# Analytical solution of  $D$ dimensional Schrödinger equation for Eckart
potential with a new improved approximation in centrifugal term

Debraj Nath  Corresponding author. Email<EMAIL_ADDRESS>Department of
Mathematics, Vivekananda College, Kolkata-700063, WB, India  Amlan K. Roy
Corresponding author. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Chemical Sciences Indian Institute of Science Education and
Research (IISER) Kolkata, Mohanpur-741246, Nadia, WB, India

###### Abstract

Analytical solutions are presented for eigenvalues, eigenfunctions of
D-dimensional Schrodinger equation having Eckart potential within Nikiforov-
Uvarov method. This uses a new, improved approximation for centrifugal term,
from a combination of Greene-Aldrich and Pekeris approximations. Solutions are
obtained in terms of hypergeometric functions. It facilitates an accurate
representation in entire domain. Its validity is illustrated for energies in
an arbitrary $\ell\neq 0$ quantum state. Results are compared for a chosen set
of potential parameters in different dimensions. In short, a simple accurate
approximation is offered for Eckart and other potentials in quantum mechanics,
in higher dimension.

PACS: 02.60.-x, 03.65.Ca, 03.65.Ge, 03.65.-w
Keywords: Eckart potential, Pekeris approximation, Greene-Aldrich
approximation, Nikiforov-Uvarov method, ro-vibrational energy.

##  I Introduction

The exponential diatomic molecular Eckart potential eckart1930 ,

$v_{e}(r)=-\frac{\alpha e^{-r/a}}{1-e^{-r/a}}+\frac{\beta
e^{-r/a}}{(1-e^{-r/a})^{2}},~{}\alpha,\beta>0.$ (1)

is used as an important model in chemical physics cooper1995 . The depth is
controlled by two positive real parameters $\alpha$ and $\beta$, while the
range is governed by a positive parameter, $\alpha$, having the dimension of
inverse length. It is related to the Schiöberg potential by a change of
parameter. It has a minimum value of $-\frac{(\alpha-\beta)^{2}}{4\beta}$ at
$r=a\ \ln\frac{(\alpha+\beta)}{(\alpha-\beta)},$ for $\alpha>\beta$. Like the
other few familiar exponential-type model potentials such as, Morse, Woods-
Saxon, Hulthén, Manning-Rosen, the relevant Schrödinger equation (SE) in this
case too offers _exact_ analytical solution _for $\ell=0$_, but not for
$\ell\neq 0$ states.  However, a number of elegant approximations have been
put forth for non-zero $\ell$ states with varied success. It may be noted that
the Eckart potential and its PT-symmetric version can be considered as special
cases of the multi-parameter exponential type potential jia2003 .

Analytical bound state energy expressions and radial wave functions for
arbitrary $\ell$ states are derived by using various approximations to the
centrifugal term dong2007 ; pekeris.ap3 ; taskin2010 ; stanek2011  in
amalgamation with a host of techniques, such as, supersymmetric shape
invariance approach diao2009 ; onate2018  combined with a functional analysis
diao2009 , asymptotic iteration falaye2012 , tridiagonal representation
zhang2012 , Feynman path integral approach diaf2015 , etc. An approximate
solution of the SE with Eckart potential and its parity-time symmetric version
was investigated zhang2009  using a Nikiforov-Uvarov (NU) method. The same for
scattering states were expressed in terms of the generalized hypergeometric
functions ${}_{2}F_{1}(a,b;c;z)$, and phase shifts were analyzed chen2008 ;
wei2008 . The relativistic bound- wei2009 ; liu2009  and scattering-state
wei2009  solutions of Klein-Gordon equation have been reported. Apart from
energies, information theoretical studies were pursued, _viz._ , Shannon
entropy pooja2016 , Rényi entropy onate2018 , etc. Variants of this potential,
like deformed hyperbolic Eckart mincang2013 , Eckart-like pekeris.ap3 , Hua
plus modified Eckart onyenegecha2020 , generalized Deng-Fan plus deformed
Eckart awoga2013  potential have been considered as well. The thermodynamic
properties of anharmonic Eckart potential have also been the subject of
investigation ortega2017 . The above discussion suggests that, a reasonable
number of good-quality methods are available for low-lying (especially,
$\ell=0$) states of this potential. But such attempts are rather limited for
high-lying states belonging to the non-trivial $\ell\neq 0$ case. Also it is
worth mentioning that a majority of the articles are devoted to eigenvalues
and eigenfunctions, mainly in 3D. A review of the available methods further
indicates that, a decent number of works resort to the approximation of
centrifugal term from physical/mathematical consideration. Herein lies the
objectives of this work, which is two-fold. At first, we would like to make an
analysis and test performance of a recently developed, simple approximation
for centrifugal term in the context of Eckart potential. So far, this has been
applied to shifted Deng-Fan, Manning-Rosen and Pöschl-Teller molecular
potentials nath2021 ; nath2021.epjp . This will extend the scope of
applicability to a wider range of physical systems. Another objective is to
report the results in D dimension, for which the references are very rare
Hulthen.greene15 .

The article is organized as follows: Sec. II gives an overview of various
approximations to the centrifugal term, including the newly proposed one
nath2021 ; nath2021.epjp , for solving the SE for Eckart potential, within the
NU method. The resultant analytic expressions are derived for radial wave
functions, eigenvalues, normalization constants. Wherever possible, the
computed energies in Sec. III are discussed and compared with available
literature results. Finally, a few remarks are made in Sec. IV, along with the
future prospect of our scheme.

##  II Theoretical analysis

Let us consider the following SE,

$-\frac{\hbar^{2}}{2\bar{\mu}}{\bf\nabla}_{D}^{2}\psi+V({\bf r})\psi({\bf
r})=E\psi({\bf r}),$ (2)

where $\overrightarrow{\bf\nabla}$ is $D$-dimensional gradient operator, ${\bf
r}=(x_{1},x_{2},...,x_{D})=(r,\theta_{1},\theta_{2},...,\theta_{D-1})=(r,\Omega_{D-1})$,
$r=|{\bf r}|=\sqrt{\sum\limits_{i=1}^{D}x_{i}^{2}}$,
$x_{i}=r\cos\theta_{i}\prod\limits_{k=1}^{i-1}\sin\theta_{k}$,
$0\leq\theta_{i}<\pi,~{}i=1,2,...,D-2$, $0\leq\theta_{D-1}<2\pi$, while,

${\bf\nabla}_{D}=\left(\frac{\partial}{\partial
r},\frac{1}{r}\frac{\partial}{\partial\theta_{1}},\frac{1}{r\sin\theta_{1}}\frac{\partial}{\partial\theta_{2}},...,\frac{1}{r\prod_{j=1}^{k-1}\sin\theta_{j}}\frac{\partial}{\partial\theta_{k}},...,\frac{1}{r\prod_{j=1}^{D-2}\sin\theta_{j}}\frac{\partial}{\partial\theta_{D-1}}\right),$
(3)

is the $D$-dimensional gradient operator. Then the Laplacian operator
${\bf\nabla}_{D}^{2}$ is defined by,

${\bf\nabla}_{D}^{2}=\frac{1}{r^{D-1}}\frac{\partial}{\partial
r}\left(r^{D-1}\frac{\partial}{\partial
r}\right)-\frac{\Lambda_{D}^{2}}{r^{2}},$ (4)

where $\Lambda_{D}$ is the $D$-dimensional generalized angular momentum
operator, given by,

$\displaystyle\Lambda_{D}^{2}=-\sum\limits_{i=1}^{D-1}\frac{(\sin\theta_{i})^{i+1-D}}{(\prod_{j=1}^{i-1}\sin\theta_{j})^{2}}\frac{\partial}{\partial\theta_{i}}\left((\sin\theta_{i})^{D-i-1}\frac{\partial}{\partial\theta_{i}}\right),$
(5)

$\bar{\mu}$ is reduced mass of diatomic molecule, and $r$ is the internuclear
distance. Let

$\psi({\bf r})=r^{-\frac{D-1}{2}}R(r)Y_{\ell,\\{\mu\\}}(\Omega),$ (6)

be the solution of Eq. (2). Then it is obtained that,

$\frac{d^{2}R}{dr^{2}}+\left[\frac{2\bar{\mu}}{\hbar^{2}}E-\frac{2\bar{\mu}}{\hbar^{2}}V(r)-\frac{L(L+1)}{r^{2}}\right]R=0,$
(7)

and

$\displaystyle\Lambda_{D}^{2}Y_{\ell,\\{\mu\\}}=\ell(\ell+D-2)Y_{\ell,\\{\mu\\}},$
(8)

where $\ell(\ell+D-2)$ is the separation constant, and $L=\ell+\frac{D-3}{2}$.
Then following dehesa.jmp.2007 ; Srivastava-Daoust ,

$Y_{\ell,\\{\mu\\}}=N_{\ell,\\{\mu\\}}e^{im\theta_{D-1}}\prod\limits_{j=1}^{D-2}C_{\mu_{j}-\mu_{j+1}}^{\alpha_{j}+\mu_{j}+1}(\cos\theta_{j})(\sin\theta_{j})^{\mu_{j}+1},$
(9)

where $C_{i}^{j}(t)$ is the Gegenbauer polynomial in $t$ of degree $i$ with
parameter $j$, whereas

$N_{\ell,\\{\mu\\}}=\frac{1}{2\pi}\prod\limits_{j=1}^{D-2}\frac{(\alpha_{j}+\mu_{j})(\mu_{j}-\mu_{j+1})!\Gamma(\alpha_{j}+\mu_{j+1})^{2}}{\pi
2^{1-2\alpha_{j}-2\mu_{j+1}}\Gamma(2\alpha_{j}+\mu_{j}+\mu_{j+1})},$ (10)

is the normalization constant. Here
$(\ell,\\{\mu\\})=(\mu_{1},\mu_{2},...,\mu_{D-1})$,
$\ell=\mu_{1}\geq\mu_{2}\geq...\geq\mu_{D-2}\geq|\mu_{D-1}|=|m|$,
$\ell=0,1,2,...$, $m=0,\pm 1,\pm 2,...$, $\alpha_{j}=(D-j-1)/2$,
$D=3,4,5,6,\cdots$. To find the general solution of resulting radial SE of Eq.
(7), we will use a transformation, $s=e^{-r/a}$, and a set of functions,
$\left\\{1,\frac{s}{1-s},\frac{s^{2}}{(1-s)^{2}}\right\\}$, for which the
centrifugal term is approximated in different forms, such as Greene-Aldrich
greene , Pekeris-type Pekeris . Recently, the authors have proposed an
intuitive combination of these two approximations nath2021 ; nath2021.epjp
with considerable success.

###  II.1 Approximation to the centrifugal term

Let us consider the following relation:

$\frac{1}{r^{2}}\approx
f_{i}(r)=\frac{1}{a^{2}}\left(x_{1i}+\frac{x_{2i}s}{1-s}+\frac{x_{3i}s^{2}}{(1-s)^{2}}\right),s=e^{-r/a},~{}i=1,2,3.$
(11)

The approximation $f_{1}(r)$, with $x_{11}=0,x_{21}=x_{31}=1$, is commonly
used in references dong2007 ; zhang2012 ; wei2008 ; greene , while for
$x_{12}=0,x_{22}=\xi_{1},x_{32}=\xi_{2}$ and $\xi_{1},\xi_{2}$ as two
adjustable dimensionless parameters, the approximation $f_{2}(r)$ is
considered by the authors of taskin2010 ; onate2018 . Apart from that, the
approximation $f_{3}(r)$ is employed in reference f3 ; f3.2 ; f3.3 ; f3.4 ;
f3.5 , having $x_{13}=\frac{1}{12},x_{23}=x_{33}=1$. The approximations
$f_{1},f_{2},f_{3}$ are good near the origin dong2007 ; taskin2010 ; onate2018
; zhang2012 ; wei2008 ; greene . Moreover, the centrifugal term
$\frac{1}{r^{2}}$ is approximated near the point $r=r_{0}$, as below,
following the references pekeris.ap3 ; Pekeris ; pekeris.badawi ; pekeris.ap4
,

$\frac{1}{r^{2}}\approx
f_{4}(r)=\frac{1}{a^{2}}\left(x_{14}+\frac{x_{24}s}{1-s}+\frac{x_{34}s^{2}}{(1-s)^{2}}\right),$
(12)

where

$\begin{array}[]{l}x_{14}=\frac{1}{u^{4}}\left[(3+u)s_{0}^{2}+(2u-6)s_{0}+(3-3u+u^{2})\right],\\\
x_{24}=\frac{2}{u^{4}}(1-s_{0})^{2}\left[3+u+\frac{2u-3}{s_{0}}\right],\\\
x_{34}=-\frac{1}{u^{4}}(1-s_{0})^{3}\left[\frac{3+u}{s_{0}}+\frac{u-3}{s_{0}^{2}}\right],\\\
s_{0}=e^{-u},u=r_{0}/a.\end{array}$ (13)

Here $r_{0}$ is a positive real number, having the dimension of length. In
particular, if $r_{0}=a\ln\left[\frac{\alpha+\beta}{\alpha-\beta}\right]$,
with $\alpha>\beta$, then the potential has a minimum value of
$-\frac{(\alpha-\beta)^{2}}{4\beta}$. By construction, $f_{4}$ approximation
is more effective around $r=r_{0}$ region pekeris.ap3 ; Pekeris ;
pekeris.badawi ; pekeris.ap4 . In this article, we propose a new approximation
to the centrifugal term, following our previous works nath2021 ; nath2021.epjp
,

$\frac{1}{r^{2}}\approx
f_{5}(r)=\sum\limits_{j=1}^{4}\lambda_{j}\,f_{j}(r)=\frac{1}{a^{2}}\left(y_{1}+\frac{y_{2}s}{1-s}+\frac{y_{3}s^{2}}{(1-s)^{2}}\right),$
(14)

where

$y_{i}=\sum\limits_{j=1}^{4}\lambda_{j}x_{ij},~{}i=1,2,3;\ \ \ \ \ \ \ \ \ \ \
\ \ \sum\limits_{j=1}^{4}\lambda_{j}=1.$ (15)

Figure 1 compares these approximations with _exact_ centrifugal term
$\frac{\ell(\ell+1)}{r^{2}}$, for $\ell=2$ having following parameters:
$(\hbar,\bar{\mu})=(1,1)$, $(a,\alpha,\beta)=(1/0.025,1/a,0.0001)$,
$(\xi_{1},\xi_{2})=(1.1,0.98)$, $D=3$. The upper and lower panels correspond
to $r\to 0$ and $r\to r_{0}$ regions. In each row, five approximations are
illustrated in five panels–the first four refer to $f_{1},f_{2},f_{3},f_{4}$,
while two rightmost panels imply $f_{5}^{(a)}$ and $f_{5}^{(b)}$ having
$(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=(0,0,0.98,0.02)$ and
$(0,0,0.02,0.98)$ respectively. Clearly, panels (C), (E) show that $f_{3}$ and
$f_{5}^{(a)}$ are good approximations near origin. Likewise, from panels (I),
(J), it is inferred that $f_{4},f_{5}^{(b)}$ are superior in the neighborhood
of $r=r_{0}$. Thus $f_{5}$ remains quite accurate over the entire effective
domain of $r$ nath2021 ; nath2021.epjp , which has been defined in our
previous communications effective.domain ; effective.domain2 . Throughout the
article, for sake of convenience, $f_{5}$ approximations will be used for
various sets of $\\{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}\\}$. These
will be indicated by alphabetic superscripts.

