# Sampling Neural Radiance Fields for Refractive Objects

Jen-I Pan<EMAIL_ADDRESS>National Tsing Hua UniversityHsinchuTaiwan ,
Jheng-Wei Su<EMAIL_ADDRESS>National Tsing Hua
UniversityHsinchuTaiwan ,  Kai-Wen Hsiao<EMAIL_ADDRESS>National Tsing
Hua UniversityHsinchuTaiwan ,  Ting-Yu Yen<EMAIL_ADDRESS>National
Tsing Hua UniversityHsinchuTaiwan and  Hung-Kuo Chu<EMAIL_ADDRESS>National Tsing Hua UniversityHsinchuTaiwan

(2022)

###### Abstract.

Recently, differentiable volume rendering in neural radiance fields (NeRF) has
gained a lot of popularity, and its variants have attained many impressive
results. However, existing methods usually assume the scene is a homogeneous
volume so that a ray is cast along the straight path. In this work, the scene
is instead a heterogeneous volume with a piecewise-constant refractive index,
where the path will be curved if it intersects the different refractive
indices. For novel view synthesis of refractive objects, our NeRF-based
framework aims to optimize the radiance fields of bounded volume and boundary
from multi-view posed images with refractive object silhouettes. To tackle
this challenging problem, the refractive index of a scene is reconstructed
from silhouettes. Given the refractive index, we extend the stratified and
hierarchical sampling techniques in NeRF to allow drawing samples along a
curved path tracked by the Eikonal equation. The results indicate that our
framework outperforms the state-of-the-art method both quantitatively and
qualitatively, demonstrating better performance on the perceptual similarity
metric and an apparent improvement in the rendering quality on several
synthetic and real scenes.

neural radiance fields, eikonal rendering

††submissionid: 139††journalyear: 2022††copyright: acmcopyright††doi:
10.1145/3550340.3564234††conference: SIGGRAPH Asia 2022 Technical
Communications; December 6–9, 2022; Daegu, Republic of Korea††ccs: Computing
methodologies Computer graphics††ccs: Computing methodologies Machine learning
\begin{overpic}[width=433.62pt,percent]{image/fig_teaser.png}
\put(11.56,23.77){\makebox[0.0pt]{\small Reference}}
\put(29.02,23.77){\makebox[0.0pt]{\small Mip-NeRF}}
\put(40.35,23.77){\makebox[0.0pt]{\small Ours}}
\put(29.02,11.0){\makebox[0.0pt]{\small LPIPS}}
\put(40.35,11.0){\makebox[0.0pt]{\small LPIPS}}
\put(62.14,23.77){\makebox[0.0pt]{\small Reference}}
\put(83.34,23.77){\makebox[0.0pt]{\small Mip-NeRF}}
\put(94.57,23.77){\makebox[0.0pt]{\small Ours}}
\put(83.34,11.0){\makebox[0.0pt]{\small LPIPS}}
\put(94.57,11.0){\makebox[0.0pt]{\small LPIPS}} \end{overpic} Figure 1. Our
framework takes multi-view images as inputs and renders novel views of both
synthetic (left) and real (right) scenes containing refractive objects. With
the benefit of considering refraction paths, our results on the surfaces (cube
and sphere) and the interior objects (torus and dolphin) are more accurately
rendered, as shown in the error maps using LPIPS index (brighter regions
indicate higher errors).

##  1\. Introduction and Related Work

\begin{overpic}[width=433.62pt]{image/fig_framework.png}
\put(10.94,0.0){\makebox[0.0pt]{\small(a) Preprocessing}}
\put(36.94,0.0){\makebox[0.0pt]{\small(b) Path sampling}}
\put(62.93,0.0){\makebox[0.0pt]{\small(c) Volume rendering}}
\put(88.92,0.0){\makebox[0.0pt]{\small(d) Optimization}}
\put(1.98,19.53){\makebox[0.0pt]{\small Image}}
\put(6.23,19.53){\makebox[0.0pt]{\small Mask}}
\put(16.43,19.53){\makebox[0.0pt]{\small Refractive index $n$}}
\put(56.81,19.53){\makebox[0.0pt]{\small$F_{\theta}$, $F_{\phi}$}}
\put(71.18,3.97){\makebox[0.0pt]{\small$F_{\psi}$}}
\put(55.1,3.97){\makebox[0.0pt]{\small$\hat{C}$}}
\put(83.24,19.53){\makebox[0.0pt]{\small Prediction}}
\put(94.72,19.53){\makebox[0.0pt]{\small Target}} \put(31.49,19.53){\small
Coarse} \put(39.49,19.53){\small Fine}
\put(79.24,3.97){\makebox[0.0pt]{\small$\mathcal{L}_{S}$}}
\put(83.24,3.97){\makebox[0.0pt]{\small$\mathcal{L}_{RGB}$}}
\put(87.24,3.97){\makebox[0.0pt]{\small$\mathcal{L}_{BD}$}} \end{overpic}
Figure 2.  Framework overview. Given the multi-view posed images and
refractive object silhouettes, we first reconstruct the refractive index $n$
of a scene from silhouettes and store it in a voxel grid (a). Next, we track
the ray of a pixel $C$ and draw the samples along the traversed path (b).
Then, we query the density and radiance of each sample from the networks,
namely $F_{\theta}$ and $F_{\phi}$, and combine them with the boundary
radiance evaluated by the boundary network $F_{\psi}$ to estimate the
resulting color (c). Finally, we optimize the three networks with respect to
the re-rendering error ($\mathcal{L}_{RGB}$) and regularizers
($\mathcal{L}_{S}$ and $\mathcal{L}_{BD}$) (d).

Refraction is ubiquitous in everyday life. For example, distorted objects seen
through the water, and a magnifying glass decreasing the field of view. Thus,
accurate rendering of refraction is crucial to improve realism. Nevertheless,
reconstructing the scene with a refractive object from multi-view images is an
ill-posed problem due to the ambiguity among geometry, material and refractive
index.

For the past two years, neural radiance field, or NeRF (Mildenhall et al.,
2020), and its variants that treat a scene as a homogeneous volume have been
widely explored. NeRF uses two multi-layer perceptrons (MLPs), one coarse
$F_{\theta}$ and one fine $F_{\phi}$, to represent a volumetric scene. The MLP
takes the encoded position x of a sample and view direction d by positional
encoding as inputs; it outputs the density $\sigma$ and radiance c. The pixel
value $\hat{C}$ is estimated by the differentiable volume rendering equation
in Eqn. 1 with all the samples $\mathbbm{r}$ along a ray cast from the camera
origin o to the pixel.

(1)
$\displaystyle\begin{split}&\hat{C}(\mathbbm{r})=\sum_{i=1}^{N}T_{i}\left(1-\text{exp}\left(-\sigma_{i}\delta_{i}\right)\right)\textbf{c}_{i}\end{split}\text{,}$

where $T_{i}=\text{exp}\left(-\sum_{j=1}^{i-1}\sigma_{j}\delta_{j}\right)$ is
the accumulated transmittance, and $\delta$ is the distance between adjacent
samples. Although NeRF can replace the background with white via the mask of
an opaque object, the background seen through a refractive object cannot be
easily removed. Thus, the original equation in NeRF (Eqn. 1) is insufficient
for our problem due to the lack of a boundary term. The samples used to
estimate the pixel value consist of two sets, one coarse $\mathbbm{r}_{c}$ and
one fine $\mathbbm{r}_{f}$, for the coarse and fine networks. Specifically,
the fine network takes the union of both sets after sorting. The coarse
samples $\mathbbm{r}_{c}={\\{(\textbf{x}_{i},\textbf{d})\\}}_{i=0}^{N_{c}}$
are drawn uniformly from the evenly-spaced bins between the near $t_{N}$ and
far $t_{F}$ planes with a stratified sampling of $t_{i}$ in Eqn. 2 for a
continuous representation; the fine samples
$\mathbbm{r}_{f}={\\{(\textbf{x}_{i},\textbf{d})\\}}_{i=0}^{N_{c}}$ are then
allocated to visible regions that most likely contribute to the pixel value
based on the coarse network with a hierarchical sampling of $t_{i}$ in Eqn. 3,
respectively by the ray equation $\textbf{x}_{i}=\textbf{o}+t_{i}\textbf{d}$.

