ZFC-, which is ZFC minus power set, is modelled by $ L\_{\delta}$ where $\delta$ is an admissible ordinal larger than
any least $\Sigma\_{n}$-admissible ordinal for n a natural number. Can some provide more information? Does it have a name? What is its most natural omega sequence?



A question on an ordinal for ZFC-
The least such ordinal $\beta$, often written $\beta\_0$, for which $L\_\beta$ is a $ZF^-$ is also characterised as the "ordinal of ramified analysis". This is because the ramified analytical hierarchy, which builds up cumulative second order number theoretic structures of the form $$\underline{P\_\alpha}=( P\_\alpha,\mathbb{N}, +, \times, \ldots)$$ with $P\_\alpha\subseteq P\_{\alpha +1}\subseteq
\mathcal{P}(\mathbb{N})$ by looking at all sets of integers definable over $\underline{P\_\alpha}$ by using instances of second order comprehensionto to obtain $P\_{\alpha+1}$. (Unions are taken at limits). This hierarchy has height exactly $\beta\_0$ (meaning $P\_{\beta\_0}=P\_{\beta\_0+1}$.) Alternatively put: $\underline{P\_{\beta\_0}}$ is then the least model of $Z\_2$, or full second order comprehension (sometimes just abbreviated to "analysis", or ``second order number theory'').


${P\_{\beta\_0}}$ is then $\mathcal{P}(\mathbb{N}) \cap L\_{\beta\_0}$.


$\beta\_0$ of course is $\Sigma\_n$-admissible for all $n$. $L\_{\beta\_0+1}$ sees that $\beta\_0$ is countable, and so I suppose one may define a particular $\omega$ sequence in $L\_{\beta\_0+1}$ cofinal in it by just taking the $<\_L$- least such.


One can also paraphrase the assertion that $L\_{\beta\_0}$ is the least $\beta$ with $L\_\beta$ a $ZF^-$ model, as being the least such that for no $n$ is the fine-structural projectum $\rho^n\_\beta$ less than $\beta$. (But this is really only dressing up one concept in the jargon of another.)


$L\_{\beta\_0}$ is (I believe) the least level of the $L$ hierarchy whose reals form a model of $\Delta^0\_4$-Determinacy (Martin - unpublished).


For the Ramified Analytical Hierarchy: see Boyd, Hensel, Putnam "An intrinsic characterisation of the ramified analytical hierarchy", JSL, late 60's I believe. This hierarchy is not much studied these days, but was of interest in the pre-Jensen fine-structural era of the analysis of levels of $L$ (the latter's work now surpassing it).
For connections between models of subsystems of second order comprehension and constructible sets, see also S. Simpson's book "Subsystems of second order arithmetic".