Figure 1:  Plot of the difference,
$\ell(\ell+1)\left(\frac{1}{r^{2}}-f_{i}(r)\right)$, for $i=1,2,3,4$. In (A),
(F) $i=1$; (B), (G) $i=2$; (C), (H) $i=3$; (D), (I) $i=4$. (E) and (J) refer
to $f_{5}^{(a)}$ and $f_{5}^{(b)}$ respectively. The parameters are: $\ell=2$,
$(\hbar,\bar{\mu})=(1,1)$, $(a,\alpha,\beta)=(1/0.025,1/a,0.0001)$,
$(\xi_{1},\xi_{2})=(1.1,0.98)$, $D=3$. See text for details.

###  II.2 Ro-vibrational energy by NU method

This is a well-established mathematical tool for various eigenvalue problems
NU . It has been utilized by a number of researchers for a broad range central
and non-central potentials in non-relativistic and relativistic quantum
mechanics ahmadov2014 ; ahmadov2017 ; ahmadov2019 ; ahmadov2020 . Under the
transformation, $s=e^{-r/a}$, Eq.(7) becomes,

$\begin{array}[]{l}\displaystyle\frac{d^{2}R}{ds^{2}}+\frac{\widetilde{\tau}(s)}{\sigma(s)}\frac{dR}{ds}+\frac{\widetilde{\sigma}(s)}{\left[\sigma(s)\right]^{2}}R=0,\end{array}$
(16)

where

$\begin{array}[]{ll}\widetilde{\tau}(s)&=1-s,~{}\sigma(s)=s-s^{2},~{}\widetilde{\sigma}(s)=-As^{2}+Bs-C,\\\
A&=\frac{2\bar{\mu}a^{2}\alpha}{\hbar^{2}}-L(L+1)(y_{2}-y_{3})+C,\\\
B&=\frac{2\bar{\mu}a^{2}(\alpha-\beta)}{\hbar^{2}}-L(L+1)y_{2}+2C,\\\
C&=\epsilon_{0}^{2}+L(L+1)y_{1},\\\
\epsilon_{0}^{2}&=-\frac{2\bar{\mu}a^{2}E}{\hbar^{2}}.\end{array}$ (17)

Let us suppose,

$R(s)=\phi_{1}(s)\phi_{2}(s),$ (18)

be the solution of Eq. (16). Following nath2021 ; nath2021.epjp ; NU , it is
then obtained that,

$\sigma(s)\phi_{2}^{{}^{\prime\prime}}(s)+\tau(s)\phi_{2}^{\prime}(s)+\nu\,\phi_{2}(s)=0,$
(19)

and

$\begin{array}[]{l}\phi_{1}(s)=\displaystyle
e^{\int\frac{\pi(s)}{\sigma(s)}ds},\end{array}$ (20)

where

$\begin{array}[]{ll}\pi(s)&=\frac{\sigma^{\prime}(s)-\widetilde{\tau}(s)}{2}\pm\sqrt{\left(\frac{\sigma^{\prime}(s)-\widetilde{\tau}(s)}{2}\right)^{2}-\widetilde{\sigma}(s)+k\,\sigma(s)},\\\
\tau(s)&=\widetilde{\tau}(s)+2\pi(s),~{}\bar{\nu}=k+\pi^{\prime}(s),\end{array}$
(21)

with $\bar{\nu}$ and $k$ being real constants. Since $\pi(s)$ is a polynomial
in $s$, one has to find $k$ in such a way that,
$\left(\frac{\sigma^{\prime}(s)-\widetilde{\tau}(s)}{2}\right)^{2}-\widetilde{\sigma}(s)+k\,\sigma(s)$
is the square of a polynomial in $s$. Then the solutions of Eq. (19) can be
written as follows,

$\phi_{2,n_{r}}(s)=\frac{1}{\rho(s)}\frac{d^{n_{r}}}{ds^{n_{r}}}\left[\sigma^{n_{r}}(s)\rho(s)\right],$
(22)

and the corresponding eigenvalues are obtained as,

$\nu_{n_{r}}=-{n_{r}}\,\tau^{\prime}(s)-\frac{{n_{r}}({n_{r}}-1)}{2}\sigma^{{}^{\prime\prime}}(s),~{}{n_{r}}=0,1,2,\dots,$
(23)

where

$\begin{array}[]{l}\rho(s)=\displaystyle\left[\sigma(s)\right]^{-1}\,e^{\int\frac{\tau(s)}{\sigma(s)}ds}.\end{array}$
(24)

According to NU method, and following the prescription nath2021 ;
nath2021.epjp ; NU , one gets a pair of $k$ as,

$\begin{array}[]{ll}k_{\pm}=\frac{2\bar{\mu}a^{2}(\alpha-\beta)}{\hbar^{2}}-L(L+1)y_{2}\pm\sqrt{C(2L_{1}-1)},\end{array}$
(25)

where

$L_{1}=\frac{1}{2}+\sqrt{\frac{1}{4}+\frac{2\bar{\mu}a^{2}\beta}{\hbar^{2}}+L(L+1)y_{3}}.$
(26)

Now, application of Eq. (25) in Eq. (21) yields,

$\begin{array}[]{ll}\pi(s)&=-\frac{s}{2}\pm\left\\{\begin{array}[]{lll}\left(\sqrt{C}-L_{1}+\frac{1}{2}\right)s+\sqrt{C},&k=k_{+},&k_{+}-B>0\\\
\left(\sqrt{C}+L_{1}-\frac{1}{2}\right)s+\sqrt{C},&k=k_{-},&k_{-}-B>0\\\
\left(\sqrt{C}-L_{1}+\frac{1}{2}\right)s-\sqrt{C},&k=k_{+},&k_{+}-B<0\\\
\left(\sqrt{C}+L_{1}-\frac{1}{2}\right)s-\sqrt{C},&k=k_{-},&k_{-}-B<0\end{array}\right\\}.\end{array}$
(27)

We have chosen $k=k_{-}$ with $k_{-}-B<0$, and selected,

$\pi(s)=-\left(\sqrt{C}+L_{1}\right)s+\sqrt{C}.$ (28)

This leads to Gradshteyn ,

$\begin{array}[]{l}\rho(s)=s^{2\sqrt{C}}(1-s)^{2L_{1}-1},\\\
\phi_{1}(s)=s^{\sqrt{C}}(1-s)^{L_{1}},\end{array}$ (29)

and

$\begin{array}[]{ll}\phi_{2,n_{r}}&=({n_{r}})!\,P_{n_{r}}^{(2\sqrt{C},2L_{1}-1)}(1-2s),\\\
&=\frac{\Gamma\left(n_{r}+2\sqrt{C}+1\right)}{\Gamma\left(2\sqrt{C}+1\right)}{}_{2}F_{1}\left(-n_{r},n_{r}+2\sqrt{C}+2L_{1};2\sqrt{C}+1;s\right),\end{array}$
(30)

where $P_{n}^{(a,b)}(x)$ is the Jacobi polynomial in $x$ of degree $n$ with
parameters $a,b$, while ${}_{2}F_{1}(a,b;c;x)$ is the Hypergeometric function,
defined by
${}_{2}F_{1}(a,b;c;x)=\sum\limits_{k=0}^{\infty}\frac{(a)_{k}(b)_{k}}{(c)_{k}}\frac{x^{k}}{k!}$
and $(a)_{k}=\frac{\Gamma(a+k)}{\Gamma(a)}=a(a+1)(a+2)...(a+k-1)$ is the
Pochhammer symbol. The eigenvalues $E_{n_{r},l}$ are then given as,

$\begin{array}[]{ll}E_{n_{r},\ell}&=\frac{\hbar^{2}L(L+1)y_{1}}{2\bar{\mu}a^{2}}-\frac{\hbar^{2}}{2\bar{\mu}a^{2}}\left(\frac{\frac{\bar{\mu}a^{2}\alpha}{\hbar^{2}}-L(L+1)(y_{2}-y_{3})/2}{n_{r}+L_{1}}-\frac{n_{r}+L_{1}}{2}\right)^{2}.\end{array}$
(31)

Therefore, the radial wave function can be expressed as Gradshteyn ,

$R_{n_{r},\ell}(r)=N_{{n_{r}},\ell}\,s^{\sqrt{C_{n_{r},\ell}}}(1-s)^{L_{1}}\,{}_{2}F_{1}\left(-{n_{r}},{n_{r}}+2\sqrt{C_{n_{r},\ell}}+2L_{1};2\sqrt{C_{n_{r},\ell}}+1;s\right),$
(32)

where

$N_{{n_{r}},\ell}=\left[\frac{2\sqrt{C_{n_{r},\ell}}(n_{r}+\sqrt{C_{n_{r},\ell}}+L_{1})\Gamma(n_{r}+2\sqrt{C_{n_{r},\ell}}+1)\Gamma(n_{r}+2\sqrt{C_{n_{r},\ell}}+2L_{1})}{a({n_{r}})!(n_{r}+\sqrt{C_{n_{r},\ell}})\Gamma(n_{r}+2L_{1})\left[\Gamma(2\sqrt{C_{n_{r},\ell}}+1)\right]^{2}}\right]^{\frac{1}{2}},$
(33)

is the normalization constant, and
$C_{n_{r},\ell}=-\frac{2\bar{\mu}a^{2}E_{n_{r},\ell}}{\hbar^{2}}+(\ell+\frac{D-3}{2})(\ell+\frac{D-1}{2})x_{1}$.
Finally, the explicit form of eigenfunctions can be written as,

$\begin{array}[]{lr}\psi_{n_{r},\ell,\\{\mu\\}}({\bf
r})&=N_{{n_{r}},\ell}r^{\frac{1-D}{2}}\,s^{\sqrt{C_{n_{r},\ell}}}(1-s)^{L_{1}}\,{}_{2}F_{1}\left(-{n_{r}},{n_{r}}+2\sqrt{C_{n_{r},\ell}}+2L_{1};2\sqrt{C_{n_{r},\ell}}+1;s\right)Y_{\ell,\\{\mu\\}}(\Omega),\\\
&s=e^{-r/a}.\end{array}$ (34)

It is to be noted that the approximation (14) is well defined for convex
combination $(0\leq\lambda_{j}\leq
1,~{}j=1(1)4,~{}\sum\limits_{j=1}^{4}\lambda_{j}=1)$ for Eckart potential, and
one may consider a linear combination (may assume negative values) for
different values of $\lambda_{j},~{}j=1(1)4$, if they satisfy the following
relations,

$\begin{array}[]{l}\sum\limits_{j=1}^{4}\lambda_{j}=1,\\\
\sum\limits_{j=1}^{4}\lambda_{j}x_{1j}\geq 0,\\\
\frac{1}{4}+\frac{2\bar{\mu}a^{2}\beta}{\hbar^{2}}+(\ell+\frac{D-3}{2})(\ell+\frac{D-1}{2})\sum\limits_{j=1}^{4}\lambda_{j}x_{3j}\geq
0.\end{array}$ (35)

Table 1:  Energies, $E_{n_{r},\ell}$, within different approximations, for
$\hbar=\bar{\mu}=1,\alpha=\frac{1}{a},a=1/0.025,D=3$. The upper and lower
portions refer to $\beta=0.0001$ and 0.0005. For details, see text.