(2)
$\displaystyle\mathbbm{t}_{c}=\left\\{t_{i}\sim\mathcal{U}\left[(i-1)/N_{c},i/N_{c}\right]\cdot\left(t_{F}-t_{N}\right)+t_{N}\right\\}_{i=1}^{N_{c}}$
(3)
$\displaystyle\begin{split}&\mathbbm{t}_{f}=\left\\{t_{i}\sim\text{InverseTransformSampling}\left(\hat{w}_{i}\right)\right\\}_{i=1}^{N_{f}}\end{split}$

where $\hat{w}_{i}=w_{i}/\sum_{j=1}^{N_{c}}w_{j}$, and
$w_{i}=T_{i}\left(1-\text{exp}\left(-\sigma_{i}\delta_{i}\right)\right)$.
However, these two sampling techniques in NeRF cannot be used with only the
distance $t$ and view direction d if a path is curved due to the refraction.
To this end, we combine light transport simulation based on the _Eikonal
equation_ with NeRF for the refraction, and we extend the original sampling
techniques in NeRF to curved paths.

In terms of the rendering quality and refraction, mip-NeRF (Barron et al.,
2021) proposes an integrated positional encoding and achieves the best quality
for both single- and multi-scale contents. Ref-NeRF (Verbin et al., 2021) uses
a concept of environment mapping to enable a sharp view-dependent effect.
CompLum (Zhu et al., 2021) brings a surface light field to avoid the costly
evaluation of light paths inside the complex refractive geometry. Still, none
of them takes refraction paths into account. Furthermore, Matusik et al.
(2002) introduce an image-based rendering method with the 3D scanner for
refractive objects.

Moreover, instead of a bounded scene, a boundary can be treated as different
representations (Zhang et al., 2020; Hao et al., 2021). To handle the leftover
transmittance or density, GANcraft (Hao et al., 2021) proposes a regularizer.
Overall, we represent the boundary as a skybox and bring a proper regularizer
to resolve the ambiguity of color blending between refractive object and
boundary. Supplementary materials, codes, and datasets are released for
academic usage at https://github.com/alexkeroro86/SampleNeRFRO.

In summary, we make the following contributions:

  * •

A NeRF-based framework for generating high-quality novel view rendering of
refractive objects presenting the refraction and total reflection effects.

  * •

A tailor-made hierarchical path sampling technique for both straight and
curved paths.

### Concurrent work

Eikonal Fields (Bemana et al., 2022) aims at the same problem as ours.
Compared to the concurrent work, both methods follow the ray equation of
geometric optics derived from the Eikonal equation for a volumetric scene by
Eikonal Rendering (Ihrke et al., 2007). We use the piecewise-linear
approximation in (Sun et al., 2008) (Eqn. 4) to construct refraction paths:

(4) $\displaystyle\begin{split}\textbf{x}_{i+1}=\textbf{x}_{i}+\frac{\Delta
s}{n}v_{i}\text{,}\quad v_{i+1}=v_{i}+\Delta
s\nabla\textbf{n}\end{split}\text{,}$

where $\Delta s$ is step size, $v$ is defined by $n\frac{d\textbf{x}}{ds}$,
$n$ is refractive index and $\nabla\textbf{n}$ is gradient index. For the
refractive index, Eikonal Fields tackles the challenging task of
reconstructing the refractive index of a scene by the dedicated multi-stage
training strategy. We assume the object’s material is known, hence its
corresponding refractive index (e.g., 1.52 for glass and 1.33 for water). In
terms of scene complexity, instead of using a bounding box annotation, we
provide more complex interior objects inside different refractive objects for
both synthetic and real scenes to inspire the follow-up works. For the
sampling, Eikonal Fields only draws samples uniformly between the bounds, but
we further propose a hierarchical path sampling technique. Besides, NeReF
(Wang et al., 2022) aims to recover the depth and normal of a flat fluid
surface for one-time refraction of the last sample using the Snell’s law.

### Assumption

Since we consider the refractive index of a refractive object, samples behind
the interior object should be occluded eventually. Therefore, a ray does not
change its direction when crossing the interior object. In addition, we ignore
the outer surface of refractive objects, such as the glass in a glass of
water.

##  2\. Method

Fig. 2 illustrates an overview of our NeRF-based framework. To be compatible
with the Eikonal equation in Eqn. 4, we first reconstruct the proxy geometry
of the refractive object by shape from silhouette and remove the noisy
components manually. Then, we choose a voxel grid with tri-linear
interpolation to represent the refractive index of a scene. For each vertex,
its refractive index is calculated by $A/(A+B)\cdot 1.0+B/(A+B)\cdot n$, where
$A$ and $B$ is the number of samples within a voxel that are outside and
inside the proxy geometry, respectively. In addition, to eliminate the stair-
step artifacts in rendering, we smooth the voxel grid before compute the
gradient index as Sun et al. (2008).

\begin{overpic}[width=325.215pt,percent]{image/fig_sampling_borderless_tight.png}
\put(25.0,46.42){\makebox[0.0pt]{\small Stratified sampling}}
\put(75.0,46.42){\makebox[0.0pt]{\small Hierarchical sampling}}
\put(-3.5,33.0){\rotatebox[origin={c}]{90.0}{\small NeRF}}
\put(-3.5,9.0){\rotatebox[origin={c}]{90.0}{\small Ours}}
\put(9.82,25.53){\makebox[0.0pt]{\small$t_{N}$}}
\put(41.42,25.53){\makebox[0.0pt]{\small$t_{F}$}}
\put(13.52,41.32){\makebox[0.0pt]{\small$t_{i}$}}
\put(4.88,41.32){\makebox[0.0pt]{\small$\textbf{o}$}}
\put(43.86,41.32){\makebox[0.0pt]{\small$\textbf{d}$}}
\put(59.72,25.53){\makebox[0.0pt]{\small$t_{N}$}}
\put(91.22,25.53){\makebox[0.0pt]{\small$t_{F}$}}
\put(65.12,41.32){\makebox[0.0pt]{\small$t_{i}$}}
\put(62.86,17.61){\makebox[0.0pt]{\small$t_{i}$}} \end{overpic} Figure 3.
Comparison between NeRF’s and our sampling techniques. A dotted double arrow
is a bin in Eqn. 2.

