In mathematics, Rathjen's
psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals
to generate large countable ordinals.[1] A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below
is closed under
(i.e. all normal functions closed in
are closed under some regular ordinal
). Rathjen uses this to diagonalise over the weakly inaccessible hierarchy.
It admits an associated ordinal notation
whose limit (i.e. ordinal type) is
, which is strictly greater than both
and the limit of countable ordinals expressed by Rathjen's
.
, which is called the "Small Rathjen ordinal" is the proof-theoretic ordinal of
, Kripke–Platek set theory augmented by the axiom schema "for any
-formula
satisfying
, there exists an addmissible set
satisfying
". It is equal to
in Rathjen's
function.[2]
Restrict
and
to uncountable regular cardinals
; for a function
let
denote the domain of
; let
denote
, and let
denote the enumeration of
. Lastly, an ordinal
is said to be to be strongly critical if
.
For
and
:
![{\displaystyle {\begin{aligned}&\beta \cup \{0,M\}\subseteq B^{n}(\alpha ,\beta )\gamma =\gamma _{1}+\cdots +\gamma _{k}{\text{ and }}\gamma _{1},\ldots ,\gamma _{k}\in B^{n}(\alpha ,\beta )\\[5pt]&\rightarrow \gamma \in B^{n+1}(\alpha ,\beta )\gamma =\varphi _{\gamma _{0}}(\gamma _{1}){\text{ and }}\gamma _{0},\gamma _{1}\in B^{n}(\alpha ,\beta )\rightarrow \gamma \in B^{n+1}(\alpha ,\beta )\pi \in B^{n}(\alpha ,\beta )\\[5pt]&{\text{and }}\gamma <\pi \rightarrow \gamma \in B^{n+1}(\alpha ,\beta )\delta ,\eta \in B^{n}(\alpha ,\beta )\land \delta <\alpha \land \eta \in \operatorname {dom} (\chi _{\delta })\\[5pt]&\rightarrow \chi _{\delta }(\eta )\in B^{n+1}(\alpha ,\beta )B(\alpha ,\beta )\\[5pt]&\bigcup _{n<\omega }B^{n}(\alpha ,\beta )\chi _{\alpha }\\[5pt]&=\operatorname {enum} (\operatorname {cl} (\kappa :\kappa \notin B(\alpha ,\kappa )\land \alpha \in B(\alpha ,\kappa )\})).\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1977537f7bc889e90a2ea815ad5410220e710870)
If
for some
, define
using the unique
. Otherwise if
for some
, then define
using the unique
, where
is a set of strongly critical ordinals
explicitly defined in the original source.
For
:
![{\displaystyle {\begin{aligned}&\kappa ^{-}\cup \{\kappa ^{-},M\}\subset C_{\kappa }^{n}(\alpha )\gamma =\gamma _{1}+\cdots +\gamma _{k}{\text{ and }}\gamma _{1},\ldots ,\gamma _{k}\in C^{n}(\alpha )\rightarrow \gamma \in C^{n+1}(\alpha )\gamma =\varphi _{\gamma _{0}}(\gamma _{1})\land \gamma _{0},\gamma _{1}\in C^{n}(\alpha ,\beta )\\[5pt]&\rightarrow \gamma \in C^{n+1}(\alpha )\pi \in C_{\kappa }^{n}(\alpha )\cap \kappa \land \gamma <\pi \land \pi \in {\textrm {R}}\\[5pt]&\rightarrow \gamma \in C_{\kappa }^{n+1}(\alpha )\gamma =\chi _{\delta }(\eta )\land \delta ,\eta \in C_{\kappa }^{n}(\alpha )\rightarrow \gamma \in C_{\kappa }^{n+1}(\alpha )\\[5pt]&\gamma =\Phi _{\delta }(\eta )\land \delta ,\eta \in C_{\kappa }^{n}(\alpha )\land 0<\delta \land \delta ,\eta <M\rightarrow \gamma \in C_{\kappa }^{n+1}(\alpha )\beta <\alpha \land \pi ,\beta \in C_{\kappa }^{n}(\alpha )\land \beta \in C_{\pi }(\beta )\rightarrow \psi _{\pi }(\beta )\in C_{\kappa }^{n+1}(\alpha )C_{\kappa }(\alpha ):=\bigcup _{C_{\kappa }^{n}(\alpha ):n<\omega }.\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4d4ab6e4ad1a8d5bdbfe29facaef092c11609725)
![{\displaystyle \psi _{\kappa }(\alpha ):=\min(\{\xi :\xi \notin C_{\kappa }(\alpha )\}).}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/b32888135c943e53a263f59f7175f1ad67b3cf81)
- Restrict
to uncountable regular cardinals.