$n_{r}$ | $\ell$ |  $-E_{n_{r},\ell}[f_{1}]$11footnotemark: 1 |  $-E_{n_{r},\ell}[f_{2}]$22footnotemark: 2 | $-E_{n_{r},\ell}[f_{3}]$ | $-E_{n_{r},\ell}[f_{4}]$ | $-E_{n_{r},\ell}[f_{5}^{(c)}]$ | $-E_{n_{r},\ell}[f_{5}^{(d)}]$ | Ref. lucha1999  | GPS33footnotemark: 3
---|---|---|---|---|---|---|---|---|---
$\beta=0.0001$
0 | 1 | 0.1008879 | 0.1015944 | 0.1008358 | 0.1008358 | 0.1010119 | 0.1008410 | 0.1008358 | 0.1008359
0 | 2 | 0.0415198 | 0.0414791 | 0.0413635 | 0.0413635 | 0.0414643 | 0.0413792 | 0.0413642 | 0.0413642
0 | 3 | 0.0193308 | 0.0189461 | 0.0190183 | 0.0190183 | 0.0191600 | 0.0190495 | 0.0190220 | 0.0190220
1 | 1 | 0.0401768 | 0.0403163 | 0.0401247 | 0.0401247 | 0.0401887 | 0.0401299 | 0.0401250 | 0.0401250
1 | 2 | 0.0190752 | 0.0189774 | 0.0189190 | 0.0189190 | 0.0190087 | 0.0189346 | 0.0189216 | 0.0189216
1 | 3 | 0.0091303 | 0.0088858 | 0.0088178 | 0.0088178 | 0.0089875 | 0.0088491 | 0.0088297 | 0.0088297
2 | 1 | 0.0185142 | 0.0185468 | 0.0184622 | 0.0184622 | 0.0185050 | 0.0184674 | 0.0184632 | 0.0184632
2 | 2 | 0.0090066 | 0.0089303 | 0.0088504 | 0.0088504 | 0.0089444 | 0.0088660 | 0.0088576 | 0.0088576
2 | 3 | 0.0040362 | 0.0038908 | 0.0037237 | 0.0037237 | 0.0039132 | 0.0037550 | 0.0037525 | 0.0037525
$\beta=0.0005$
0 | 1 | 0.0704527 | 0.0706770 | 0.0704006 | 0.0704006 | 0.0704815 | 0.0704058 | $\cdots$ | 0.0704007
0 | 2 | 0.0342158 | 0.0340877 | 0.0340595 | 0.0340596 | 0.0341431 | 0.0340752 | $\cdots$ | 0.0340604
0 | 3 | 0.0169482 | 0.0165709 | 0.0166357 | 0.0166359 | 0.0167790 | 0.0166671 | $\cdots$ | 0.0166401
1 | 1 | 0.0301503 | 0.0301782 | 0.0300982 | 0.0300982 | 0.0301402 | 0.0301034 | $\cdots$ | 0.0300987
1 | 2 | 0.0159649 | 0.0158461 | 0.0158087 | 0.0158088 | 0.0158942 | 0.0158244 | $\cdots$ | 0.0158120
1 | 3 | 0.0079688 | 0.0077346 | 0.0076563 | 0.0076564 | 0.0078281 | 0.0076877 | $\cdots$ | 0.0076699
2 | 1 | 0.0141676 | 0.0141623 | 0.0141156 | 0.0141156 | 0.0141509 | 0.0141208 | $\cdots$ | 0.0141170
2 | 2 | 0.0074840 | 0.0074025 | 0.0073277 | 0.0073278 | 0.0074208 | 0.0073434 | $\cdots$ | 0.0073363
2 | 3 | 0.0034413 | 0.0033048 | 0.0031288 | 0.0031290 | 0.0033201 | 0.0031602 | $\cdots$ | 0.0031612

11footnotemark: 1These compare with those from dong2007 . |  22footnotemark: 2These compare with those from taskin2010 , where $\xi=1.10$ and $\lambda=0.98$.
---|---
33footnotemark: 3These are calculated here for this work, using GPS gps1 ;
gps2 ; gps3 .

##  III Results and discussion

At first, Table 1 reports 9 low-lying energies corresponding to first three
non-zero-$\ell$ (1-3) values of first three $n_{r}$ (1–3). These are given for
$D=3$ having a fixed $a=1/0.025$; the upper and lower portions correspond to
$\beta=0.0001$ and 0.0005 respectively. The four columns 3–6 employ four
different approximations _viz._ , $f_{1},f_{2},f_{3},f_{4}$ from Eqs. (11) and
(12), while columns 7 and 8 provide those for approximations $f_{5}^{(c)}$ and
$f_{5}^{(d)}$ in Eq. (14), with fixed $(\xi_{1},\xi_{2})=(1.1,0.98)$ set, but
having different $\lambda_{i}$’s, namely,
$(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4})=(0.5,0.2,0.2,0.1)$ and
$(0.1,0,0,0.9)$. These are compared with accurate numerical results of
dong2007 ; taskin2010 ; lucha1999  and from the generalized pseudospectral
(GPS) method gps1 ; gps2 ; gps3 . The latter has been found to be very
successful for a variety of potentials of physical and chemical relevance. In
case of $\beta=0.0005$, no reference energies could be found except the GPS
gps1 ; gps2 ; gps3 . The approximated energies quite favorably compare with
literature results.

In our analysis, for $D=3$, under $f_{1}$ approximation, the zero-energy
states characterized by quantum numbers $(n_{r},\ell)$, for a particular
$a=a_{0}$, can be obtained from the following equation,

$\begin{array}[]{l}\displaystyle\left[\frac{8\bar{\mu}a_{0}^{2}(\alpha_{0}-\beta)}{\hbar^{2}}-\left(2n_{r}+1\right)^{2}-\left(2\ell+1\right)^{2}\right]^{2}=\displaystyle(2n_{r}+1)^{2}\left[(2\ell+1)^{2}+\frac{8\bar{\mu}a_{0}^{2}\beta}{\hbar^{2}}\right],\end{array}$
(36)

where $\alpha_{0}$ is a function of $a_{0}$. Our analysis also reveals that,
for a given $a=a_{12}$, two states identified by quantum numbers
$({n_{r1},\ell_{1}})$ and $({n_{r2},\ell_{2}})$ are degenerate possessing same
energy. This can be found from following relation,

$\begin{array}[]{l}\displaystyle\frac{2\bar{\mu}a_{12}^{2}\alpha_{12}}{\hbar^{2}}\left(\frac{1}{n_{r1}+\frac{1}{2}+\sqrt{\ell_{1}(\ell_{1}+1)+\frac{1}{4}+\frac{2\bar{\mu}a_{12}^{2}\beta}{\hbar^{2}}}}\mp\frac{1}{n_{r2}+\frac{1}{2}+\sqrt{\ell_{2}(\ell_{2}+1)+\frac{1}{4}+\frac{2\bar{\mu}a_{12}^{2}\beta}{\hbar^{2}}}}\right)\\\
\\\ \displaystyle=\frac{1\mp 1}{2}+n_{r1}\mp
n_{r2}+\sqrt{\ell_{1}(\ell_{1}+1)+\frac{1}{4}+\frac{2\bar{\mu}a_{12}^{2}\beta}{\hbar^{2}}}\mp\sqrt{\ell_{2}(\ell_{2}+1)+\frac{1}{4}+\frac{2\bar{\mu}a_{12}^{2}\beta}{\hbar^{2}}}.\end{array}$
(37)

Considering the positive sign, $a_{12}$ seems to satisfy the equation as
below,

$\begin{array}[]{l}\displaystyle\frac{2\bar{\mu}a_{12}^{2}\alpha_{12}}{\hbar^{2}}\displaystyle=\left(n_{r1}+\frac{1}{2}+\sqrt{\left(\ell_{1}+\frac{1}{2}\right)^{2}+\frac{2\bar{\mu}a_{12}^{2}\beta}{\hbar^{2}}}\right)\left(n_{r2}+\frac{1}{2}+\sqrt{\left(\ell_{2}+\frac{1}{2}\right)^{2}+\frac{2\bar{\mu}a_{12}^{2}\beta}{\hbar^{2}}}\right).\end{array}$
(38)

The degenerate energy levels can be obtained numerically under $f_{5}$
approximation. Using Eq. (38), one can find these levels,
$E_{n_{r1},2}=E_{n_{r2},2}$ which are equal at the points, $a=a_{12}$.

Table 2:  Energies, $E_{n_{r},\ell}$ for $\hbar=\bar{\mu}=1,\alpha=1/a,\beta=0.0001,a=1/0.025,(\xi_{1},\xi_{2})=(1.1,0.98)$, within $f_{5}^{(d)}$ approximation. The columns (3,8), (4,9), (5,10) refer to $D=3,4,5$. See text for details. $n_{r}$ | $\ell$ | $-E_{n_{r},\ell}(3)$ | $-E_{n_{r},\ell}(4)$ | $-E_{n_{r},\ell}(5)$ | $n_{r}$ | $\ell$ | $-E_{n_{r},\ell}(3)$ | $-E_{n_{r},\ell}(4)$ | $-E_{n_{r},\ell}(5)$
---|---|---|---|---|---|---|---|---|---
0 | 1 | 0.1008410 | 0.0631369 | 0.0413635 | 1 | 3 | 0.0088491 | 0.0058207 | 0.0035513
0 | 2 | 0.0413792 | 0.0278917 | 0.0190183 | 1 | 4 | 0.0036034 | 0.0019722 | 0.0006987
0 | 3 | 0.0190495 | 0.0130013 | 0.0086797 | 2 | 1 | 0.0184674 | 0.0129021 | 0.0088504
0 | 4 | 0.0087318 | 0.00564956 | 0.0033141 | 2 | 2 | 0.0088660 | 0.0059195 | 0.0037237
1 | 1 | 0.0401299 | 0.0274586 | 0.0189190 | 2 | 3 | 0.0037550 | 0.0021629 | 0.0009477
1 | 2 | 0.0189346 | 0.0130356 | 0.0088178 | 2 | 4 | 0.0009998 | 0.0001657 | $\cdots$

In Table 2, we offer representative state energies, for different dimensions
(3–5) for particular values of $a=1/0.025,\ \beta=0.0001$, within the
$f_{5}^{(d)}$ approximation, having $\lambda_{1}-\lambda_{4}$ as
$(0.1,0,0,0.9)$. The $D=3$ results are slightly different from those in
previous table, as the $\lambda_{i}$’s employed for $f_{5}$ are different. No
reference energies can be found for comparison. Note that, the state $E_{2,4}$
does not exist for $D=5$, for the $a$ considered in this table. Furthermore,
under $f_{1}$ approximation, for $D\geq 3$, there occurs an inter-dimensional
degeneracy, corresponding to energy levels
$E_{n_{r1},\ell_{1}}(D_{i})=E_{n_{r2},\ell_{2}}(D_{j})$ at the point
$a=a_{ij}$, given by the equation,

$\begin{array}[]{l}\displaystyle\frac{2\bar{\mu}\left(a_{12}^{D_{ij}}\right)^{2}\alpha_{12}^{D_{ij}}}{\hbar^{2}}\left(\frac{1}{n_{r1}+\frac{1}{2}+\sqrt{\Big{(}\ell_{1}+\frac{D_{i}-2}{2}\Big{)}^{2}++\frac{2\bar{\mu}\left(a_{12}^{D_{ij}}\right)^{2}\beta}{\hbar^{2}}}}\mp\frac{1}{n_{r2}+\frac{1}{2}+\sqrt{\Big{(}\ell_{2}+\frac{D_{j}-2}{2}\Big{)}^{2}+\frac{2\bar{\mu}\left(a_{12}^{D_{ij}}\right)^{2}\beta}{\hbar^{2}}}}\right)\\\
\\\ \displaystyle=\frac{1\mp 1}{2}+n_{r1}\mp
n_{r2}+\sqrt{\Big{(}\ell_{1}+\frac{D_{i}-2}{2}\Big{)}^{2}+\frac{2\bar{\mu}\left(a_{12}^{D_{ij}}\right)^{2}\beta}{\hbar^{2}}}\mp\sqrt{\Big{(}\ell_{2}+\frac{D_{j}-2}{2}\Big{)}^{2}+\frac{2\bar{\mu}\left(a_{12}^{D_{ij}}\right)^{2}\beta}{\hbar^{2}}}\end{array}$
(39)

where $\alpha_{12}^{D_{ij}}$ is again a function of $a_{12}^{D_{ij}}$.
Similarly, considering the positive sign, one obtains,

$\begin{array}[]{l}\displaystyle\frac{2\bar{\mu}\left(a_{12}^{D_{ij}}\right)^{2}\alpha_{12}^{D_{ij}}}{\hbar^{2}}=\displaystyle\left(\frac{1}{2}+n_{r1}+\sqrt{\left(\ell_{1}+\frac{D_{i}-2}{2}\right)^{2}+\frac{2\bar{\mu}\left(a_{12}^{D_{ij}}\right)^{2}\beta}{\hbar^{2}}}\right)\\\
\left(\frac{1}{2}+n_{r2}+\sqrt{\left(\ell_{2}+\frac{D_{j}-2}{2}\right)^{2}+\frac{2\bar{\mu}\left(a_{12}^{D_{ij}}\right)^{2}\beta}{\hbar^{2}}}\right).\end{array}$
(40)

The value of $a=a_{0}^{D}$, corresponding to a zero-energy state occurs for
$(n_{r},\ell)$ quantum numbers, which for $D>3$, under $f_{1}$ approximation,
can be estimated from,

$\begin{array}[]{l}\displaystyle\left[\frac{8\bar{\mu}(a_{0}^{D})^{2}(\alpha_{0}^{D}-\beta)}{\hbar^{2}}-\left(2n_{r}+1\right)^{2}-\left(2\ell+D-2\right)^{2}\right]^{2}=\displaystyle(2n_{r}+1)^{2}\left[\left(2\ell+D-2\right)^{2}+\frac{8\bar{\mu}(a_{0}^{D})^{2}\beta}{\hbar^{2}}\right].\end{array}$
(41)

From Eqs. (40) and (41) we see that, $a_{12}^{D_{ij}}=a_{12},$ if
$D_{i}=D_{j}=3$ and $a_{0}^{D}=a_{0}$ if $D=3$.

##  IV Conclusion

Analytical eigenvalues and eigenfunctions are obtained for  Eckart potential
from a solution of $D$-dimensional SE within the rubric of Nikiforov-Uvarov
method. A new, improved approximation to the centrifugal term is proposed for
this potential, which is based on physical grounds.  This general form is
intuitively arrived from an amalgamation of two well-known existing
approximations for centrifugal potential, namely, Greene-Aldrich and Pekeris-
type. This required a linear combination of four approximating functions. The
condition on the involved parameters is also discussed. It offers an accurate
representation of the potential throughout the entire effective domain of $r$.
Energy states corresponding to arbitrary quantum numbers ($\ell\neq 0$) are
presented and compared critically amongst each other, as well as, with
available theoretical reference results.  The approach can be easily extended
to other exponential-type potentials of interest, such as Deng-Fan, Manning-
Rosen, Pöschl-Teller, Hulthén etc. Some of these may be taken up in future
communications.

## Acknowledgement

Financial support from MATRICS, DST-SERB, New Delhi (sanction order:
MTR/2019/000012) is gratefully acknowledged. Partial funding from DST SERB
(sanction order: CRG/2019/000293) is appreciated.

## ORCID

Debraj Nath: https://orcid.org/0000-0001-9937-7032
Amlan K. Roy: https://orcid.org/0000-0001-5555-8915

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# The Axiom of Choice in the $\kappa$-Mantle

Andreas Lietz111Institut für Mathematische Logik und Grundlagenforschung,
Universität Münster, Einsteinstrasse 62, 48149 Münster, FRG.  222Current
address: Institut für Diskrete Mathematik und Geometrie, TU Wien, Wiedner
Hauptstrasse 8-10/104, 1040 Wien, AT
This paper is part of the authors PhD thesis.