To construct a path according to the refractive index and gradient index, we
leverage the Eikonal equation in Eqn. 4 to bend a piecewise-linear path at
each step $i$. We choose the step size $\Delta s=(t_{F}-t_{N})/(N_{c}\times
N_{e})$ to track the path in $N_{c}*N_{e}$ steps. Here all the steps are
denoted as the _Eikonal samples_ , and we also collect the distance of each
step. However, if we use all the Eikonal samples to estimate the pixel value,
it is costly to evaluate by using the coarse network. Therefore, we randomly
draw one sample in every $N_{e}$ samples to reduce the times of network
evaluation, and these drawn samples are called the _coarse samples_
$\mathbbm{r}_{c}={\\{(\textbf{x}_{i},\textbf{d}_{i})\\}}_{i=0}^{N_{c}}$, where
$\textbf{d}_{i}=v_{i}/||v_{i}||$. For the _fine samples_
$\mathbbm{r}_{f}={\\{(\textbf{x}_{i},\textbf{d}_{i})\\}}_{i=0}^{N_{f}}$, we
should not only allocate the fine samples to visible regions as NeRF but also
make sure they are still along the piecewise-linear path by extending the
hierarchical sampling. With Eqn. 3, we transform the set of fine distances
$\mathbbm{t}_{f}$ to the fine samples $\mathbbm{r}_{f}$ by assigning the
direction of the nearest former Eikonal sample $\textbf{d}_{\lfloor t\rfloor}$
to $\textbf{d}_{i}$ according to the distance $t$. Then, we re-calculate the
position $\textbf{x}_{i}$ based on the position $\textbf{x}_{\lfloor
t\rfloor}$ and distance $\lfloor t\rfloor$ of the nearest former Eikonal
sample along the direction $\textbf{d}_{i}$ by $\textbf{x}_{\lfloor
t\rfloor}+\textbf{d}_{i}(t-\lfloor t\rfloor)$. We illustrate our sampling
techniques and compare to NeRF in Fig. 3.

After the network evaluation of samples $\mathbbm{r}$, we gather the
corresponding density $\sigma$ and radiance c by the following volume
rendering equation with boundary term:

(5) $\displaystyle\hat{C}(\mathbbm{r})$
$\displaystyle=\sum_{i=1}^{N}T_{i}(1-\text{exp}(-\sigma_{i}\delta_{i}))\textbf{c}_{i}+T_{N+1}C^{\prime}(\textbf{d}_{N})\text{,}$

where $C^{\prime}$ is a skybox represented as a small MLP
$F_{\psi}:\textbf{d}\rightarrow\textbf{c}$ whose architecture is based on the
normal field in NeRFactor (Zhang et al., 2021), and $\textbf{d}_{N}$ is the
leaving direction from the bounded volume.

Finally, we optimize the three MLPs, namely $F_{\theta}$, $F_{\phi}$ and
$F_{\psi}$, with respect to the following objective function:

(6)
$\displaystyle\begin{split}\mathcal{L}=\lambda_{RGB}\mathcal{L}_{RGB}+\lambda_{BD}\mathcal{L}_{BD}+\lambda_{S}\mathcal{L}_{S}\end{split}\text{,}$

where $\lambda_{RGB}$, $\lambda_{BD}$ and $\lambda_{S}$ are the weighted
hyper-parameters. We illustrate these terms in Fig. 2(d).

### Re-rendering error

We use a L2 loss to compare the coarse $\hat{C}_{c}$ and fine $\hat{C}_{f}$
pixel values with the ground truth $C(\textbf{r})$ as NeRF:

(7)
$\displaystyle\begin{split}\mathcal{L}_{RGB}=\|C(\textbf{r})-\hat{C}_{c}(\mathbbm{r}_{c})\|_{2}^{2}+\|C(\textbf{r})-\hat{C}_{f}\left(\text{sort}(\mathbbm{r}_{c}\cup\mathbbm{r}_{f})\right)\|_{2}^{2}\end{split}\text{.}$

### Boundary regularizer

It is calculated based on the re-rendering error but only updates the density
$\sigma$ evaluated by the fine network to preserve the visual quality and
eliminate the blurry artifacts on the refractive surface:

(8)
$\displaystyle\begin{split}\mathcal{L}_{BD}=\mathbbm{1}(T_{f,N_{c}+N_{f}+1})\cdot\|C(\textbf{r})-T_{f,N_{c}+N_{f}+1}C^{\prime}(\textbf{d}_{f,N_{c}+N_{f}})\|_{1}\end{split}\text{,}$

where $\mathbbm{1}(\cdot)$ is an indicator function to ignore the error of
occluded region by the threshold $0.5$ on the last accumulated transmittance.

### Smoothness regularizer

We add an L2 gradient penalty to boundary as NeRFactor (Zhang et al., 2021) on
a tile of directions $\textbf{d}^{\prime}$:

(9)
$\displaystyle\begin{split}\mathcal{L}_{S}=\left(0.5\cdot\|\begin{bmatrix}-1&1\end{bmatrix}*C^{\prime}(\textbf{d}^{\prime})\|_{2}^{2}+0.5\cdot\|\begin{bmatrix}-1\\\
1\end{bmatrix}*C^{\prime}(\textbf{d}^{\prime})\|_{2}^{2}\right)\end{split}\text{.}$

##  3\. Result

### Dataset

We rendered four synthetic scenes, namely Ship, Torus, DeerGlobe and StarLamp,
from viewpoints sampled on a full sphere with refraction and total reflection
effects. The viewpoints are 100, 100, and 200 views of size $800\times 800$
for training, validation, and testing splits, respectively. We resize all the
images by half for experiments. Moreover, we captured one real scene (Dolphin)
from viewpoints sampled upon a hemisphere. The viewpoints are 100, 50, and 100
views of size $2560\times 1920$ for training, validation, and testing splits,
respectively, and the camera poses are calibrated with AprilTag (Krogius et
al., 2019). We resize all the images by half and crop the center for
experiments. We also select three real scenes from Eikonal Fields (Bemana et
al., 2022), namely Ball, Glass and Pen, and compare to the provided video
sequences.

\begin{overpic}[width=390.25534pt,percent]{image/fig_qualitative_wo.png}
\par\put(-1.5,22.41){\rotatebox[origin={c}]{90.0}{\small{Torus}}}
\put(-1.5,6.88){\rotatebox[origin={c}]{90.0}{\small{Dolphin}}}
\put(49.33,22.41){\rotatebox[origin={c}]{90.0}{\small{Ship}}}
\put(49.33,6.88){\rotatebox[origin={c}]{90.0}{\small{Pen}}}
\par\put(6.88,29.8){\makebox[0.0pt]{\scriptsize Ours}}
\put(17.47,29.8){\makebox[0.0pt]{\scriptsize Mip-NeRF}}
\put(24.53,29.8){\makebox[0.0pt]{\scriptsize Ours w/o BD}}
\put(31.59,29.8){\makebox[0.0pt]{\scriptsize Ours w/o H}}
\put(38.65,29.8){\makebox[0.0pt]{\scriptsize Ours}}
\put(45.71,29.8){\makebox[0.0pt]{\scriptsize Reference}}
\par\put(57.71,29.8){\makebox[0.0pt]{\scriptsize Ours}}
\put(68.3,29.8){\makebox[0.0pt]{\scriptsize Mip-NeRF}}
\put(75.36,29.8){\makebox[0.0pt]{\scriptsize Ours w/o BD}}
\put(82.42,29.8){\makebox[0.0pt]{\scriptsize Ours w/o H}}
\put(89.48,29.8){\makebox[0.0pt]{\scriptsize Ours}}
\put(96.54,29.8){\makebox[0.0pt]{\scriptsize Reference}}
\par\put(6.88,14.03){\makebox[0.0pt]{\scriptsize Ours}}
\put(17.47,14.03){\makebox[0.0pt]{\scriptsize Mip-NeRF}}
\put(24.53,14.03){\makebox[0.0pt]{\scriptsize Ours w/o BD}}
\put(31.59,14.03){\makebox[0.0pt]{\scriptsize Ours w/o H}}
\put(38.65,14.03){\makebox[0.0pt]{\scriptsize Ours}}
\put(45.71,14.03){\makebox[0.0pt]{\scriptsize Reference}}
\par\put(61.26,14.03){\makebox[0.0pt]{\scriptsize Ours}}
\put(75.36,14.03){\makebox[0.0pt]{\scriptsize Mip-NeRF}}
\put(82.42,14.03){\makebox[0.0pt]{\scriptsize Eikonal Fields}}
\put(89.48,14.03){\makebox[0.0pt]{\scriptsize Ours}}
\put(96.54,14.03){\makebox[0.0pt]{\scriptsize Reference}} \par\end{overpic}
Figure 4.  Qualitative comparison of the selected synthetic (top) and real
(bottom) scenes.

