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?

  • 1
    $\begingroup$ If you only want some $\delta$ for which $L_\delta\models$ZFC-, then you can take $\delta=\omega_1$, or any infinite successor cardinal, for that matter. But probably you want to characterize the least $\delta$ such that $L_\delta\models$ZFC-? $\endgroup$ – Joel David Hamkins Jul 31 '13 at 23:53
  • 1
    $\begingroup$ Your ordinal $\delta$ is smaller than $\delta^1_2$, the supremum of the $\Delta^1_2$-definable pre-wellorder relations on $\mathbb{N}$, but larger of course than $\delta^1_1=\omega_1^{ck}$. $\endgroup$ – Joel David Hamkins Aug 1 '13 at 2:00
  • $\begingroup$ Yes, I am interested in the least such ordinal. $\endgroup$ – Frode Alfson Bjørdal Aug 1 '13 at 8:16
  • $\begingroup$ Question: Is $L_{\beta_0}$ then also a model of $\Sigma_n$-$KP$ (that is, $KP$+Infinity+$\Sigma_n$-Collection+$\Sigma_n$-Separation? $\endgroup$ – Thomas Benjamin Feb 6 '15 at 13:37
  • $\begingroup$ Yes, that would be so. $\endgroup$ – Frode Alfson Bjørdal Feb 6 '15 at 21:52

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".

| cite | improve this answer | |
  • $\begingroup$ Can you explain what work Jensen did and how it superseded the need to study the ramified analytical hierarchy? $\endgroup$ – Keshav Srinivasan May 14 '19 at 6:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.