(September 2023)

###### Abstract

Usuba has asked whether the $\kappa$-mantle, the intersection of all grounds
that extend to $V$ via a forcing of size ${<}\kappa$, is always a model of
$\mathrm{ZFC}$. We give a negative answers by constructing counterexamples
where $\kappa$ is a Mahlo cardinal, $\kappa=\omega_{1}$ and where $\kappa$ is
the successor of a regular uncountable cardinal.

##  1 Introduction

Set-Theoretic Geology is the study of the structure of grounds, that is inner
models of $\mathrm{ZFC}$ that extend to $V$ via forcing, and associated
concepts. Motivated by the hope to uncover canonical structure hidden
underneath generic sets, the mantle was born.

######  Definition 1.1.

The mantle, denoted $\mathbb{M}$, is the intersection of all grounds.

This definition only makes sense due to the uniform definability of grounds.

######  Fact 1.2.

There is a first order $\in$-formula $\varphi(x,y)$ such that

$W_{r}=\\{x|\varphi(x,r)\\}$

defines a ground for all $r\in V$ and all grounds are of this form. Moreover,
if $\kappa$ is a cardinal and $W$ extends to $V$ via a forcing of size
${<}\kappa$ then there is $r\in V_{\kappa}$ with $W=W_{r}$.

This was proven independently by Woodin [Woo11] [Woo04], Laver [Lav07] and was
later strengthened by Hamkins, see [FHR15].

This allows us to quantify freely over grounds as we will frequently do.
It was quickly realized that every model of $\mathrm{ZFC}$ is the mantle of
another model of $\mathrm{ZFC}$, see [FHR15], which eradicated any chance of
finding nontrivial structure in the mantle. However, the converse question
remained open for some while, namely whether the mantle is provably a model of
$\mathrm{ZFC}$. This tough nut was cracked by Toshimichi Usuba.

######  Fact 1.3 (Usuba,[Usu17]).

The mantle is always a model of $\mathrm{ZFC}$.

Thereby the mantle was established as a well behaved canonical object in the
theory of forcing. Fuchs-Hamkins-Reitz [FHR15] suggested to study restricted
forms of the mantle.

######  Definition 1.4.

Let $\Gamma$ be a class333In this case, we think of $\Gamma$ as a definition,
possibly with ordinal parameters, so that $\Gamma$ can be evaluated grounds of
$V$. $\Gamma$ of forcings.

  1. $(i)$

A $\Gamma$-ground is a ground $W$ that extends to $V$ via a forcing
$\mathbb{P}\in\Gamma^{W}$.

  2. $(ii)$

The $\Gamma$-mantle $\mathbb{M}_{\Gamma}$ is the intersection of all
$\Gamma$-grounds.

  3. $(iii)$

We say that the $\Gamma$-grounds are downwards directed if for any two
$\Gamma$-grounds $W_{0},W_{1}$ there is a $\Gamma$-ground $W_{\ast}\subseteq
W_{0},W_{1}$.

  4. $(iv)$

We say that the $\Gamma$-grounds are downwards set-directed if for any set-
indexed collection of $\Gamma$-grounds $\langle W_{r}\mid r\in X\rangle$ there
is a $\Gamma$-ground $W_{\ast}$ contained in all $W_{r}$ for $r\in X$.

  5. $(v)$

We say that $\Gamma$ is ground absolute if the $\Gamma$-grounds of a
$\Gamma$-ground $W$ are exactly those common grounds of $V$ and $W$ that are
$\Gamma$-grounds from the perspective of $V$, i.e. being a $\Gamma$-ground is
absolute between $V$ and all $\Gamma$-grounds.

######  Remark 1.5.

Note that if $\Gamma$ is provably (in $\mathrm{ZFC}$) closed under quotients
and two-step iterations then $\Gamma$ is ground absolute.

Fuchs-Hamkins-Reitz [FHR15] have shown abstractly that if $\Gamma$ is ground
absolute and has directed grounds then
$\mathbb{M}_{\Gamma}\models\mathrm{ZF}$. To prove
$\mathbb{M}\models\mathrm{AC}$ they seemingly need the stronger assumption
that the $\Gamma$-grounds are downwards set-directed, the argument is as
follows: Suppose $X\in\mathbb{M}$ is not wellordered in $\mathbb{M}$. Then for
every wellorder $\prec$ of $X$, we choose $W_{\prec}$ a $\Gamma$-ground from
which $\prec$ is missing. By downwards set directedness, there is a
$\Gamma$-ground $W$ contained in all such grounds $W_{\prec}$, but then $X\in
W$ is not wellordered in $W$ either, contradiction. The main result of this
part shows that indeed simple downwards directedness does not suffice to prove
choice in $\mathbb{M}_{\Gamma}$ in general.

We will be interested in $\mathbb{M}_{\Gamma}$ for $\Gamma$ the class of all
forcings of size ${<}\kappa$, where $\kappa$ is some given cardinal. In this
case, we denote the $\Gamma$-mantle by $\mathbb{M}_{\kappa}$ and call it the
$\kappa$-mantle. The associated grounds are the $\kappa$-grounds. The interest
of the $\kappa$-mantle arose in different contexts.

The following is known:

######  Fact 1.6 (Usuba, [Usu18]).

If $\kappa$ is a strong limit then $\mathbb{M}_{\kappa}\models\mathrm{ZF}$.

Usuba proved this by showing that the $\kappa$-grounds are directed in this
case. Usuba subsequently asked:

######  Question 1.7 (Usuba, [Usu18]).

Is $\mathbb{M}_{\kappa}$ always a model of $\mathrm{ZFC}$?

We will answer this question in the negative by providing counterexamples for
three different types of cardinals $\kappa$.

We also mention that Fuchs-Hamkins-Reitz demonstrated that
$\mathbb{M}_{\Gamma}$ can fail to be a model of choice for a different class
of forcings, $\Gamma=\\{\sigma\text{-closed forcings}\\}$.

######  Fact 1.8 (Fuchs-Hamkins-Reitz, [FHR15]).

If $\Gamma$ is the class of all $\sigma$-closed forcings it is consistent that
$\mathbb{M}_{\Gamma}\models\mathrm{ZF}\wedge\neg\mathrm{AC}$.

It turns out that there is an interesting tension between large cardinal
properties of $\kappa$ and the failure of choice in $\mathbb{M}_{\kappa}$. On
the one side, Usuba has shown:

######  Fact 1.9 (Usuba, [Usu18]).

If $\kappa$ is extendible then $\mathbb{M}_{\kappa}=\mathbb{M}$. In particular
$\mathbb{M}_{\kappa}$ is a model of $\mathrm{ZFC}$.

Indeed, this result was the initial motivation of investigating the
$\kappa$-mantle. Sargsyan-Schindler [SS18] showed that a similar situation
arises in the least iterable inner model with a strong cardinal above a Woodin
cardinal for $\kappa$ the unique strong cardinal in this universe. See also
[SSS21] and [Sch22b] for further results in this direction.
On another note, Schindler has proved the following.

######  Fact 1.10 (Schindler, [Sch18]).

If $\kappa$ is measurable then $\mathbb{M}_{\kappa}\models\mathrm{ZFC}$.

The big difference to Fact 1.9 is that the existence of a measurable is
consistent with the failure of the Bedrock Axiom444The Bedrock Axiom states
that the universe has a minimal ground, which turns out to be equivalent to
“$\mathbb{M}$ is a ground”.. Particularly, we might have
$\mathbb{M}_{\kappa}\neq\mathbb{M}$ for $\kappa$ measurable.
If we go even lower in the large cardinal hierarchy then even less choice
principles seem to be provable in the corresponding mantle. The relevant
results here are due to Farmer Schlutzenberg.

######  Fact 1.11 (Schlutzenberg, [Sch22a]).

Suppose that $\kappa$ is weakly compact. Then

  1. $(i)$

$\mathbb{M}_{\kappa}\models\kappa\text{-}\mathrm{DC}$ and

  2. $(ii)$

for every $A\in H_{\kappa^{+}}\cap\mathbb{M}_{\kappa}$,

$\mathbb{M}_{\kappa}\models``A\in H_{\kappa^{+}}\text{ is wellorderable}".$

######  Definition 1.12.

Suppose $\alpha$ is an ordinal and $X$ is a set. $({<}\alpha,X)$-choice holds
if for any $\beta<\alpha$ and any sequence $\vec{x}\coloneqq\langle
x_{\gamma}\mid\gamma<\beta\rangle$ of nonempty elements of $X$ there is a
choice sequence for $\vec{x}$, that is a sequenece $\langle
y_{\gamma}\mid\gamma<\beta\rangle$ with $y_{\gamma}\in x_{\gamma}$ for all
$\gamma<\beta$.

######  Fact 1.13 (Schlutzenberg, [Sch22a]).

Suppose $\kappa$ is inaccessible. Then we have

  1. $(i)$

$V_{\kappa}\cap\mathbb{M}_{\kappa}\models\mathrm{ZFC}$ and

  2. $(ii)$

$\mathbb{M}_{\kappa}\models``({<}\kappa,H_{\kappa^{+}})\text{-choice}"$.

### Acknowledgments

The author thanks Ralf Schindler and Farmer Schlutzenberg for valuable
discussions on the present topic. Moreover, the author thanks Ralf Schindler
for his support and guidance during his work on this project.
Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy EXC 2044 390685587,
Mathematics Münster: Dynamics - Geometry - Structure.

##  2 Overview

In Section 3.1, we will argue that “$\kappa$ is measurable” cannot be replaced
by “$\kappa$ is Mahlo” in Fact 1.10, as wells as that
$({<}\kappa,H_{\kappa^{+}})$-choice cannot be strengthened to
$({<}\kappa+1,H_{\kappa^{+}})$-choice in Fact 1.13.

######  Theorem 2.1.

If $\mathrm{ZFC}$ is consistent with the existence of a Mahlo cardinal, then
it is consistent with $\mathrm{ZFC}$ that there is a Mahlo cardinal $\kappa$
so that $\mathbb{M}_{\kappa}$ fails to satisfy the axiom of choice. In fact we
may have

$\mathbb{M}_{\kappa}\models``({<}\kappa+1,H_{\kappa^{+}})\text{-choice
fails}".$

In Section 3.2, we will investigate the $\kappa$-mantle for
$\kappa=\omega_{1}$, as well as the $\Gamma$-mantle where
$\Gamma=\\{\text{Cohen forcing}\\}$, denoted by $\mathbb{M}_{\mathsf{C}}$. We
will first proof that these mantles are always models of $\mathrm{ZF}$ and
will go on to provide a result analogous to Theorem 2.1.

######  Theorem 2.2.

It is consistent relative to a Mahlo cardinal that both
$\mathbb{M}_{\omega_{1}}$ and $\mathbb{M}_{\mathsf{C}}$ fail to satisfy the
axiom of choice.

In Section 3.3, we will generalize this to any successor of a regular
cardinal.

######  Theorem 2.3.

Suppose that

  1. $(i)$

$\mathrm{GCH}$ holds,

  2. $(ii)$

the Ground Axiom555The Ground Axiom states that there is no nontrivial ground.
See [Rei06] for more information on this axiom. holds and

  3. $(iii)$

$\kappa$ is a regular uncountable cardinal.

Then there is a cardinal preserving generic extension in which the
$\kappa^{+}$-mantle fails to satisfy the axiom of choice.

In this case however, it is not known if the $\kappa^{+}$-mantle is a model of
$\mathrm{ZF}$ in general. The proof of all these three theorems follows a
similar pattern, though the details differ from case to case and it seems that
we cannot employ a fully unified approach.

##  3 The Axiom of Choice May Fail in $\mathbb{M}_{\kappa}$

###  3.1 The case “$\kappa$ is Mahlo”

Here, we will construct a model where the $\kappa$-mantle for a Mahlo cardinal
$\kappa$ does not satisfy the axiom of choice. We will start with $L$ and
assume that $\kappa$ is the least Mahlo there. The final model will be a
forcing extension of $L$ by

$\mathbb{P}=\prod_{\lambda\in
I\cap\kappa}^{\text{${<}\kappa$-support}}\mathrm{Add}(\lambda,1)$

where $I$ is the class of all inaccessible cardinals. We define $\mathbb{P}$
to be a product forcing and not an iteration (in the usual sense), as we want
to generate many $\kappa$-grounds. Let $G$ be $\mathbb{P}$-generic over $L$.
We will show that $\kappa$ is still Mahlo in $L[G]$ and that
$\mathbb{M}_{\kappa}^{L[G]}$ does not satisfy the axiom of choice. We remark
that, would we start with a model in which $\kappa$ is measurable,
$\mathbb{P}$ would provably force $\kappa$ to not be measurable.
First, let’s fix notation. For $\lambda<\kappa$, we may factor $\mathbb{P}$ as
$\mathbb{P}_{\leq\lambda}\times\mathbb{P}_{>\lambda}$ where in each case we
only take a product over all $\gamma\in I\cap\kappa$ with $\gamma\leq\lambda$
and $\gamma>\lambda$ respectively. Observe that $\mathbb{P}_{>\lambda}$ is a
${<}\kappa$-support product while $\mathbb{P}_{\leq\lambda}$ is a full support
product. We also factor $G$ as $G_{\leq\lambda}\times G_{>\lambda}$
accordingly. For $\lambda\in I\cap\kappa$ we denote the generic for
$\mathrm{Add}(\lambda,1)^{L}$ induced by $G$ as $g_{\lambda}$. In addition to
this, for $\alpha\leq\kappa$ we denote the $\alpha$-th inaccessible cardinal
by $I_{\alpha}$.
For $\alpha<\kappa$ let $E_{\alpha}\colon\kappa\rightarrow 2$ be the function
induced by $g_{I_{\alpha}}$. It will be convenient to think of $G$ as a
$\kappa\times\kappa$-matrix $M$ which arises by stacking the maps
$(E_{\alpha})_{\alpha<\kappa}$ on top of each other, starting with $E_{I_{0}}$
and proceeding downwards, and then filling up with $0$’s to produce rows of
equal length $\kappa$. Let us write

$e_{\alpha,\beta}=\begin{cases}E_{\alpha}(\beta)&\text{if }\beta<I_{\alpha}\\\
0&\text{else.}\\\ \end{cases}$

The $(e_{\alpha,\beta})_{\alpha,\beta<\kappa}$ are the entries of $M$:

${e_{0,0}}$${e_{0,1}}$${e_{0,2}}$${\cdots}$${0}$${\cdots}$${0}$${\cdots}$${0}$${\cdots}$${e_{1,0}}$${e_{1,1}}$${e_{1,2}}$${\cdots}$${e_{1,I_{0}}}$${\cdots}$${0}$${\cdots}$${0}$${\cdots}$${\vdots}$${\vdots}$${\vdots}$${\ddots}$${\vdots}$${\ddots}$${0}$${\cdots}$${0}$${\cdots}$${e_{\alpha,0}}$${e_{\alpha,1}}$${e_{\alpha,2}}$${\cdots}$${e_{\alpha,I_{0}}}$${\cdots}$${e_{\alpha,I_{1}}}$${\cdots}$${0}$${\cdots}$${\vdots}$${\vdots}$${\vdots}$${\ddots}$${\vdots}$${\vdots}$${\vdots}$${\ddots}$${0}$${\cdots}$$\left(\vbox{\hrule
height=60.14885pt,depth=60.14885pt,width=0.0pt}\right.$$\left.\vbox{\hrule
height=60.14885pt,depth=60.14885pt,width=0.0pt}\right)$$r_{1}$$c_{2}$$M_{\geq\alpha}$$M=$

We will give the $\alpha$-th row of $M$ the name $r_{\alpha}$ and we denote
the $\beta$-th column of $M$ by $c_{\beta}$. One trivial but key observation
is that $r_{\alpha}$ carries the same information as $g_{I_{\alpha}}$.