### Experimental setting

We choose PSNR and SSIM for low-level image similarity, and LPIPS for better
mimicking human preference as our evaluation metrics. We set $N_{c}=64$,
$N_{f}=128$ and 200k training iterations with batch rays 1024 for mip-NeRF
(Barron et al., 2021) and ours. Moreover, during the first 2.5k warm-up
iterations, only the re-rendering error in Eqn. 7 is used. Note that we crop
the object region of an image for evaluating the real scenes.

### Ablation study

We validate our design choices with two experiments on the synthetic scenes.
_Ours w/o H_ uses no hierarchical sampling but with $N_{c}=256$. _Ours w/o BD_
uses no additional boundary regularizer. The result in Fig. 4 with boundary
regularizer shows the better refraction on the surface such as the green box
of Ship, and the hierarchical sampling further preserves the details such as
the wave in the green box of Dolphin.

### Competing method

We compare our method with mip-NeRF (Barron et al., 2021) and Eikonal Fields
(Bemana et al., 2022). The results compared with mip-NeRF show that our method
achieves a better performance of LPIPS across all the scenes (see Table 1).
For the comparison against mip-NeRF in Fig. 4, our method preserves much more
details (Ship) and generates less blurry results (Dolphin). Then, we compare
Eikonal Fields on the selected real scenes. Our method obtains a comparable
LPIPS, and Eikonal Fields cannot reconstruct the refractive index of Dolphin
scene (see Table 1). As shown in Fig. 4, our method could faithfully generate
plausible results with better clearness than Eikonal Fields (Pen).

Table 1.  Quantitative comparison of the selected synthetic and real scenes.
The top three methods of each metric for a scene are marked by gold, silver
and bronze.

| Torus | Ship | DeerGlobe | StarLamp
---|---|---|---|---
| PSNR | SSIM | LPIPS | PSNR | SSIM | LPIPS | PSNR | SSIM | LPIPS | PSNR | SSIM | LPIPS
Mip-NeRF | 22.32 | 0.759 | 0.268 | 23.93 | 0.828 | 0.151 | 26.79 | 0.881 | 0.134 | 22.04 | 0.885 | 0.092
Ours | 25.46 | 0.853 | 0.130 | 24.76 | 0.840 | 0.122 | 27.43 | 0.896 | 0.109 | 22.08 | 0.878 | 0.086
Ours w/o H | 25.57 | 0.852 | 0.136 | 24.77 | 0.838 | 0.127 | 27.58 | 0.894 | 0.108 | 22.08 | 0.876 | 0.091
Ours w/o BD | 25.87 | 0.847 | 0.133 | 25.01 | 0.838 | 0.127 | 30.03 | 0.906 | 0.089 | 21.83 | 0.866 | 0.102

| Dolphin | Ball | Glass | Pen
---|---|---|---|---
| PSNR | SSIM | LPIPS | PSNR | SSIM | LPIPS | PSNR | SSIM | LPIPS | PSNR | SSIM | LPIPS
Mip-NeRF | 18.48 | 0.459 | 0.479 | 16.29 | 0.523 | 0.418 | 18.44 | 0.475 | 0.414 | 19.13 | 0.487 | 0.428
Eikonal Fields | - | - | - | 18.38 | 0.583 | 0.239 | 17.89 | 0.436 | 0.303 | 18.83 | 0.485 | 0.335
Ours | 18.35 | 0.430 | 0.416 | 17.62 | 0.491 | 0.275 | 18.35 | 0.440 | 0.306 | 18.95 | 0.494 | 0.315

##  4\. Discussion and Future Work

We present a NeRF-based framework that synthesizes the refraction in novel
views and achieves better human perception performance in several scenes. The
results show that explicitly tracking curved paths traversing through
different refractive indices can produce more visually plausible refraction.
Furthermore, with the help of sampling techniques and a boundary regularizer,
our framework can further improve surface details and clarity. Our method
still has limitations. The blurry geometric details in real scenes result from
the imperfect camera poses compared to the synthetic data, and the foggy
artifacts appear on refractive surfaces. In the future, we plan to tackle
relighting via environment mapping to enable novel views under a new
illumination and optimizing a voxel grid of refractive index to handle more
complex refractive objects.

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# Isomorphic limit ultrapowers for infinitary logic

Saharon Shelah  Einstein Institute of Mathematics
Edmond J. Safra Campus, Givat Ram
The Hebrew University of Jerusalem
Jerusalem, 9190401, Israel
and
Department of Mathematics
Hill Center - Busch Campus
Rutgers, The State University of New Jersey
110 Frelinghuysen Road
Piscataway, NJ 08854-8019 USA<EMAIL_ADDRESS>http://shelah.logic.at

(Date: July 10, 2017)

###### Abstract.

The logic ${\mathbb{L}}^{1}_{\theta}$ introduced in [Sh:797]; it is the
maximal logic below ${\mathbb{L}}_{\theta,\theta}$ in which a well ordering is
not definable. We investigate it for $\theta$ a compact cardinal. We prove it
satisfies several parallel of classical theorems on first order logic,
strengthening the thesis that it is a natural logic. In particular, two models
are ${\mathbb{L}}^{1}_{\theta}$-equivalent iff for some $\omega$-sequence of
$\theta$-complete ultrafilters, the iterated ultra-powers by it of those two
models are isomorphic.

Also for strong limit $\lambda>\theta$ of cofinality $\aleph_{0}$, every
complete ${\mathbb{L}}^{1}_{\theta}$-theory hence a special model of
cardinality $\lambda$, a parallel of saturated. It is unique, all reducts are
special, too, so we have another proof of interpolation in this case.

###### Key words and phrases:

model theory, infinitary logics, compact cardinals, ultrapowers, ultra limits,
stability, saturated models, classification theory, isomorphic ultralimits

###### 2010 Mathematics Subject Classification:

Primary: 03C45; Secondary: 03C30, 03C55

The author would like to thank the Israel Science Foundation for partial
support of this research (Grant No. 1053/11). References like [Sh:797,
2.11=La18] means we cite from [Sh:797], Claim 2.11 which has label La18, this
to help if [Sh:797] will be revised. The author thanks Alice Leonhardt for the
beautiful typing. This was separated from [Sh:1019] which was first typed May
10, 2012; so was IJM 7367. Paper 1101

Anotated Content

§0 Introduction, pg.  ‣ 0\. Introduction

§(0A) Background and results, (label v), pg.0(A)

§(0B) Preliminaries, (label w,x), pg. 0(B)

§1 Characterizing equivalence by $\omega$-limit ultra-powers, (label d), pg.1

  1. [We characterize ${\mathbb{L}}^{1}_{<\theta}$-equivalence of $M_{1},M_{2}$ by having isomorphic ultralimits by a sequence of length $\omega$ of $\theta$-complete ultrafilters. This logic, ${\mathbb{L}}^{1}_{\theta}$, is from [Sh:797] except that here we restrict ourselves to $\theta$ is a compact cardinal.]

§2 Special Models, pg.2

  1. [We investigate model of cardinality a strong limit cardinal $>\theta$ of cofinality $\aleph_{0}$. We define $\lambda$-special model of complete theory $T\subseteq{\mathbb{L}}^{1}_{\theta}(\tau_{T})$, for $\lambda$ as above and prove existence and uniqueness. We generalize some classical theorems in model theory.]

