User blog:Simply Beautiful Art/Feferman's theta extended to weakly compact cardinals

Recently I've been working on an ordinal collapsing variation of Feferman's \(\theta\) involving weakly compact cardinals, and the end result is something that functions very nicely in my opinion. First we define the skeletal structure involving sets of additive principals, then regulars, then Mahlos, and so on. Then we will define the main ordinal collapsing function which returns things like large computable ordinals.

First, let \(K\) be the first weakly compact ordinal and \(K'\) be the first hyper-Mahlo after \(K\). We then define \(S(A)\), a proposition with \(A\subset K'\), and all the following as subsets of \(K'\) over arguments that are elements of \(K'\):

\(\displaystyle S(A)\Leftrightarrow\forall f:\sup A\mapsto\sup A,\exists\alpha\in A,\forall\eta\in\alpha(f(\eta)\in\alpha)\)

\(\displaystyle\mathrm B(\alpha,\kappa)_0=\kappa\cup\{0,K\}\)

\(\displaystyle\mathrm B(\alpha,\kappa)_{n+1}=\{\gamma+\delta~|~\gamma,\delta\in\mathrm B(\alpha,\kappa)_n\}\)

\(\displaystyle\hphantom{\mathrm B(\alpha,\kappa)_{n+1}={}}{}\cup\{\Theta_\eta(\mu)~|~\mu,\eta\in\mathrm B(\alpha,\kappa)_n\land\eta\in\alpha\}\)

\(\displaystyle\mathrm B(\alpha,\kappa)=\bigcup_{n\in\mathbb N}\mathrm B(\alpha,\kappa)_n\)

\(\displaystyle\Xi(\alpha)=\bigg\{\kappa,K~|~\kappa\notin\mathrm B(\alpha,\kappa)\land\alpha\in\operatorname{cl}(\mathrm B(\alpha,\kappa))\land S\bigg(\bigcap_{\eta\in\mathrm B(\alpha,\kappa)\cap\alpha}\Xi(\eta)\bigg)\bigg\}\)

\(\displaystyle\Theta_\alpha=\operatorname{enum}(\Xi(\alpha))\)

Here we have \(S(A)\), a restatement of "\(A\) is stationary" in a possibly more intuitive form for the purposes here. It's relevance is that we want sets such that every function gets "stuck" at a point, so our ordinal collapsing functions will always get stuck, as they usually do.

There's not much to say about \(\mathrm B\), it's simply the standard part of any ordinal collapsing function.

\(\Xi\) is the thing we will take with us to the next ordinal collapsing function, as we will define soon below. It is essentially sets of ordinals which have the intersection of previous \(\Xi\)'s stationary in it. A few values are:

\(\Xi(0)=\{\omega^\gamma~|~\gamma\in K'\}\) is the set of additive principals.

\(\Xi(1)\) is the set of regular ordinals.

\(\Xi(2)\) is the set of Mahlo ordinals.

\(\Xi(3)\) is the set of 1-Mahlo ordinals, where the set of Mahlos less than them are stationary.

\(\Xi(K)\) is the set of hyper-Mahlo ordinals, where every element \(\kappa\) is \(\kappa\)-Mahlo.

It is interesting to note that \(\Xi\) includes ordinals larger than \(K\), which is how it produces expressions such as \(K^K=\Theta_0(\Theta_0(K+K))\).

\(\Theta_\alpha\) is then the enumeration of \(\Xi(\alpha)) i.e. \(\Theta_\alpha(0)\) is the first element of \(\Xi(\alpha)\), \(\Theta_\alpha(1)\) is the second element of \(\Xi(\alpha)\), etc.

We now define our main ordinal collapsing function of interest, with the same restriction to subsets of \(K'\) and arguments as elements of \(K'\):

\(\displaystyle\mathrm C(\alpha,\kappa)_0=\kappa\cup\{0,K\}\)

\(\displaystyle\mathrm C(\alpha,\kappa)_{n+1}=\{\gamma+\delta~|~\gamma,\delta\in\mathrm C(\alpha,\kappa)_n\}\)

\(\displaystyle\hphantom{\mathrm C(\alpha,\kappa)_{n+1}={}}{}\cup\{\theta_\xi^\eta(\mu)~|~\mu,\xi,\eta\in\mathrm C(\alpha,\kappa)_n\land\eta\in\alpha\}\)

\(\displaystyle\mathrm C(\alpha,\kappa)=\bigcup_{n\in\mathbb N}\mathrm C(\alpha,\kappa)_n\)

\(\displaystyle\theta_\pi^\alpha=\operatorname{enum}\{\kappa,K\in\Xi(\pi)~|~\kappa\notin\mathrm C(\alpha,\kappa)\land\alpha\in\operatorname{cl}(\mathrm C(\alpha,\kappa))\}\)

There's not much to say about \(\mathrm C\), it's simply the standard part of any ordinal collapsing function.

\(\theta_\pi^\alpha\) is then the enumeration of elements of \(\Xi(\pi)\) in a manner akin to Feferman's \(\theta\). A few values are given as:

\(\theta_\pi^0(\mu)=\Theta_\pi(\mu)\) e.g. \(\theta_1^0(\mu)\) is the \(\mu\)th regular ordinal.

\(\theta_\pi^1(\mu)\) are (almost all) the fixed-points of \(\theta_\pi^0\) e.g. \(\theta_1^1(\mu)\) is the \(\mu\)th inaccessible ordinal that isn't Mahlo (i.e. not fixed-points that are already given by another combination of arguments).

\(\theta_\pi^2(\mu)\) are (almost all) the fixed-points of \(\theta_\pi^1\).

\(\theta_0^\Omega(\mu)=\Gamma_\mu=\theta_0^{\Gamma_\mu}(0)\) for \(\mu\in\Omega\), where \(\Omega=\theta_1^0(0)) is the first regular ordinal.

\(\theta_1^{\theta_2^1(0)}(\mu)\) are the regular limits of Mahlo ordinals for \(\mu\in\theta_2^1(0)\), where \(\theta_2^1(0)\) is the first Mahlo fixed-point i.e. a Mahlo limit of Mahlo ordinals.

The smallest ordinal this notation can't write is given by \(\theta_0^{K'}(0)=\theta_0^\alpha(0)\), where \(\alpha=\sup\{0,\Theta_0(0),\Theta_{\Theta_0(0)}(0),\dots\}\) is the limit of everything that can be written in this notation.