\gamma<{\mathfrak t}$ (as desired) or $\sigma$ is not the identity, in which case $L_{{\rm crit}(\sigma)} \prec L_{{\sigma}({\rm crit}(\sigma))}$, so that ${\mathfrak t} \leq \sigma({\rm crit}(\sigma)) < {\mathfrak t}$ by the other direction (giving a contradiction). In particular, $L_{\mathfrak t}$ is a model of ZFC, but ${\mathfrak t}$ is not the least $\beta$ such that $L_\beta$ is a model of ZFC and it is much bigger than $\Sigma$, the supremum of the accidentally writable ordinals.

${\mathfrak t}$ is the least $\beta$ such that there is a $\gamma<\beta$ with $L_\gamma \prec L_\beta$. That ${\mathfrak t} \leq$ the least such $\beta$ is obvious. On the other hand, if $X \subset L_{\mathfrak t}$ is $\subseteq$-least with $X \prec L_{\mathfrak t}$, then $X \not= L_{\mathfrak t}$; hence if $\sigma \colon L_\gamma \cong X$, then either $\sigma$ is the identity and thus $\gamma<{\mathfrak t}$ (as desired) or $\sigma$ is not the identity, in which case $L_{{\rm crit}(\sigma)} \prec L_{{\sigma}({\rm crit}(\sigma))}$, so that ${\mathfrak t} \leq \sigma({\rm crit}(\sigma)) < {\mathfrak t}$ by the other direction (giving a contradiction). In particular, $L_{\mathfrak t}$ is a model of ZFC${}^-$ (ZFC w/o the power set axiom), in fact of ZFC${}^-$ plus ``every set is at most countable,'' but ${\mathfrak t}$ is not the least $\beta$ such that $L_\beta$ is a model of ZFC${}^-$ and it is much bigger than $\Sigma$, the supremum of the accidentally writable ordinals.

${\mathfrak t}$ is the least $\beta$ such that there is a $\gamma<\beta$ with $L_\gamma \prec L_\beta$. That ${\mathfrak t} \leq$ the least such $\beta$ is obvious. On the other hand, if $X \subset L_{\mathfrak t}$ is $\subseteq$-least with $X \prec L_{\mathfrak t}$, then $X \not= L_{\mathfrak t}$; hence if $\sigma \colon L_\gamma \cong X$, then either $\sigma$ is the identity and thus $\gamma<{\mathfrak t}$ (as desired) or $\sigma$ is not the identity, in which case $L_{{\rm crit}(\sigma)} \prec L_{{\sigma}({\rm crit}(\sigma))}$, so that ${\mathfrak t} \leq \sigma({\rm crit}(\sigma)) < {\mathfrak t}$ by the other direction (giving a contradiction). In particular, $L_{\mathfrak t}$ is a model of ZFC, but ${\mathfrak t}$ is not the least $\beta$ such that $L_\beta$ is a model of ZFC.

${\mathfrak t}$ is the least $\beta$ such that there is a $\gamma<\beta$ with $L_\gamma \prec L_\beta$. That ${\mathfrak t} \leq$ the least such $\beta$ is obvious. On the other hand, if $X \subset L_{\mathfrak t}$ is $\subseteq$-least with $X \prec L_{\mathfrak t}$, then $X \not= L_{\mathfrak t}$; hence if $\sigma \colon L_\gamma \cong X$, then either $\sigma$ is the identity and thus $\gamma<{\mathfrak t}$ (as desired) or $\sigma$ is not the identity, in which case $L_{{\rm crit}(\sigma)} \prec L_{{\sigma}({\rm crit}(\sigma))}$, so that ${\mathfrak t} \leq \sigma({\rm crit}(\sigma)) < {\mathfrak t}$ by the other direction (giving a contradiction). In particular, $L_{\mathfrak t}$ is a model of ZFC, but ${\mathfrak t}$ is not the least $\beta$ such that $L_\beta$ is a model of ZFC and it is much bigger than $\Sigma$, the supremum of the accidentally writable ordinals.

${\mathfrak t}$ is the least $\beta$ such that there is a $\gamma<\beta$ with $L_\gamma \prec L_\beta$. That ${\mathfrak t} \leq$ the least such $\beta$ is obvious. On the other hand, if $X \subset L_{\mathfrak t}$ is $\subseteq$-least with $X \prec L_{\mathfrak t}$, then $X \not= L_{\mathfrak t}$; hence if $\sigma \colon L_\gamma \cong X$, then either $\sigma$ is the identity and thus $\gamma<{\mathfrak t}$ (as desired) or $\sigma$ is not the identity, in which case $L_{{\rm crit}(\sigma)} \prec L_{{\sigma}({\rm crit}(\sigma))}$, so that ${\mathfrak t} \leq \sigma({\rm crit}(\sigma)) < {\mathfrak t}$ by the other direction (giving a contradiction).

${\mathfrak t}$ is the least $\beta$ such that there is a $\gamma<\beta$ with $L_\gamma \prec L_\beta$. That ${\mathfrak t} \leq$ the least such $\beta$ is obvious. On the other hand, if $X \subset L_{\mathfrak t}$ is $\subseteq$-least with $X \prec L_{\mathfrak t}$, then $X \not= L_{\mathfrak t}$; hence if $\sigma \colon L_\gamma \cong X$, then either $\sigma$ is the identity and thus $\gamma<{\mathfrak t}$ (as desired) or $\sigma$ is not the identity, in which case $L_{{\rm crit}(\sigma)} \prec L_{{\sigma}({\rm crit}(\sigma))}$, so that ${\mathfrak t} \leq \sigma({\rm crit}(\sigma)) < {\mathfrak t}$ by the other direction (giving a contradiction). In particular, $L_{\mathfrak t}$ is a model of ZFC, but ${\mathfrak t}$ is not the least $\beta$ such that $L_\beta$ is a model of ZFC.