##  0\. Introduction

###  0(A). Background and results

In the sixties, ultra-products were very central in model theory. Recall
Keisler [Kei61], solving the untrue problem of the time, assuming an instance
of GCH characterizes elementary equivalence by proving: for any two models
$M_{1},M_{2}$ (of vocabulary $\tau$ of cardinality $\leq\lambda$ and) of
cardinality $\leq\lambda^{+}$, they have isomorphic ultrapowers, even
$M^{\lambda}_{1}/D\approx M^{\lambda}_{2}/D$ for some ultrafilter $D$ on
$\lambda$ iff $M_{1},M_{2}$ are elementarily equivalent. Kochen [Koc61] uses
iteration on taking ultra-powers (on a well ordered index set) to characterize
elementary equivalence. Gaifman [Gai74] uses ultra-powers on
$\aleph_{1}$-complete ultrafilters iterated along linear ordered index set.
Keisler [Kei63] uses general $(\aleph_{0},\aleph_{0})-{\rm l.u.p.}$, see
below, Definition 0.10(4) for $\kappa=\aleph_{0}$. Shelah [Sh:13] proves this
in ZFC (but the ultrafilter is on $2^{\|M_{1}\|+\|M_{2}\|}$).

Hodges-Shelah [HoSh:109] is closer to the present work and see there on
earlier works, it deals with isomorphic ultrapowers (and isomorphic reduced
powers) for the $\theta$-complete ultrafilter (and filter) case, but note that
having isomorphic ultra-powers by $\theta$-complete ultrafilters is not an
equivalence relation. In particular assume $\theta>\aleph_{0}$ is a compact
cardinal and little more (we can get it by forcing over a universe with a
supercompact cardinal and a class of measurable cardinals). Then two models
have isomorphic ultrapowers for some $\theta$-complete ultrafilter iff in all
relevant games the isomorphism player does not lose. Those relevant games are
of length $\zeta<\theta$ and deal with the reducts to a sub-vocabulary of
cardinality $<\theta$.

The characterization [HoSh:109] of having isomorphic ultra-powers by
$\theta$-complete ultra-filters, necessarily is not so “nice” because this
relation is not an equivalence relation. Hence having isomorphic ultra-powers
is not connected to having the same theory in some logic. But [Sh:797]
suggests a logic
${\mathbb{L}}^{1}_{\theta}\subseteq{\mathbb{L}}_{\theta,\theta}$ with some
good properties (like well ordering not characterizable, interpolation)
maximal under such properties. We may wonder, do we have a characterization of
models being ${\mathbb{L}}^{1}_{\theta}$-equivalent?

In §1 we characterize ${\mathbb{L}}^{1}_{\theta}$-equivalence of models by
having isomorphic iterated ultra-powers of length $\omega$. We then prove some
generalizations of classical model theoretic theorems, like the existence and
uniqueness of special models in $\lambda$ when $\lambda>\theta+|T|$ is strong
limit of cofinality $\aleph_{0}$. All this seems to strengthen the thesis of
[Sh:797] that ${\mathbb{L}}^{1}_{\theta}$ is a natural logic.

###  0(B). Preliminaries

######  Hypothesis 0.1.

$\theta$ is a compact uncountable cardinal (of course, we use only restricted
versions of this).

######  Notation 0.2.

1) Let $\varphi(\bar{x})$ mean: $\varphi$ is a formula of
${\mathbb{L}}_{\theta,\theta},\bar{x}$ is a sequence of variables with no
repetitions including the variables occuring freely in $\varphi$ and $\ell
g(\bar{x})<\theta$ if not said otherwise. We use $\varphi,\psi,\vartheta$ to
denote formulas and $\varphi^{{\rm st}}$ or $\varphi^{[{\rm st}]}$ or
$\varphi^{{\rm if}({\rm st})}$ is $\varphi$ if ${\rm st}$ is true or 1 and
$\neg\varphi$ if ${\rm st}$ is false or 0.

2) For a set $u$, usually of ordinals, let $\bar{x}_{[u]}=\langle
x_{\varepsilon}:\varepsilon\in u\rangle$, now $u$ may be an ordinal but, e.g.
if $u=[\alpha,\beta)$ we may write $\bar{x}_{[\alpha,\beta)}$; similarly for
$\bar{y}_{[u]},\bar{z}_{[u]}$; let $\ell g(\bar{x}_{[u]})=u$.

3) $\tau$ denotes a vocabulary, i.e. a set of predicates and function symbols
each with $<\theta$ places (but in §3 the number of places is finite).

4) $T$ denotes a theory in ${\mathbb{L}}_{\theta,\theta}$ or
${\mathbb{L}}^{1}_{\theta}$ (see below); usually complete in the vocabulary
$\tau_{T}$ and with a model of cardinality $\geq\theta$ if not said otherwise.

5) Let ${\rm Mod}_{T}$ be the class of models of $T$.

6) For a model $M$ let its vocabulary be $\tau_{M}$.

######  Notation 0.3.

$\varepsilon,\zeta,\xi$ are ordinals $<\theta$.

######  Definition 0.4.

1) Let ${\rm uf}_{\theta}(I)$ be the set of $\theta$-complete ultrafilters on
$I$, non-principal if not said otherwise. Let ${\rm fil}_{\theta}(I)$ be the
set of $\theta$-complete filters on $I$; mainly we use
$(\theta,\theta)$-regular ones (see below).

2) $D\in{\rm fil}_{\theta}(I)$ is called $(\lambda,\theta)$-regular when there
is a witness $\bar{w}=\langle w_{t}:t\in I\rangle$ which means:
$w_{t}\in[\lambda]^{<\theta}$ for $t\in I$ and
$\alpha<\lambda\Rightarrow\\{t:\alpha\in w_{t}\\}\in D$.

3) Let ${\rm ruf}_{\lambda,\theta}(I)$ be the set of
$(\lambda,\theta)$-regular $D\in{\rm uf}_{\theta}(I)$; let ${\rm
rfil}_{\lambda,\theta}(I)$ be the set of $(\lambda,\theta)$-regular $D\in{\rm
fil}_{\theta}(I)$; when $\lambda=|I|$ we may omit $\lambda$.

######  Definition 0.5.

1) ${\mathbb{L}}_{\theta,\theta}(\tau)$ is the set of formulas of
${\mathbb{L}}_{\theta,\theta}$ in the vocabulary $\tau$.

2) For $\tau$-models $M,N$ let $M\prec_{{\mathbb{L}}_{\theta,\theta}}N$ means:
if $\varphi(\bar{x})\in{\mathbb{L}}_{\theta,\theta}(\tau_{M})$ and
$\bar{a}\in{}^{\ell g(\bar{x})}M$ then
$M\models\varphi[\bar{a}]\Leftrightarrow N\models\varphi[\bar{a}]$.

And, of course

######  Fact 0.6.

For a complete $T\subseteq{\mathbb{L}}_{\theta,\theta}(\tau)$.

$({\rm Mod}_{T},\prec_{{\mathbb{L}}_{\theta,\theta}})$ has amalgamation and
the joint embedding property (JEP), that is:

  1. $(a)$

amalgamation: if $M_{0}\prec_{{\mathbb{L}}_{\theta,\theta}}M_{\ell}$ for
$\ell=1,2$ then there are $M_{3},f_{1},f_{2},M^{\prime}_{1},M^{\prime}_{2}$
such that

    1. $\bullet$

$M_{0}\prec_{{\mathbb{L}}_{\theta,\theta}}M_{3}$

    2. $\bullet$

for $\ell=1,2,f_{\ell}$ is a $\prec_{{\mathbb{L}}_{\theta,\theta}}$-embedding
of $M_{\ell}$ into $M_{3}$ over $M_{0}$, that is, for some $\tau_{T}$-models
$M^{\prime}_{\ell}$ for $\ell=1,2$ we have
$M^{\prime}_{\ell}\prec_{{\mathbb{L}}_{\theta,\theta}}M_{3}$ and $f_{\ell}$ is
an isomorphism from $M_{\ell}$ onto $M^{\prime}_{\ell}$ over $M_{0}$;

  2. $(b)$

JEP: if $M_{1},M_{2}$ are ${\mathbb{L}}_{\theta,\theta}$-equivalent
$\tau$-models then there is a $\tau$-model $M_{3}$ and
$\prec_{{\mathbb{L}}_{\theta,\theta}}$-embedding $f_{\ell}$ of $M_{\ell}$ into
$M_{3}$ for $\ell=1,2$.