We will be frequently interested in the matrix $M$ with its first $\alpha$
rows deleted for some $\alpha<\kappa$, so we will give this matrix the name
$M_{\geq\alpha}$. Note that $M_{\geq\alpha}$ corresponds to the generic
$G_{\geq I_{\alpha}}$. Finally observe that we may think of conditions in
$\mathbb{P}$ as partial matrices that approximate such a matrix $M$ in the
sense that they already have the trivial $0$’s in the upper right corner, in
any row $\alpha<\kappa$ they have information for ${<}I_{\alpha}$ many
$\beta<I_{\alpha}$ on whether $e_{\alpha,\beta}$ is $0$ or $1$ and they
contain non-trivial information in less than $\kappa$-many rows.

######  Lemma 3.1.

$L$ and $L[G]$ have the same inaccessibles.

###### Proof.

First, we show that all limit cardinals of $L$ are limit cardinals in $L[G]$.
It is enough to prove that all double successors $\delta^{++}$ are preserved.
This is obvious for $\delta\geq\kappa$ as $\mathbb{P}$ has size $\kappa$. For
$\delta<\kappa$, $\mathbb{P}_{>\delta}$ is ${\leq}\delta^{++}$-closed so that
all cardinals $\leq\delta^{++}$ are preserved in $L[G_{>\delta}]$.
Furthermore, $\mathbb{P}_{<\delta}$ has size at most $\delta^{+}$ in
$L[G_{>\delta}]$ by $\mathrm{GCH}$ in $L$. Hence $\delta^{++}$ is still a
cardinal in $L[G]$.
Now we have to argue that all $\lambda\in I$ remain regular. Again, this is
clear if $\lambda>\kappa$. On the other hand, assume
$\delta\coloneqq\operatorname{cof}(\lambda)^{L[G]}<\lambda$. As
$\mathbb{P}_{>\delta}$ is ${\leq}\delta$-closed, $\lambda$ is still regular in
$L[G_{>\delta}]$. Hence, a witness to $\operatorname{cof}(\lambda)=\delta$
must be added in the extension of $L[G_{{>}\delta}]$ by
$\mathbb{P}_{\leq\delta}$. But this forcing has size $<\lambda$ in
$L[G_{{>}\delta}]$ and thus could not have added such a sequence. ∎

In fact, $\mathbb{P}$ does not collapse any cardinals (if $V=L$), but some
more work is required to prove this. This is, however, not important for our
purposes. Next, we aim to show that $\kappa$ remains Mahlo in $L[G]$.

To prove this, it is convenient to introduce a generalization of Axiom A.

######  Definition 3.2.

For $\kappa$ an ordinal, $\lambda$ a cardinal we say that a forcing
$\mathbb{Q}$ satisfies Axiom A$(\kappa,\lambda)$, abbreviated by
$\mathrm{AA}(\kappa,\lambda)$, if there is a sequence
$\langle\leq_{\alpha}\mid\alpha<\kappa\rangle$ of partial orders on
$\mathbb{Q}$ so that

  1. $(\mathrm{AA}.i)$

$\forall\alpha\leq\beta<\kappa\
\leq_{\beta}\subseteq\leq_{\alpha}\subseteq\leq_{\mathbb{Q}}$,

  2. $(\mathrm{AA}.ii)$

for all antichains $A$ in $\mathbb{Q}$, $\alpha<\kappa$ and $p\in\mathbb{Q}$
there is $q\leq_{\alpha}p$ so that $|\\{a\in A\mid a\|q\\}|<\lambda$ and

  3. $(\mathrm{AA}.iii)$

for all $\beta<\kappa$ if $\vec{p}=\langle p_{\alpha}\mid\alpha<\beta\rangle$
satisfies $p_{\gamma}\leq_{\alpha}p_{\alpha}$ for all $\alpha<\gamma<\beta$
then there is a fusion $p_{\beta}$ of $\vec{p}$, that is
$p_{\beta}\leq_{\alpha}p_{\alpha}$ for all $\alpha<\beta$.

######  Remark 3.3.

The usual Axiom A is thus Axiom A$(\omega+1,\omega_{1})$.

######  Proposition 3.4.

Suppose $\lambda$ is regular uncountable cardinal and $\mathbb{Q}$ satisfies
$\mathrm{AA}(\lambda,\lambda)$. Then $\mathbb{Q}$ preserves stationary subsets
of $\lambda$.

###### Proof.

Suppose $S\subseteq\lambda$ is stationary, $\dot{C}$ is a $\mathbb{Q}$-name
for a club in $\lambda$ and $p\in\mathbb{P}$. We will imitate the standard
proof that a ${<}\kappa$-closed forcing preserves stationary sets. Let
$\langle\leq_{\alpha}\mid\alpha<\lambda\rangle$ witness that $\mathbb{Q}$
satisfies $\mathrm{AA}(\lambda,\lambda)$.

######  Claim 3.5.

For any $q\in\mathbb{Q},\alpha<\lambda$ there is $r\leq_{\alpha}q$ and some
$\alpha<\gamma<\lambda$ with $q\Vdash\check{\gamma}\in\dot{C}$.

###### Proof.

Construct a sequence $\langle q_{\alpha}\mid\alpha<\omega\rangle$ of
conditions in $\mathbb{Q}$ and an ascending sequence $\langle\gamma_{n}\mid
n<\omega\rangle$ of ordinals with

  1. $(i)$

$q_{0}=q$, $\gamma_{0}=\alpha$,

  2. $(ii)$

$q_{n+1}\leq_{\alpha+n}q_{n}$ for all $n<\omega$ and

  3. $(iii)$

$q_{n+1}\Vdash``\dot{C}\cap(\check{\gamma}_{n},\check{\gamma}_{n+1})\neq\emptyset$

for all $n<\omega$. The construction is immediate using that $\lambda$ is
regular uncountable and $(\mathrm{AA}.iii)$. Then by $(\mathrm{AA}.ii)$, there
is $q_{\ast}\leq_{\alpha}q$ which is below all $q_{n}$, $n<\omega$. It follows
that

$q_{\ast}\Vdash\check{\gamma}_{\ast}\in\dot{C}$

where $\gamma_{\ast}=\sup_{n<\omega}\gamma_{n}$. ∎

Suppose toward a contradiction that $p\Vdash\dot{C}\cap\check{S}=\emptyset$.
By the claim above, we can build sequences $\langle
p_{\alpha}\mid\alpha<\lambda\rangle$ of conditions in $\mathbb{Q}$ and an
increasing sequence $\langle\gamma_{\alpha}\mid\alpha<\lambda\rangle$ of
ordinals below $\lambda$ so that

  1. $(i)$

$p_{0}=p$,

  2. $(ii)$

$p_{\beta}\leq_{\alpha}p_{\alpha}$ for all $\alpha\leq\beta<\lambda$ and

  3. $(iii)$

$p_{\alpha+1}\Vdash\check{\gamma}_{\alpha}\in\dot{C}$ for all
$\alpha<\lambda$.

Let $D$ be the set of all limit points ${<}\lambda$ of
$\\{\gamma_{\alpha}\mid\alpha<\lambda\\}$. For any $\alpha<\lambda$, we have

$p_{\alpha+1}\Vdash\check{D}\cap\gamma_{\alpha}\subseteq\dot{C}$

which shows that $D\cap S=\emptyset$, contradiction. ∎

######  Lemma 3.6.

$\mathbb{P}$ satisfies $\mathrm{AA}(\kappa,\kappa)$.

###### Proof.

For $\gamma<\kappa$ define $\leq_{\gamma}$ by $r\leq_{\gamma}q$ if $r\leq q$
and $r\upharpoonright\gamma=q\upharpoonright\gamma$ for $q,r\in\mathbb{P}$. We
will only show that $(\mathrm{AA}.ii)$ holds. So let $p\in\mathbb{P}$,
$\gamma<\kappa$ and $A\subseteq\mathbb{P}$ a maximal antichain. Let $\langle
q_{\alpha}|\alpha<\delta\rangle$ be an enumeration of all conditions in
$\mathbb{P}_{\leq gamma}$ below $p\upharpoonright\gamma+1$ with
$\delta=|\mathbb{P}_{\leq\gamma}|$. We construct a $\leq_{\gamma}$-descending
sequence $\langle p_{\alpha}|\alpha\leq\delta\rangle$ of conditions in
$\mathbb{P}$ starting with $p_{0}=p$ as follows: If $\alpha\leq\delta$ then
choose some $\leq_{\gamma}$-bound of $\langle
p_{\beta}\mid\beta<\alpha\rangle$. This is possible as $\mathbb{P}_{>\gamma}$
is ${\leq}\delta$-closed, as the next forcing only appears at the next
inaccessible. Moreover, if possible and $\alpha<\delta$ make sure that

$q_{\alpha}^{\frown}p_{\alpha}\upharpoonright(\gamma,\kappa)$

is below a condition in $A$. This completes the construction. Set $q\coloneqq
q_{\kappa}$, we will show that $q$ is compatible with at most $\delta$-many
elements of $A$. Toward this goal, suppose $a\in A$ and $q$ is compatible with
$a$. We may find some $\alpha<\kappa$ so that
$a\upharpoonright\gamma+1=q_{\alpha}$. It follows that we must have succeeded
in the construction of $p_{\alpha}$ with the additional demand that

$q_{\alpha}^{\frown}p_{\alpha}\upharpoonright(\gamma,\kappa)$

is below a condition in $A$, but this can only be true for $a$. We have shown
that for any $a\in A$ compatible with $q$ there is $\alpha<\delta$ with
$q_{\alpha}^{\frown}q\upharpoonright(\gamma,\kappa)\leq a$ and note that no
single $\alpha$ can witness this for more than one element of $A$. ∎

######  Corollary 3.7.

$\kappa$ is Mahlo in $L[G]$.

###### Proof.

This follows immediately from Lemma 3.1, Lemma 3.6 and Proposition 3.4. ∎

Next, we aim to find an easier description of $\mathbb{M}_{\kappa}^{L[G]}$.
Recall the $\lambda$-approximation property introduced by Hamkins [Ham03]:

######  Definition 3.8.

Let $W\subseteq V$ be an inner model, $\lambda$ an infinite cardinal.

  1. $(i)$

For $x\in V$, a $\lambda$-approximation of $x$ by $W$ is of the form $x\cap y$
where $y\in W$ is of size ${\leq}\lambda$.

  2. $(ii)$

$W\subseteq V$ satisfies the $\lambda$-approximation property if whenever
$x\in V$ and all $\lambda$-approximations of $x$ by $W$ are in $W$, then $x\in
W$.

All $\kappa$-grounds satisfy the $\kappa$-approximation property (cf.
[FHR15]).

######  Lemma 3.9.

$\mathbb{M}_{\kappa}^{L[G]}=\bigcap_{\lambda\in I\cap\kappa}L[G_{>\lambda}]$.

###### Proof.

Suppose $W$ is a $\kappa$-ground of $L[G]$. It is enough to find $\lambda\in
I\cap\kappa$ such that $L[G_{>\lambda}]\subseteq W$. Clearly, $\mathbb{P}\in
L\subseteq W$. As $\kappa$ is a limit of inaccessibles, we may take some
$\lambda<\kappa$ inaccessible so that $W$ is a $\lambda$-ground. Thus
$W\subseteq L[G]$ satisfies the $\lambda$-approximation property. We will show
$G_{>\lambda}\in W$ (even $G_{\geq\lambda}\in W$). Find $\alpha$ with
$\lambda=I_{\alpha}$, it is thus enough to show $M_{\geq\alpha}\in W$. To any
$\lambda$-approximation $M_{\geq\alpha}\cap a$ of $M_{\geq\alpha}$ by $W$
corresponds some $a^{\prime}\subseteq\kappa\setminus\alpha\times\kappa$,
$a^{\prime}\in W$ of size ${<}\lambda$ so that

$M_{\geq\alpha}\cap a=M_{\geq\alpha}\upharpoonright a^{\prime}\coloneqq\langle
e_{\gamma,\beta}\mid(\gamma,\beta)\in a^{\prime}\rangle.$

We will show that all such restrictions of $M_{\geq\alpha}$ are in $W$. So let
$a\in W$, $a\subseteq\kappa\setminus\alpha\times\kappa$, $|a|<\lambda$. As
$0^{\\#}$ does not exist in $W$, there is $b\in L$,
$b\subseteq\kappa\setminus\alpha\times\kappa$ of size $<\lambda$ with
$a\subseteq b$. For all $\alpha\leq\gamma<\kappa$, the set of
$\beta<I_{\gamma}$ with $(\gamma,\beta)\in b$ is bounded in $I_{\gamma}$. As
described earlier, we may think of conditions in $\mathbb{P}$ as partial
$\kappa\times\kappa$ matrices. With this in mind, the conditions
$p\in\mathbb{P}$ that contain information on the entry $e_{\gamma,\beta}$ for
all $(\gamma,\beta)\in b$ form a dense set of $\mathbb{P}$. Thus
$M\upharpoonright b=\langle e_{\gamma,\beta}\mid(\gamma,\beta)\in b\rangle$ is
essentially a condition $p\in\mathbb{P}\subseteq W$ and hence
$M\upharpoonright a=(M\upharpoonright b)\upharpoonright a\in W$. As
$W\subseteq L[G]$ satisfies the $\lambda$-approximation property, we have
$M_{\geq\alpha}\in W$.