is a unique increasing function such that the range of
is exactly
.
is the closure of
, i.e.
, where
denotes the class of non-zero limit ordinals.
![{\displaystyle B_{0}(\alpha ,\beta )=\beta \cup \{0,M\}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e19831c93339dcd67c1b1619fa1c6aa30bfd0ffc)
![{\displaystyle B_{n+1}(\alpha ,\beta )=\{\gamma +\delta ,\varphi _{\gamma }(\delta ),\chi _{\mu }(\delta )|\gamma ,\delta ,\mu \in B_{n}(\alpha ,\beta )\land \mu <\alpha \}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/df53467aac240c20f902cc9e8344142397fcbd9e)
![{\displaystyle B(\alpha ,\beta )=\bigcup _{n<\omega }B_{n}(\alpha ,\beta )}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/43a456d0fedca36d94a5e647e2dfb8a7b428c76c)
![{\displaystyle \chi _{\alpha }(\beta )=\operatorname {enum} (\operatorname {cl} (\{\pi :B(\alpha ,\pi )\cap M\subseteq \pi \land \alpha \in B(\alpha ,\pi )\}))=\operatorname {enum} (\{\beta \in \operatorname {Lim} \mid \sup\{\pi :B(\alpha ,\pi )\cap M\subseteq \pi \land \alpha \in B(\alpha ,\pi )\}\cap \beta )=\beta \}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/487c2c36b2056b42e097b89a67b075c14bc15776)
![{\displaystyle C_{0}(\alpha ,\beta )=\beta \cup \{0,M\}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4e729220ec301d3f59dd70771e1f26721a61705b)
![{\displaystyle C_{n+1}(\alpha ,\beta )=\{\gamma +\delta ,\varphi _{\gamma }(\delta ),\chi _{\mu }(\delta ),\psi _{\pi }(\mu )|\gamma ,\delta ,\mu ,\pi \in B_{n}(\alpha ,\beta )\land \mu <\alpha \}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/31800764df31e1da3ca9a2e9e1d0c0ed5708bad1)
![{\displaystyle C(\alpha ,\beta )=\bigcup _{n<\omega }C_{n}(\alpha ,\beta )}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/cb9623458724f0283c8f5954652d9f31108e2752)
![{\displaystyle \psi _{\pi }(\alpha )=\min(\{\beta :C(\alpha ,\beta )\cap \pi \subseteq \beta \land \alpha \in C(\alpha ,\beta )\})}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/f50eadbbc9e39c27fc027f6dda78c0113284604a)
Rathjen originally defined the
function in more complicated a way in order to create an ordinal notation associated to it. Therefore, it is not certain whether the simplified OCF above yields an ordinal notation or not. The original
functions used in Rathjen's original OCF are also not so easy to understand, and differ from the
functions defined above.
Rathjen's
and the simplification provided above are not the same OCF. This is partially because the former is known to admit an ordinal notation, while the latter isn't known to admit an ordinal notation.[citation needed] Rathjen's
is often confounded with another of his OCFs which also uses the symbol
, but they are distinct notions. The former one is a published OCF, while the latter one is just a function symbol in an ordinal notation associated to an unpublished OCF.[3]