The well known generalization of Łos theorem is:

######  Theorem 0.7.

1) If
$\varphi(\bar{x}_{[\zeta]})\in{\mathbb{L}}_{\theta,\theta}(\tau),D\in{\rm
uf}_{\theta}(I)$ and $M_{s}$ is a $\tau$-model for $s\in I$ and
$f_{\varepsilon}\in\prod\limits_{s\in I}M_{s}/D$ for $\varepsilon<\zeta$ then
$M\models\varphi[\ldots,f_{\varepsilon}/D,\ldots]_{\varepsilon<\zeta}$ iff the
set $\\{s\in
I:M_{s}\models\varphi[\ldots,f_{\varepsilon}(s),\ldots]_{\varepsilon<\zeta}\\}$
belongs to $D$.

2) Similarly $M\prec_{{\mathbb{L}}_{\theta,\theta}}M^{I}/D$.

######  Definition 0.8.

0) We say $X$ respects $E$ when for some set $I,E$ is an equivalence relation
on $I$ and $X\subseteq I$ and $sEt\Rightarrow(s\in X\leftrightarrow t\in X)$.

1) We say $\mathbb{x}=(I,D,{\mathscr{E}})$ is a $(\kappa,\sigma)-{\rm
l.u.f.t.}$ (limit-ultra-filter-iteration triple) when :

  1. $(a)$

$D$ is a filter on the set $I$

  2. $(b)$

${\mathscr{E}}$ is a family of equivalence relations on $I$

  3. $(c)$

$({\mathscr{E}},\supseteq)$ is $\sigma$-directed, i.e. if $\alpha(*)<\sigma$
and $E_{i}\in{\mathscr{E}}$ for $i<\alpha(*)$ then there is
$E\in{\mathscr{E}}$ refining $E_{i}$ for every $i<\alpha(*)$

  4. $(d)$

if $E\in{\mathscr{E}}$ then $D/E$ is a $\kappa$-complete ultrafilter on $I/E$
where $D/E:=\\{X/E:X\in D$ and $X$ respects $E\\}$.

1A) Let $\mathbb{x}$ be a $(\kappa,\theta)-{\rm l.f.t.}$ mean that above we
weaken (d) to

  1. $(d)^{\prime}$

if $E\in{\mathscr{E}}$ then $D/E$ is a $\kappa$-complete filter.

2) Omitting “$(\kappa,\sigma)$” means $(\theta,\aleph_{0})$, recalling
$\theta$ is our fixed compact cardinal.

3) Let
$(I_{1},D_{1},{\mathscr{E}}_{1})\leq^{1}_{h}(I_{2},D_{2},{\mathscr{E}}_{2})$
mean that:

  1. $(a)$

$h$ is a function from $I_{2}$ onto $I_{1}$

  2. $(b)$

if $E\in{\mathscr{E}}_{1}$ then $h^{-1}\circ E\in{\mathscr{E}}_{2}$ where
$h^{-1}\circ E=\\{(s,t):s,t\in I_{2}$ and $h(s)Eh(t)\\}$

  3. $(c)$

if $E_{1}\in{\mathscr{E}}_{1}$ and $E_{2}=h^{-1}\circ E_{1}$ then
$D_{1}/E_{1}=h^{\prime\prime}(D_{2}/E_{2})$.

######  Remark 0.9.

Note that in 0.8(3), if $h={\rm id}_{I_{2}}$ then $I_{1}=I_{2}$.

######  Definition 0.10.

Assume $\mathbb{x}=(I,D,{\mathscr{E}})$ is a $(\kappa,\sigma)$-l.u.f.t.

1) For a function $f$ let ${\rm eq}(f)=\\{(s_{1},s_{2}):f(s_{1})=f(s_{2})\\}$.
If $\bar{f}=\langle f_{i}:i<i_{*}\rangle$ and $i<i_{*}\Rightarrow{\rm
dom}(f_{i})=I$ then ${\rm eq}(\bar{f})=\cap\\{{\rm eq}(f_{i}):i<i_{*}\\}$.

2) For a set $U$ let $U^{I}|{\mathscr{E}}=\\{f\in{}^{I}U:{\rm eq}(f)$ is
refined by some $E\in{\mathscr{E}}\\}$.

3) For a model $M$ let ${\rm
l.r.p.}_{\mathbb{x}}(M)=M^{I}_{D}|{\mathscr{E}}=(M^{I}/D){\restriction}\\{f/D:f\in{}^{I}M$
and ${\rm eq}(f)$ is refined by some $E\in{\mathscr{E}}\\}$, pedantically (as
${\rm arity}(\tau_{M})$ may be $>\aleph_{0}$),
$M^{I}_{D}|{\mathscr{E}}=\cup\\{M^{I}_{D}{\restriction}E:E\in{\mathscr{E}}\\}$;
${\rm l.r.p.}$ stands for limit reduced power.

4) If $\mathbb{x}$ is ${\rm l.u.f.t.}$ we may in (3) write ${\rm
l.u.p.}_{\mathbb{x}}(M)$.

We now give the generalization of Keisler [Kei63]; Hodges-Shelah [HoSh:109,
Lemma 1,pg.80] in the case $\kappa=\sigma$.

######  Theorem 0.11.

1) If $\sigma\leq\kappa$ and $(I,D,{\mathscr{E}})$ is $(\kappa,\sigma)-{\rm
l.u.f.t.},\varphi=\varphi(\bar{x}_{[\zeta]})\in{\mathbb{L}}_{\kappa,\sigma}(\tau)$
so $\zeta<\sigma,f_{\varepsilon}\in M^{I}|{\mathscr{E}}$ for
$\varepsilon<\zeta$ then
$M^{I}_{D}|{\mathscr{E}}\models\varphi[\ldots,f_{\varepsilon}/D,\ldots]$ iff
$\\{s\in
I:M\models\varphi[\ldots,f_{\varepsilon}(s),\ldots]_{\varepsilon<\zeta}\\}\in
D$.

2) Moreover $M\prec_{{\mathbb{L}}_{\kappa,\sigma}}M^{I}_{D}/{\mathscr{E}}$,
pedantically $\mathbb{j}=\mathbb{j}_{M,\mathbb{x}}$ is a
$\prec_{{\mathbb{L}}_{\kappa,\sigma}}$-elementary embedding of $M$ into
$M^{I}_{D}/{\mathscr{E}}$ where $\mathbb{j}(a)=\langle a:s\in I\rangle/D$.

3) We define $(\prod\limits_{s\in I}M_{s})^{I}_{D}|{\mathscr{E}}$ similarly
when ${\rm eq}(\langle M_{s}:s\in I\rangle)$ is refined by some
$E\in{\mathscr{E}}$, may use this more in end of the proof of 1.2.

######  Convention 0.12.

1) Abusing a notation in $\prod\limits_{s\in I}M_{s}/D$ we allow $f/D$ for
$f\in\prod\limits_{s\in S}M_{s}$ when $S\in D$.