∎

######  Remark 3.10.

The above argument shows that for any $\lambda\in I\cap\kappa$

$\mathbb{M}_{\kappa}^{L[G_{>\lambda}]}=\mathbb{M}_{\kappa}^{L[G]}.$

In fact, whenever $\delta$ is a strong limit, the $\delta$-mantle is always
absolute to any $\delta$-ground. The use of Jensen’s covering lemma in the
above argument is not essential, in fact a model in which the $\kappa$-mantle
does not satisfy choice for $\kappa$ Mahlo can be analogously constructed in
the presence of $0^{\sharp}$. However, the absence of $0^{\sharp}$ simplifies
the proof.

We will later show that $\mathcal{P}(\kappa)^{\mathbb{M}_{\kappa}^{L[G]}}$
does not admit a wellorder in $\mathbb{M}_{\kappa}^{L[G]}$. First, we analyze
which subsets of $\kappa$ $\mathbb{M}_{\kappa}^{L[G]}$ knows of. We call
$a\subseteq\kappa$ fresh if $a\cap\lambda\in L$ for all $\lambda<\kappa$.

######  Proposition 3.11.

The subsets of $\kappa$ in $\mathbb{M}_{\kappa}^{L[G]}$ are exactly the fresh
subsets of $\kappa$ in $L[G]$.

###### Proof.

First suppose $a\subseteq\kappa$, $a\in\mathbb{M}_{\kappa}^{L[G]}$. If
$\lambda<\kappa$ then $a\in L[G_{>\lambda}]$. As $\mathbb{P}_{>\lambda}$ is
${\leq}\lambda$-closed in $L$, $a\cap\lambda\in L$.
For the other direction assume $a\in L[G]$ is a fresh subset of $\kappa$ and
assume $W$ is a $\kappa$-ground of $L[G]$. There is $\lambda<\kappa$ so that
$W\subseteq L[G]$ satisfies the $\lambda$-approximation property. As $a$ is
fresh, all the $\lambda$-approximations of $a$ in $W$ are in $W$. Thus $a\in
W$. ∎

The columns $c_{\beta}$, $\beta<\kappa$, of $M$ are the fresh subsets of
$\kappa$ relevant to our argument.

######  Proposition 3.12.

All $c_{\beta}$, $\beta<\kappa$, are $\mathrm{Add}(\kappa,1)$-generic over
$L$.

###### Proof.

The map $\pi\colon\mathbb{P}\rightarrow\mathrm{Add}(\kappa,1)$ that maps
$p\in\mathbb{P}$ to the information that $p$ has on $c_{\beta}$ is well-
defined as $\mathbb{P}$ is a bounded support iteration of length $\kappa$.
Clearly, $\pi$ is a projection. ∎

This is exactly the reason we chose bounded support in the definition of
$\mathbb{P}$.

We are now in good shape to complete the argument.

######  Theorem 3.13.

$({<}\kappa+1,H_{\kappa^{+}})$-choice fails in $\mathbb{M}_{\kappa}^{L[G]}$.

###### Proof.

Note that any generic for $\mathrm{Add}(\kappa,1)^{L}$ is the characteristic
function of a fresh subset of $\kappa$ so that
$c_{\beta}\in\mathbb{M}_{\kappa}^{L[G]}$ for any $\beta<\kappa$. Of course,
the sequence $\langle c_{\beta}|\beta<\kappa\rangle$ is not in
$\mathbb{M}_{\kappa}^{L[G]}$, as one can compute the whole matrix $M$ (and
thus the whole generic $G$) from this sequence. However, we can make this
sequence fuzzy to result in an element of $\mathbb{M}_{\kappa}^{L[G]}$. Let
$\sim$ be the equivalence relation of eventual coincidence on
$({}^{\kappa}2)^{\mathbb{M}_{\kappa}^{L[G]}}$, i.e.

$x\sim y\Leftrightarrow\exists\delta<\kappa\
x\upharpoonright[\delta,\kappa)=y\upharpoonright[\delta,\kappa).$

We call $\langle[c_{\beta}]_{\sim}|\beta<\kappa\rangle$ the fuzzy sequence.

######  Claim 3.14.

The fuzzy sequence is an element of $\mathbb{M}_{\kappa}^{L[G]}$.

###### Proof.

By Lemma 3.9, it is enough to show that for every $\alpha<\kappa$, $L[G_{\geq
I_{\alpha}}]$ knows of this sequence. But $L[G_{>I_{\alpha}}]$ contains the
matrix $M_{\geq\alpha}$ and thus the sequence

$\langle c_{\beta}\upharpoonright(\kappa\setminus\alpha)|\beta<\kappa\rangle$

so that $L[G_{\geq\alpha}]$ can compute the relevant sequence of equivalence
classes from this parameter. ∎

Finally, we argue that $\mathbb{M}_{\kappa}^{L[G]}$ does not contain a choice
sequence for the fuzzy sequence666That is, there is no sequence $\langle
x_{\beta}\mid\beta<\kappa\rangle\in\mathbb{M}_{\kappa}^{L[G]}$ with
$x_{\beta}\in[c_{\beta}]_{\sim}$ for all $\beta<\kappa$.. Heading toward a
contradiction, let us assume that

$\langle x_{\beta}|\beta<\kappa\rangle\in\mathbb{M}_{\kappa}^{L[G]}$

is such a sequence. $L[G]$ knows about the sequence

$\langle\delta_{\beta}|\beta<\kappa\rangle$

where $\delta_{\beta}$ is the least $\delta$ with
$x_{\beta}\upharpoonright(\kappa\setminus\delta)=c_{\beta}\upharpoonright(\kappa\setminus\delta)$.
The set of $\lambda<\kappa$ that are closed under the map
$\beta\longmapsto\delta_{\beta}$ is club in $\kappa$. As $\kappa$ is Mahlo in
$L[G]$, there is an inaccessible $\alpha=I_{\alpha}<\kappa$ that is closed
under $\beta\longmapsto\delta_{\beta}$. Now observe that

$x_{\beta}(\alpha)=1\Leftrightarrow c_{\beta}(\alpha)=1\Leftrightarrow
r_{\alpha}(\beta)=1$

holds for all $\beta<I_{\alpha}$, so that
$r_{\alpha}\in\mathbb{M}_{\kappa}^{L[G]}$. But this is impossible as clearly
$r_{\alpha}$ is not fresh.

∎

Theorem 2.1 follows.

######  Remark 3.15.

The only critical property of $L$ that we need to make sure that
$\mathbb{M}_{\kappa}$ is not a model of choice in $L[G]$ is that $L$ has no
nontrivial grounds, i.e. $L$ satisfies the ground axiom. $\mathrm{GCH}$ is
convenient and implies that no cardinals are collapsed, but it is not
necessary. The use of Jensen’s covering lemma can also be avoided, as
discussed earlier.

###  3.2 The $\omega_{1}$-mantle

Up to now, we have focused on the $\kappa$-mantle for strong limit $\kappa$.
We will get similar results for the $\omega_{1}$-mantle. There is some
ambiguity in the definition of the $\omega_{1}$-mantle, depending on whether
or not $\omega_{1}$ is considered as a parameter or as a definition. In the
former case, it is the intersections of all grounds $W$ so that $W$ extends to
$V$ via a forcing so that $W\models|\mathbb{P}|<\omega_{1}^{V}$, where in the
latter case we would require $W\models|\mathbb{P}|<\omega_{1}^{W}$. These
mantles are in general not equal. To make the distinction clear, we give the
latter version the name “Cohen mantle” and denote it by
$\mathbb{M}_{\mathsf{C}}$. The reason for the name is, of course, that all
non-trivial countable forcings are forcing-equivalent to Cohen forcing.

######  Lemma 3.16.

$\mathbb{M}_{\omega_{1}}\models\mathrm{ZF}$ and
$\mathbb{M}_{\mathsf{C}}\models\mathrm{ZF}$.

###### Proof.

First let us do it for $\mathbb{M}_{\mathsf{C}}$. Clearly,
$\mathbb{M}_{\mathsf{C}}$ is closed under the Gödel operations. It is thus
enough to show that $\mathbb{M}_{\mathsf{C}}\cap
V_{\alpha}\in\mathbb{M}_{\mathsf{C}}$ for all $\alpha\in\mathrm{Ord}$. Let $W$
be any Cohen-ground. As Cohen-forcing is homogeneous,
$\mathbb{M}_{\mathsf{C}}^{V}$ is a definable class in $W$. Hence,
$\mathbb{M}_{\mathsf{C}}\cap V_{\alpha}=\mathbb{M}_{\mathsf{C}}\cap
V_{\alpha}^{W}\in W$. As $W$ was arbitrary, this proves the claim.

Now onto $\mathbb{M}_{\omega_{1}}$. The above argument shows that all we need
to do is show that $\mathbb{M}_{\omega_{1}}$ is a definable class in all
associated grounds. So let $W$ be such a ground. There are two cases. First,
assume that $\omega_{1}^{W}=\omega_{1}^{V}$. Then $W$ extends to $V$ via Cohen
forcing, so $\mathbb{M}_{\omega_{1}}$ is definable in $W$. Next, suppose that
$\omega_{1}^{W}<\omega_{1}^{V}$. This can only happen if $\omega_{1}^{V}$ is a
successor cardinal in $W$, say $W\models\omega_{1}^{V}=\mu^{+}$. In this case,
$W$ extends to $V$ via a forcing of $W$-size $\leq\mu$ and which collapses
$\mu$ to be countable. It is well known that in this situation, $W$ extends to
$V$ via $\mathrm{Col}(\omega,\mu)$, which is homogeneous as well, so once
again, $\mathbb{M}_{\omega_{1}}$ is a definable class in $W$. ∎

Once again, choice can fail.

######  Theorem 3.17.

Relative to the existence of a Mahlo cardinal, it is consistent that there is
no wellorder of $\mathcal{P}(\omega_{1}^{V})^{\mathbb{M}_{\omega_{1}}}$ in
$\mathbb{M}_{\omega_{1}}$.

We remark that the Mahlo cardinal is used in a totally different way than in
the last section. In the model we will construct, $\omega_{1}$ will be
inaccessible in $\mathbb{M}_{\omega_{1}}$. Let us once again assume $V=L$ for
the rest of the section and let $\kappa$ be Mahlo. Let $\mathbb{P}$ be the
“${<}\kappa$-support version of $\mathrm{Col}(\omega,{<}\kappa)$”, that is

$\mathbb{P}=\prod_{\alpha<\kappa}^{<\kappa-\text{support}}\mathrm{Col}(\omega,\alpha).$

Let us pick a $\mathbb{P}$-generic filter $G$ over $V$. From now on,
$\mathbb{M}_{\omega_{1}}$ will denote $\mathbb{M}_{\omega_{1}}^{V[G]}$ and
$\mathbb{M}_{\mathsf{C}}$ will denote $\mathbb{M}_{\mathsf{C}}^{V[G]}$.

######  Proposition 3.18.

Suppose $\mathbb{Q}$ is a forcing, $\gamma<\lambda$ and $\lambda$ is a
cardinal. If $\mathbb{Q}$ is $\mathrm{AA}(\gamma,\lambda)$ then in
$V^{\mathbb{Q}}$ there is no surjection from any $\beta<\gamma$ onto
$\lambda$.

###### Proof.

This is a straightforward adaptation of the proof that Axiom A forcings
preserve $\omega_{1}$. ∎

The following lemma is the only significant use of the Mahloness of $\kappa$.

######  Lemma 3.19.

$\mathbb{P}$ satisfies $\mathrm{AA}(\kappa,\kappa)$.

###### Proof.

We define $\leq_{\alpha}$ independent of $\alpha<\kappa$ as the order
$\leq^{\ast}$: Let $p\leq^{\ast}q$ iff $p\leq q$ and
$p\upharpoonright\mathrm{supp}(q)=q$. The only nontrivial part is showing that
for any antichain $A$ and any $p\in\mathbb{P}$ there is $q\leq^{\ast}p$ with

$|\\{a\in A\mid a\|q\\}|<\kappa.$

Let

$\mathbb{P}\upharpoonright\alpha\coloneqq\\{p\in\mathbb{P}\mid\sup\mathrm{supp}(p)<\alpha\\}$

for all $\alpha<\kappa$. We will proceed to find some $q$ with the desired
property. For convenience, we may assume that $A$ is a maximal antichain. As
$\kappa$ is Mahlo, there is a regular $\lambda<\kappa$ so that
$p\mathbb{P}\upharpoonright\lambda$ and any
$r\in\mathbb{P}\upharpoonright\lambda$ is compatible with some $a\in
A\cap\mathbb{P}\upharpoonright{\lambda}$. As $V=L$, $\diamondsuit_{\lambda}$
holds. Thus there is a sequence $\vec{d}\coloneqq\langle
d_{\alpha}\mid\alpha<\lambda\rangle$ with

  1. $(\vec{d}.i)$

$d_{\alpha}\in\mathbb{P}_{\leq\alpha}$ and

  2. $(\vec{d}.ii)$

for all $r\in\mathbb{P}_{\leq\lambda}$ there are stationarily many
$\alpha<\lambda$ with $d_{\alpha}=r\upharpoonright\alpha$.