2) For $\bar{c}\in{}^{\gamma}(\prod\limits_{s\in I}M_{s}/D)$ we can find
$\langle\bar{c}_{s}:s\in I\rangle$ such that
$\bar{c}_{s}\in{}^{\gamma}(M_{s})$ and $\bar{c}=\langle\bar{c}_{s}:s\in
I\rangle/D$ which means: if $i<\ell g(\bar{c})$ then $c_{s,i}\in M_{s}$ and
$c_{i}=\langle c_{s,i}:s\in I\rangle/D$.

######  Remark 0.13.

1) Why the “pedantically” in 0.10(3)? Otherwise if $\mathbb{x}$ is a
$(\theta,\sigma)-{\rm l.u.f.t.},({\mathscr{E}}_{\mathbb{x}},\supseteq)$ is not
$\kappa^{+}$-directed, $\kappa<{\rm arity}(\tau)$ then defining ${\rm
l.u.p.}_{\mathbb{x}}(M)$, we have freedom: if $R\in\tau,{\rm
arity}_{\tau}(R)\geq\kappa$, i.e. on
$R^{N}{\restriction}\\{\bar{a}:\bar{a}\in{}^{{\rm arity}(P)}N$ and no
$E\in{\mathscr{E}}$ refines ${\rm eq}(\bar{a})\\}$ so we have no restrictions.

2) So, e.g. for categoricity we better restrict ourselves to vocabularies
$\tau$ such that ${\rm arity}(\tau)=\aleph_{0}$.

######  Definition 0.14.

We say $M$ is a $\theta$-complete model when for every
$\varepsilon<\theta,R_{*}\subseteq{}^{\varepsilon}M$ and
$F_{*}:{}^{\varepsilon}M\rightarrow M$ there are $R,F\in\tau_{M}$ such that
$R^{M}=R_{*}\wedge F^{M}=F_{*}$.

######  Observation 0.15.

1) If $M$ is a $\tau$-model of cardinality $\lambda$ then there is a
$\theta$-complete expansion $M^{+}$ of $M$ so $\tau(M^{+})\supseteq\tau(M)$
and $\tau(M^{+})$ has cardinality $|\tau_{M}|+2^{(\|M\|^{<\theta})}$.

2) For models $M\prec_{{\mathbb{L}}_{\theta,\theta}}N$ and $M^{+}$ as above
the following conditions are equivalent:

  1. $(a)$

$N={\rm l.u.p.}_{\mathbb{x}}(M)$ identifying $a\in M$ with
$\mathbb{j}_{\mathbb{x}}(a)\in N$, for some $(\theta,\theta)-{\rm
l.u.f.t.}\,\mathbb{x}$

  2. $(b)$

there is $N^{+}$ such that $M^{+}\prec_{{\mathbb{L}}_{\theta,\theta}}N^{+}$
and $N^{+}{\restriction}\tau_{M}$ is isomorphic to $N$ over $M$.

3) For a model $M$, if $(P^{M},<^{M})$ is a $\theta$-directed partial order
and $\chi={\rm cf}(\chi)\geq\theta$ and $\lambda=\lambda^{\|M\|}+\chi$ then
for some $(\theta,\theta)-{\rm l.u.f.t.}\,\mathbb{x}$, the model $N:={\rm
l.u.p.}_{\mathbb{x}}(M)$ satisfies $(P^{N},<^{N})$ has a cofinal increasing
sequence of length $\chi$ and $|P^{N}|=\lambda$.

###### Proof..

Easy, e.g.

3) Let $M^{+}$ be as in part (1). Note that $M^{+}$ has Skolem functions and
let $T^{\prime}$ be the following set of formulas: ${\rm
Th}_{{\mathbb{L}}_{\theta,\theta}}(M^{+})\cup\\{P(x_{\varepsilon}):{\varepsilon}<\lambda\cdot\chi\\}\cup\\{P(\sigma(x_{\varepsilon_{0}},\ldots,x_{\varepsilon_{i}},\ldots)_{i<i(*)})\rightarrow\sigma(x_{\varepsilon_{0}},\dotsc,x_{\varepsilon_{i}},\ldots)_{i<i(*)}<x_{\varepsilon}:\sigma$
is a $\tau(M^{+})$-term so $i(*)<\theta$ and
$i<i(*)\Rightarrow\varepsilon_{i}<\varepsilon<\lambda\cdot\chi\\}$.

Clearly

  1. $(*)$

$T^{\prime}$ is $(<\theta)$-satisfiable in $M^{+}$.

[Why? Because if $T^{\prime\prime}\subseteq T^{\prime}$ has cardinality
$<\theta$ then the set $u=\\{\varepsilon<\lambda\cdot\chi:x_{\varepsilon}$
appears in $T^{\prime\prime}\\}$ has cardinality $<\theta$ and let $i(*)={\rm
otp}(u)$; clearly for each $\varepsilon\in u$ the set
$\Gamma_{\varepsilon}=T^{\prime}\cap\\{P(\sigma(x_{\varepsilon_{0}},\ldots))\rightarrow\sigma(x_{\varepsilon_{0}},\dotsc,x_{\varepsilon_{i}},\ldots)_{i<i(*)}<x_{\varepsilon}:i(*)<\theta$
and $\varepsilon_{i}<\varepsilon$ for $i<i(*)\\}$ has cardinality $<\theta$.
Now we choose $c_{\varepsilon}\in M$ by induction on $\varepsilon\in u$ such
that the assignment $x_{\zeta}\mapsto c_{\zeta}$ for $\zeta\in\varepsilon\cap
u$ in $M^{+}$ satisfies $\Gamma_{\varepsilon}$, possible because
$|\Gamma_{\varepsilon}|<\theta$ and $(P^{M},<^{M})$ is $\theta$-directed. So
the $M^{+}$ with the assignment $x_{\varepsilon}\mapsto c_{\varepsilon}$ for
$\varepsilon\in u$ is a model of $T^{\prime\prime}$, so $T^{\prime}$ is
($<\theta)$-satisfiable indeed.]

Recalling that $|M|=\\{c^{M^{+}}:c\in\tau(M^{+})$ an individual constant$\\}$,
$T^{\prime}$ is realized in some
$\prec_{{\mathbb{L}}_{\theta,\theta}}$-elementary extension $N^{+}$ of $M^{+}$
by the assignment $x_{\varepsilon}\mapsto
a_{\varepsilon}(\varepsilon<\lambda\cdot\chi)$. Without loss of generality
$N^{+}$ is the Skolem hull of
$\\{a_{\varepsilon}:\varepsilon<\lambda\cdot\chi\\}$, so
$N:=N^{+}{\restriction}\tau(M)$ is as required by the choice of $T^{\prime}$.
Now $\mathbb{x}$ is as required exists by part (2) of the claim. ∎

######  Observation 0.16.

1) If $\mathbb{x}$ is a non-trivial $(\theta,\theta)-{\rm l.u.f.t.}$ and
$\chi={\rm cf}({\rm l.u.p.}(\theta<))$ then $\chi=\chi^{<\theta}$.

2) Also $\mu=\mu^{<\theta}$ when $\mu$ is the cardinality of ${\rm
l.u.p.}(\theta,<)$.

###### Proof..

1) By the choice of $\mathbb{x}$ clearly $\chi\geq\theta$. As $\chi$ is
regular $\geq\theta$ by a theorem of Solovay [Sol74] we have
$\chi^{<\theta}=\chi$.