Construct a sequence

$\langle q_{\alpha}\mid\alpha<\lambda\rangle$

of conditions in $\mathbb{P}\upharpoonright\lambda$ with
$q_{\alpha}\leq^{\ast}q_{\beta}$ for all $\alpha<\beta<\lambda$ as follows:
Set $q_{0}=p$. If $q_{\beta}$ is defined for all $\beta<\alpha$, let first
$q^{\prime}_{\alpha}=\bigcup_{\beta<\alpha}q_{\beta}$ and note that this is a
condition. Let $\gamma_{\alpha}=\sup\mathrm{supp}(q_{\alpha}^{\prime})$. Now
find $a\in A\cap\mathbb{P}\upharpoonright\lambda$ that is compatible with
$d_{\gamma_{\alpha}}$ and let

$q_{\alpha}\coloneqq
q_{\alpha}^{\prime\frown}a\upharpoonright[\gamma_{\alpha},\lambda).$

Finally, set $q=\bigcup_{\alpha<\lambda}q_{\alpha}$. We have to show that $q$
is compatible with only a few elements of $A$, so suppose $b\in A$ is
compatible with $q$. The properties of $\vec{d}$ guarantee that there is
$\alpha<\lambda$ so that

  1. $(\alpha.i)$

$\gamma_{\alpha}=\alpha$ and

  2. $(\alpha.ii)$

$d_{\alpha}=b\upharpoonright\alpha$.

Hence in the construction of $q_{\alpha+1}$ we found some $a\in
A\cap\mathbb{P}\upharpoonright\lambda$ compatible with
$b\upharpoonright\alpha$ and have
$q_{\alpha+1}\upharpoonright[\alpha,\lambda)\leq
a\upharpoonright[\alpha,\lambda)$. If $a\neq b$, then $a\perp b$ and the
incompatibility must lie in the interval $[\alpha,\lambda)$. But then
$q_{\alpha+1}$ and $b$ are incompatible as well, contradiction. Thus $b=a$ and
it follows that $q$ is compatible with at most $\lambda$-many elements of $A$.
∎

######  Corollary 3.20.

We have

  1. $(G.i)$

$\omega_{1}^{L[G]}=\kappa$ and

  2. $(G.ii)$

if $g\colon\omega\rightarrow\mathrm{Ord}\in L[G]$ then there is some
$\alpha<\kappa$ so that $g\in V[G_{\leq\alpha}]$.

###### Proof.

To see $(G.i)$, note that $\mathbb{P}$ collapses all cardinals ${<}\kappa$ to
$\omega$, so $\omega_{1}^{L[G]}\geq\kappa$. As $\mathbb{P}$ satisfies
$\mathrm{AA}(\kappa,\kappa)$, there is no surjection from $\omega$ onto
$\kappa$ in $L[G]$.
Next, let us prove $(G.ii)$. Let $\dot{g}\in L$ be a name for $g$. In $L[G]$,
find a decreasing sequence of conditions $\langle p_{n}\mid n<\omega\rangle$
in $G$ so that $p_{n}$ decides the value of $\dot{g}(\check{n})$ (from the
perspective of $L$). Let $\alpha=\sup_{n<\omega}\sup\mathrm{supp}(p_{n})$. By
$(G.i)$, $\alpha<\kappa$. But then $L[G_{\leq\alpha}]$ can compute the whole
of $g$. ∎

From now on, $\mathbb{M}_{\omega_{1}}$ denotes
$\mathbb{M}_{\omega_{1}}^{L[G]}$ and $\mathbb{M}_{\mathsf{C}}$ is
$\mathbb{M}_{\mathsf{C}}^{L[G]}$. Let us define an auxiliary model
$N=\bigcap_{\alpha<\kappa}L[G_{>\alpha}]$. It is clear that
$\mathbb{M}_{\omega_{1}}\subseteq N$.

Recall the following fact due to Solovay.

######  Fact 3.21 (Solovay, [Sol70]).

If $G,H$ are mutually generic filters over $V$ (for any forcings) then
$V[G]\cap V[H]=V$.

######  Proposition 3.22.

We have that

  1. $(N.i)$

$N\models\mathrm{ZF}$ and

  2. $(N.ii)$

$N\cap\mathcal{P}(\kappa)=\mathbb{M}_{\omega_{1}}\cap\mathcal{P}(\kappa)=\mathbb{M}_{\mathsf{C}}\cap\mathcal{P}(\kappa)=\\{a\subseteq\kappa\mid\forall\beta<\kappa\
a\cap\beta\in V\\}$.

###### Proof.

First, we will prove $(N.i)$. Once again it is enough to show that $N$ is
definable in all models of the form $L[G_{>\alpha}]$ for $\alpha<\kappa$. But
this is clear from the definition of $N$.
Next, we show $(N.ii)$.
$\mathbb{M}_{\omega_{1}}\cap\mathcal{P}(\kappa)\subseteq\mathbb{M}_{\mathsf{C}}\cap\mathcal{P}(\kappa)\subseteq
N\cap\mathcal{P}(\kappa)$ is trivial. If $a\in N\cap\mathcal{P}(\kappa)$ and
$\beta<\kappa$ then $a\cap\beta\in L[G_{\leq\alpha}]$ for some $\alpha$ by
clause $(G.ii)$ of Corollary 3.20. As $a\in N$, $a\cap\beta\in
L[G_{>\alpha}]$, too. Thus by Fact 3.21

$a\in L[G_{\leq\alpha}]\cap L[G_{>\alpha}]=L.$

The final inclusion
$N\cap\mathcal{P}(\kappa)\subseteq\mathbb{M}_{\omega_{1}}\cap\mathcal{P}(\kappa)$
holds since if $W$ is a ground of $L[G]$ which extends to $L[G]$ via
$\mathbb{Q}$ of size $<\kappa$ then $\mathbb{Q}$ cannot add a fresh subset of
$\kappa$. ∎

###### Proof of Theorem 3.17..

We will show that in $L[G]$, neither $\mathbb{M}_{\omega_{1}}$ nor
$\mathbb{M}_{\mathsf{C}}$ possess a wellorder of its version of
$\mathcal{P}(\kappa)$. In fact, we will show that $N$ does not have such a
wellorder, which is enough by $(N.ii)$ of the above proposition. Once again,
let $\sim$ be the equivalence relation on functions
$f\colon\kappa\rightarrow\kappa\in N$ of eventual coincidence. For $n<\omega$,
let

$d_{n}\colon\kappa\rightarrow\kappa,\ d_{n}(\alpha)=g_{\alpha}(n)$

where $g_{\alpha}$ is the map $\omega\rightarrow\alpha$ induced by the slice
of $G$ generic for $\mathrm{Col}(\omega,\alpha)$. As before, we get that the
fuzzy sequence $\langle[d_{n}]_{\sim}\mid n<\omega\rangle\in N$. If $N$ had a
wellorder of $\mathcal{P}(\kappa)$, then there would be a choice sequence
$\langle x_{n}\mid n<\omega\rangle\in N$ for the fuzzy sequence. In $L[G]$,
one can define the sequence $\langle\delta_{n}\mid n<\omega\rangle$ where
$\delta_{n}$ is the least point after which $x_{n}$ and $d_{n}$ coincide. As
$\kappa=\omega_{1}$ in $L[G]$, the $\delta_{n}$ are bounded uniformly by some
$\delta<\kappa$. But this means that $G_{>\delta}\in N$, a contradiction. ∎

It is natural to conjecture that
$N=\mathbb{M}_{\mathsf{C}}=\mathbb{M}_{\omega_{1}}$, though we do not have a
proof of any of these equalities. The problem is that we cannot follow the
strategy from Section 3.1: $L[G]$ has Cohen-grounds which do not contain any
$g_{\alpha}$ for $\alpha<\kappa$, let alone a tail of the sequence
$(g_{\alpha})_{\alpha<\kappa}$.

######  Question 3.23.

Is $N=\mathbb{M}_{\mathsf{C}}=\mathbb{M}_{\omega_{1}}$?

###  3.3 The successor of a regular uncountable cardinal case

We show that, again under $V=L$, for every regular uncountable $\kappa$ there
is a forcing extension in which $\mathbb{M}_{\kappa^{+}}$ is not a model of
$\mathrm{ZFC}$. The upside here is that we do not need any large cardinals at
all in our construction, however we pay a price: We do not know whether
$\mathbb{M}_{\kappa^{+}}$ is a model of $\mathrm{ZF}$ in general.

######  Theorem 3.24.

Assume $V=L$ and suppose $\kappa$ is regular uncountable. Then after forcing
with

$\mathbb{P}\coloneqq\prod_{\alpha<\kappa^{+}}^{<\kappa^{+}-\text{support}}\mathrm{Add}(\kappa,1)$

$\mathbb{M}_{\kappa^{+}}$ is not a model of $\mathrm{ZFC}$.

First, lets do a warm-up with an initial segment of $\mathbb{P}$. We thank
Elliot Glazer for explaining (the nontrivial part of) the following argument
to the author.

######  Lemma 3.25 (Elliot Glazer).

If $\kappa$ is regular and $\diamondsuit_{\kappa}$ holds then

$\mathbb{P}_{\leq\kappa}=\prod_{\alpha<\kappa}^{\text{full
support}}\mathrm{Add}(\kappa,1)$

satisfies $\mathrm{AA}(\kappa+1,\kappa^{+})$.

An additional assumption beyond “$\kappa$ is regular” is necessary here: It is
well known that

$\prod_{n<\omega}^{\text{full support}}\mathrm{Add}(\omega,1)$

collapses $2^{\omega}$ to $\omega$.

###### Proof.

We let $p\leq_{\alpha}q$ if $p\leq q$ and
$p\upharpoonright\alpha=q\upharpoonright\alpha$. It is easy to see that
$(\mathrm{AA}.i)$ and $(\mathrm{AA}.iii)$ of Definition 3.2 hold, so let us
show $(\mathrm{AA}.ii)$. Therefore, let $\alpha<\kappa$,
$p\in\mathbb{P}_{\leq\kappa}$ and an antichain $A$ in
$\mathbb{P}_{\leq\kappa}$ be given. As $\diamondsuit_{\kappa}$ holds, there is
a sequence $\langle d_{\beta}\mid\beta<\kappa\rangle$ with
$d_{\beta}\in\mathbb{P}_{\leq\beta}$ so that for any
$q\in\mathbb{P}_{\leq\kappa}$ there is some $\beta$ with
$q\upharpoonright\beta=d_{\beta}$. We will define a sequence
$(p_{\beta})_{\alpha\leq\beta\leq\kappa}$ inductively so that
$p_{\gamma}\leq_{\beta}p_{\beta}$ for all $\beta\leq\gamma\leq\kappa$. We put
$p_{\alpha}=p$. At limit stages $\beta$ we let $p_{\beta}$ be the canonical
fusion of $\langle p_{\gamma}\mid\alpha\leq\gamma<\beta\rangle$. So assume
$p_{\beta}$ is defined. We choose $p_{\beta+1}\leq_{\beta}p_{\beta}$ so that,
if possible,

$d_{\beta}^{\frown}p_{\beta+1}\leq a$

for some $a\in A$. Otherwise, we are lazy and set $p_{\beta+1}=p_{\beta}$.
Now clearly $q\coloneqq p_{\kappa}\leq_{\alpha}p$ and we will show that $q$ is
compatible with at most $\kappa$-many conditions in $A$. To see this, suppose
$a\in A$ is compatible with $q$. We may find $\beta<\kappa$ so that
$d_{\beta}=a\upharpoonright\beta$. In the construction of $p_{\beta+1}$ from
$p_{\beta}$, we tried to achieve that

$d_{\beta}^{\frown}p_{\beta+1}\upharpoonright[\beta,\kappa)$

is below some condition in $A$, which is possible and only possible for $a$.
This shows that for any $a\in A$ that is compatible with $q$, there is
$\beta<\kappa$ so that $q\upharpoonright[\beta,\kappa)\leq
a\upharpoonright[\beta,\kappa)$. As $\mathbb{P}_{\leq\beta}$ has size
${\leq}\kappa$, it follows that there are at most $\kappa$-many such $a\in A$.
∎

######  Corollary 3.26.

Under the same assumptions as before, $\mathbb{P}_{\leq\kappa}$ preserves all
cardinals $\leq\kappa^{+}$.

###### Proof.

$\mathbb{P}_{\leq\kappa}$ is ${<}\kappa$-closed and satisfies
$\mathrm{AA}(\kappa+1,\kappa^{+})$. ∎

We aim to prove a similar result for $\mathbb{P}$.

######  Lemma 3.27.

If $\kappa$ is regular and $\diamondsuit_{\kappa}$ holds then $\mathbb{P}$
preserves all cardinals $\leq\kappa^{+}$. Moreover, if $G$ is
$\mathbb{P}$-generic and $g\colon\kappa\rightarrow\mathrm{Ord}$ is in $V[G]$
then there is $\alpha<\kappa^{+}$ with $g\in V[G_{\leq\alpha}]$.

The argument is similar, but somewhat more complicated. To do so, we introduce
a further abstraction of $\mathrm{AA}(\kappa,\lambda)$.

######  Definition 3.28.

Suppose that $\mathcal{P}=(P,\preceq)$ is a partial order, $\mathbb{Q}$ is a
forcing, $\kappa<\lambda$ are ordinals. $\mathbb{Q}$ satisfies Strategic Axiom
A$(\kappa,\lambda,\mathcal{P})$ ($\mathrm{SAA}(\kappa,\lambda,\mathcal{P})$)
if there is a family $\langle\leq_{x}\mid x\in P\rangle$ of partial orders on
$\mathbb{Q}$ so that

  1. $(\mathrm{SAA}.i)$

$\leq_{y}\subseteq\leq_{x}\subseteq\leq_{\mathbb{Q}}$ whenever $x\preceq y$
for $x,y\in P$,

  2. $(\mathrm{SAA}.ii)$

for any antichain $A\subseteq\mathbb{Q}$, any $x\in P$ and $p\in\mathbb{Q}$,
there is $q\leq_{x}p$ with

$|\\{a\in A\mid a\|p\\}|<\lambda$

and

  3. $(\mathrm{SAA}.iii)$

player II has a winning strategy in the following game we call
$\mathcal{G}(\mathbb{Q},\kappa,\mathcal{P})$:

I | $p_{0}$ |  | $p_{1}$ |  | $\dots$ | $p_{\omega}$ |  | $\dots$
---|---|---|---|---|---|---|---|---
II |  | $x_{0}$ |  | $x_{1}$ | $\dots$ |  | $x_{\omega}$ | $\dots$

The game has length $\kappa$. In an even round $\alpha\cdot 2$, Player I plays
some condition $p_{\alpha}\in\mathbb{Q}$ so that
$p_{\alpha}\leq_{x_{\beta}}p_{\beta}$ for all $\beta<\alpha$ played so far. In
an odd round $\alpha\cdot 2+1$, player II plays some $x_{\alpha}\in P$ with
$x_{\beta}\preceq x_{\alpha}$ for all $\beta<\alpha$.
Player I wins the game iff some player has no legal moves in some round
${<}\kappa$. If the game last all $\kappa$ rounds instead, II wins.

It is straightforward to generalize Proposition 3.18.

######  Proposition 3.29.

Suppose $\mathbb{Q}$ satisfies $\mathrm{SAA}(\kappa,\lambda,\mathcal{P})$.
Then in $V^{\mathbb{Q}}$, there is no surjection
$f\colon\beta\rightarrow\lambda$ for any $\beta<\kappa$.

∎

######  Lemma 3.30.

If $\kappa$ is regular and $\diamondsuit_{\kappa}$ holds then $\mathbb{P}$
satisfies

$\mathrm{SAA}(\kappa+1,\kappa^{+},\mathcal{P}_{\kappa}(\kappa^{+}))$

where $\mathcal{P}_{\kappa}(\kappa^{+})$ is ordered by inclusion.

###### Proof.

For $x\in\mathcal{P}_{\kappa}(\kappa^{+})$ we will write $p\leq_{x}q$ if
$p\leq q$ and $p\upharpoonright x=q\upharpoonright x$. It is clear that
$(\mathrm{SAA}.i)$ holds.
Next, we aim to establish $(\mathrm{SAA}.iii)$. We describe a strategy for
player II in the relevant game. We will need to do some additional
bookkeeping. Let

$h\colon\kappa\rightarrow\kappa\times\kappa$

be a surjection such that if $h(\beta)=(\alpha,\gamma)$ then
$\alpha\leq\beta$. Suppose that $p_{\alpha}$ is the last condition played by
player I and $(x_{\beta})_{\beta<\alpha}$ have been played already. In the
background, we already have chosen some surjections
$s_{\beta}\colon\kappa\rightarrow\mathrm{supp}(p_{\beta})$ for $\beta<\alpha$
and we will adjoin a surjection $s_{\alpha}\mathrel{\mathop{\mathchar
58\relax}}\kappa\rightarrow\mathrm{supp}(p_{\alpha})$ to that list. We set

$x_{\alpha}={s_{\gamma_{0}}(\gamma_{1})}\cup\bigcup_{\beta<\alpha}x_{\beta}$

where $(\gamma_{0},\gamma_{1})=h(\alpha)$. As $\kappa$ is regular,
$x_{\alpha}\in\mathcal{P}_{\kappa}(\kappa^{+})$.

######  Claim 3.31.

Player I does not run out of moves before the game ends.

###### Proof.

Suppose we reached round $2\cdot\alpha\leq\kappa$ and let
$x=\bigcup_{\beta<\alpha}x_{\beta}$. We will find a legal play $p_{\ast}$ for
player I. For
$\gamma\in\kappa^{+}\setminus\bigcup_{\beta<\alpha}\mathrm{supp}(p_{\beta})$,
let $p_{\ast}(\gamma)$ be trivial. The point is that for $\gamma\in x$,
$\langle p_{\beta}(\gamma)\mid\beta<\alpha\rangle$ stabilizes eventually to
some $p_{\ast}(\gamma)$. If $\alpha=\kappa$, then our bookkeeping made sure
that we have

$x=\bigcup_{\beta<\kappa}\mathrm{supp}(p_{\beta})$

so that $p_{\ast}$ is already fully defined and a legal play. If
$\alpha<\kappa$ instead, then there are possibly
$\gamma\in\bigcup_{\beta<\alpha}\mathrm{supp}(p_{\beta})-x$, but then $\langle
p_{\beta}(\gamma)\mid\beta<\alpha\rangle$ is a sequence of length ${<}\kappa$,
so we may pick a lower bound $p_{\ast}(\gamma)\in\mathrm{Add}(\kappa,1)$ for
it. ∎

It remains to show $(\mathrm{SAA}.ii)$ and here we will use that
$\diamondsuit_{\kappa}$ holds. Let $\langle d_{\beta}\mid\beta<\kappa\rangle$
be the “$\diamondsuit_{\kappa}$-sequence for $\mathbb{P}_{\leq\kappa}$” that
appeared in the proof of Lemma 3.25 and let $A$ be a maximal antichain in
$\mathbb{P}$. Choose $\tau$ to be a winning strategy for player II in
$\mathcal{G}(\mathbb{P},\kappa+1,\mathcal{P}_{\kappa}(\kappa^{+}))$ and we
will describe a strategy $\sigma$ for player I: Suppose $\alpha\leq\kappa$ and
$p_{\beta}$, $x_{\beta}$ have already been played for $\beta<\alpha$. This
time, we will have picked some surjections $s_{\beta}\colon\kappa\rightarrow
x_{\beta}$ for $\beta<\alpha$ in the background. Let
$x_{<\alpha}\coloneqq\bigcup_{\beta<\alpha}x_{\beta}$. Then, assuming there is
a legal move, pick some $p_{\alpha}$ so that

  1. $(p_{\alpha}.i)$

$p_{\alpha}\leq_{x_{\beta}}p_{\beta}$ for all $\beta<\alpha$ and

  2. $(p_{\alpha}.ii)$

if possible, $p_{\alpha}\upharpoonright(\kappa^{+}\setminus x_{<\alpha})\cup
e_{\alpha}\upharpoonright x_{<\alpha}$ is below a condition in $A$

where $e_{\alpha}$ is defined by

$e_{\alpha}(s_{\gamma_{0}}(\gamma_{1}))=d_{\alpha}(\gamma)$

whenever $\gamma<\alpha$ and $h(\gamma)=(\gamma_{0},\gamma_{1})$ (and
$e_{\alpha}$ is trivial where we did not specify a value)777$e_{\alpha}$ may
fail to be a function, in which case $(p_{\alpha}.ii)$ is void..
Let $\langle p_{\alpha}\mid\alpha\leq\kappa\rangle$, $\langle
x_{\alpha}\mid\alpha<\kappa\rangle$ be the sequences of moves played by player
I and II in a game where player I follows $\sigma$ and player II follows
$\tau$. As $\tau$ is a winning strategy, the sequence must be of length
$\kappa+1$. We will show that $q\coloneqq p_{\kappa}$ is compatible with at
most $\kappa$-many elements of $A$. So let $a\in A$ and assume that $q$ is
compatible with $a$.

######  Claim 3.32.

There is $\alpha<\kappa$ so that $e_{\alpha}\in\mathbb{P}$ and
$e_{\alpha}\upharpoonright x_{<\alpha}=a\upharpoonright x_{<\alpha}$.

###### Proof.

We define $b\in\mathbb{P}_{\leq\kappa}$ by
$b(\gamma)=a(s_{\gamma_{0}}(\gamma_{1}))$ whenever
$h(\gamma)=(\gamma_{0},\gamma_{1})$. Then there is $\alpha<\kappa$ with

  1. $(\alpha.i)$

$b\upharpoonright\alpha=d_{\alpha}$ and

  2. $(\alpha.ii)$

$x_{<\alpha}=\\{s_{\gamma_{0}}(\gamma_{1})\mid\exists\gamma<\alpha\
h(\gamma)=(\gamma_{0},\gamma_{1})\\}$.

It is easy to see now that $\alpha$ is as desired. ∎

Thus in round $\alpha\cdot 2$ in the game, player I tried to make sure that

$a\upharpoonright x_{<\alpha}\cup
p_{\alpha}\upharpoonright(\kappa^{+}\setminus x_{<\alpha})$

is below some condition in $A$. This is possible for $a$, and only for $a$ as
$q$ and $a$ are compatible.
We have shown that for any $a\in A$ that is compatible with $q$, there is
$\alpha<\kappa$ such that $q\upharpoonright(\kappa^{+}\setminus
x_{<\alpha})\leq a\upharpoonright(\kappa^{+}\setminus x_{<\alpha})$. As there
are only ${\leq}\kappa$-many $r\in\mathbb{P}$ with support contained in
$x_{<\alpha}$, this implies that there are at most $\kappa$-many such $a\in
A$. ∎

Lemma 3.27 follows from Lemma 3.30 and Proposition 3.29 similarly to how we
proved Corollary 3.20.

######  Remark 3.33.

If additionally $\mathrm{GCH}$ holds at $\kappa^{+}$ then $\mathbb{P}$ does
not collapse any cardinals at all by a standard $\Delta$-system argument.

###### Proof of Theorem 3.24..

Let $G$ be $\mathbb{P}$-generic over $L$. By Lemma 3.27, all $L$-cardinals
$\leq\kappa^{+}$ are still cardinals in $L[G]$ (in fact, all cardinals are
preserved). Let $N=\bigcap_{\alpha<\kappa^{+}}L[G_{>\alpha}]$. Using that $N$
is definable in every model of the form $L[G_{>\alpha}]$, it is easy to check
that $N$ is a model of $\mathrm{ZF}$. Once again, we call
$A\subseteq\kappa^{+}$ fresh if $A\cap\alpha\in L$ for all
$\alpha<\kappa^{+}$.

######  Claim 3.34.

$\mathcal{P}(\kappa^{+})^{\mathbb{M}_{\kappa^{+}}}=\mathcal{P}(\kappa^{+})^{N}=\\{A\subseteq\kappa^{+}\mid
A\text{ is fresh}\\}^{L[G]}$.

###### Proof.

$\mathcal{P}(\kappa^{+})^{\mathbb{M}_{\kappa^{+}}}\subseteq\mathcal{P}(\kappa^{+})^{N}$
is trivial. Suppose $A\subseteq\kappa^{+},A\in N$. Given $\alpha<\kappa^{+}$,
by Lemma 3.27, there is $\beta<\kappa^{+}$ so that $A\cap\alpha\in
L[G_{\leq\beta}]$ so that

$A\cap\alpha\in L[G_{\leq\beta}]\cap L[G_{>\beta}]=L$

by Fact 3.21. For the last inclusion assume $A\in L[G]$ is a fresh subset of
$\kappa^{+}$ and $W$ is any $\kappa^{+}$-ground of $L[G]$. It follows that
$W\subseteq L[G]$ satisfies the $\kappa^{+}$-approximation property so that
$A\in W$ as any bounded subset of $A$ is in $L\subseteq W$. ∎

We will show that there is no wellorder of
$\mathcal{P}(\kappa^{+})^{\mathbb{M}_{\kappa^{+}}}$ in
$\mathbb{M}_{\kappa^{+}}$. So assume otherwise. Let $\sim$ be the equivalence
relation of eventual coincidence on ${}^{\kappa^{+}}2$ in $N$. We can realise
$G$ as a matrix where the $\alpha$-th row is $\mathrm{Add}(\kappa,1)$-generic
over $L$. Now the columns are in fact $\mathrm{Add}(\kappa^{+},1)$-generic
over $L$. Let us write $c_{\alpha}$ for the $\alpha$-th column
($\alpha<\kappa^{+})$ and $d_{\beta}$ for the $\beta$-th row ($\beta<\kappa$).
For any $\alpha<\kappa^{+}$ we have that $\langle
d_{\beta}\upharpoonright[\alpha,\kappa^{+})\mid\beta<\kappa\rangle\in
L[G_{>\alpha}]$. Thus

$\langle[d_{\beta}]_{\sim}\mid\beta<\kappa\rangle\in N$

and by our assumption there must be a choice function, say $\langle
x_{\beta}\mid\beta<\kappa\rangle$, in $N$. In $L[G]$, we can define the
sequence $\langle\delta_{\beta}\mid\beta<\kappa\rangle$, where
$\delta_{\beta}$ is the least point after which $x_{\beta}$ and $d_{\beta}$
coincide. As $\kappa^{+}$ is not collapsed by $\mathbb{P}$, we can strictly
bound all $\delta_{\beta}$ by some $\delta_{\ast}<\kappa^{+}$. But then

$\langle x_{\beta}(\delta_{\ast})\mid\beta<\kappa\rangle\in N$

is $\mathrm{Add}(\kappa,1)$-generic over $L$, which contradicts that $N$ and
$L$ have the same subsets of $\kappa$. ∎

Note that Fact 1.6 does not apply in the situation here, so we may ask:

######  Question 3.35.

Is $\mathbb{M}_{\kappa^{+}}$ a model of $\mathrm{ZF}$? Is
$\mathbb{M}_{\kappa^{+}}=N$?

##  4 Conclusion

There are a number of open questions regarding the interplay between large
cardinal properties of $\kappa$ and the $\kappa$-mantle. The following table
summarizes what is known as presented in the introduction.

Large cardinal property of $\kappa$ | Theory of $\mathbb{M}_{\kappa}$ extends…
---|---
extendible | $\mathrm{ZFC}+\mathrm{GA}$
measurable | $\mathrm{ZFC}$
weakly compact | $\mathrm{ZF}+\kappa\text{-}\mathrm{DC}$
inaccessible | $\mathrm{ZF}+(<\kappa,H_{\kappa^{+}})\text{-choice}$

There is certainly much more to discover here. How optimal are these results?
Optimality has only been proven for one of them, namely the first. This is due
to Gabriel Goldberg.

######  Fact 4.1 (Goldberg, [Gol21]).

Suppose $\kappa$ is an extendible cardinal. Then there is a class forcing
extension in which $\kappa$ remains extendible and $\mathbb{M}_{\kappa}$ is
not a $\kappa$-ground. In particular, if $\lambda<\kappa$ and
$\mathbb{M}_{\lambda}\models\mathrm{ZFC}$ then $\mathbb{M}_{\lambda}$ has a
nontrivial ground.

The most interesting question seems to be up to when exactly the axiom of
choice can fail to hold in $\mathbb{M}_{\kappa}$. Since this can happen at a
Mahlo cardinal, the natural next test question is whether this is possible at
a weakly compact cardinal.

######  Question 4.2.

Suppose that $\kappa$ is weakly compact. Must
$\mathbb{M}_{\kappa}\models\mathrm{ZFC}$?

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