2) See the proof of [Sh:1019, 2.20(3)=La27(3)]. ∎

######  Definition 0.17.

For a vocabulary $\tau,\tau$-models $M_{1},M_{2}$, a set $\Gamma$ of formulas
in the vocabulary $\tau$ in any logic (each with finitely many free variables
if not said otherwise; see [Sh:1019, 2.9=La10(4)]), cardinal $\theta$ and
ordinal $\alpha$ we define a game
$\Game=\Game_{\Gamma,\theta,\alpha}[M_{1},M_{2}]$ as follows, and using
$(M_{1},\bar{b}_{1}),(M_{2},\bar{b}_{2})$ with their natural meaning when
${\rm Dom}(\bar{b}_{1})={\rm Dom}(\bar{b}_{2})$.

  1. $(A)$

The moves are indexed by $n<\omega$ (but every actual play is finite), just
before the $n$-th move we have a state
$\mathbb{s}_{n}=(A^{1}_{n},A^{2}_{n},h^{1}_{n},h^{2}_{n},g_{n},\beta_{n},n)$

  2. $(B)$

$\mathbb{s}=(A^{1},A^{2},h^{1},h^{2},g,\beta,n)=(A^{1}_{\mathbb{s}},A^{2}_{\mathbb{s}},h^{1}_{\mathbb{s}},h^{2}_{\mathbb{s}},g_{\mathbb{s}},\beta_{\mathbb{s}},n_{\mathbb{s}})$
is a state (or $n$-state or $(\theta,n)$-state or $(\theta,<\omega)$-state)
when:

    1. $(a)$

$A^{\ell}\in[M_{\ell}]^{\leq\theta}$ for $\ell=1,2$

    2. $(b)$

$\beta\leq\alpha$ is an ordinal

    3. $(c)$

$h^{\ell}$ is a function from $A^{\ell}$ into $\omega$

    4. $(d)$

$g$ is a partial one-to-one function from $M_{1}$ to $M_{2}$ and let
$g^{1}_{\mathbb{s}}=g^{1}=g_{\mathbb{s}}=g$ and let
$g^{2}_{\mathbb{s}}=g^{2}=(g^{1}_{\mathbb{s}})^{-1}$,

    5. $(e)$

Dom$(g^{\ell})\subseteq A^{\ell}$ for $\ell=1,2$

    6. $(f)$

$g$ preserves satisfaction of the formulas in $\Gamma$ and their negations,
i.e. for $\varphi(\bar{x})\in\Gamma$ and $\bar{a}\in{}^{\ell
g(\bar{x})}\text{Dom}(g)$ we have $M_{1}\models\varphi[\bar{a}]\Leftrightarrow
M_{2}\models\varphi[g(\bar{a})]$

    7. $(g)$

if $a\in\text{ Dom}(g^{\ell})$ then $h^{\ell}(a)<n$

  3. $(C)$

we define the state $\mathbb{s}=\mathbb{s}_{0}=\mathbb{s}^{0}_{\alpha}$ by
letting
$n_{\mathbb{s}}=0,A^{1}_{\mathbb{s}}=\emptyset=A^{2}_{\mathbb{s}},\beta_{\mathbb{s}}=\alpha,h^{1}_{\mathbb{s}}=\emptyset=h^{2}_{\mathbb{s}},g_{\mathbb{s}}=\emptyset$;
so really $\mathbb{s}$ depends only on $\alpha$ (but in general, this may not
be a state for our game as possibly for some sentence $\psi\in\Gamma$ we have
$M_{1}\models\psi\Leftrightarrow M_{2}\models\neg\psi$)

  4. $(D)$

we say that a state $\mathbb{t}$ extends a state $\mathbb{s}$ when
$A^{\ell}_{\mathbb{s}}\subseteq
A^{\ell}_{\mathbb{t}},h^{\ell}_{\mathbb{s}}\subseteq h^{\ell}_{\mathbb{t}}$
for $\ell=1,2$ and $g_{\mathbb{s}}\subseteq
g_{\mathbb{t}},\beta_{\mathbb{s}}>\beta_{\mathbb{t}},n_{\mathbb{s}}<n_{\mathbb{t}}$;
we say $\mathbb{t}$ is a successor of $\mathbb{s}$ if in addition
$n_{\mathbb{t}}=n_{\mathbb{s}}+1$

  5. $(E)$

in the $n$-th move

the anti-isomorphism player (AIS) chooses
$(\beta_{n+1},\iota_{n},A^{\prime}_{n})$ such that:

$\iota_{n}\in\\{1,2\\},\beta_{n+1}<\beta_{n}$ and $A^{\iota_{n}}_{n}\subseteq
A^{\prime}_{n}\in[M_{\iota_{n}}]^{\leq\theta}$,

the isomorphism player (ISO) chooses a state $\mathbb{s}_{n+1}$ such that

    1. $\bullet$

$\mathbb{s}_{n+1}$ is a successor of $\mathbb{s}_{n}$

    2. $\bullet$

$A^{\iota_{n}}_{\mathbb{s}_{n+1}}=A^{\prime}_{n}$

    3. $\bullet$

$A^{3-\iota_{n}}_{\mathbb{s}_{n+1}}=A^{3-\iota_{n}}_{\mathbb{s}_{n}}\cup{\rm
Dom}(g^{3-\iota_{n}}_{\mathbb{s}_{n+1}})$

    4. $\bullet$

if $a\in A^{\prime}_{n}\backslash A^{\iota_{n}}_{\mathbb{s}_{n}}$ then
$h^{\iota_{n}}_{\mathbb{s}_{n+1}}(a)\geq n+1$

    5. $\bullet$

${\rm Dom}(g^{\iota_{n}}_{\mathbb{s}_{n+1}})=\\{a\in
A^{\iota_{n}}_{\mathbb{s}_{n}}:h^{\iota_{n}}_{\mathbb{s}_{n}}(a)<n+1\\}$ so it
includes ${\rm Dom}(g^{\iota_{n}}_{\mathbb{s}_{n}})$

    6. $\bullet$

$\beta_{\mathbb{s}_{n+1}}=\beta_{n+1}$.

  6. $(F)$

     * •

the play ends when one of the players has no legal moves (always occur as
$\beta_{n}<\beta_{n-1}$) and then this player loses, this may occur for $n=0$

     * •

for $\alpha=0$ we stipulate that ISO wins iff $\mathbb{s}^{0}_{\alpha}$ is a
state.

######  Definition 0.18.

1) Let ${\mathscr{E}}^{0,\tau}_{\Gamma,\theta,\alpha}$ be the class
$\\{(M_{1},M_{2}):M_{1},M_{2}$ are $\tau$-models and in the game
$\Game_{\Gamma,\theta,\alpha}[M_{1},M_{2}]$ the ISO player has a winning
strategy$\\}$ where $\Gamma$ is a set of formulas in the vocabulary $\tau$,
each with finitely many free variables.

2) ${\mathscr{E}}^{1,\tau}_{\Gamma,\theta,\alpha}$ is the closure of
${\mathscr{E}}^{0,\tau}_{\Gamma,\theta,\alpha}$ to an equivalence relation (on
the class of $\tau$-models).

3) Above we may replace $\Gamma$ by qf$(\tau)$ which means $\Gamma=$ the set
at$(\tau)$ or bs$(\tau)$ formulas in the vocabulary $\tau$.

4) Above if we omit $\tau$ we mean $\tau=\tau_{\Gamma}$ and if we omit
$\Gamma$ we mean bs$(\tau)$. Abusing notation we may say $M_{1},M_{2}$ are
${\mathscr{E}}^{0,\tau}_{\Gamma,\theta,\alpha}$-equivalent.

######  Definition 0.19.

For a vocabulary $\tau$, the $\tau$-models $M_{1},M_{2}$ are
${\mathbb{L}}^{1}_{<\theta}$-equivalent iff for every $\mu<\theta$ and
$\alpha<\mu^{+}$ and $\tau_{1}\subseteq\tau$ of cardinality $\leq\mu$, letting
$\Gamma=$ the quantifier first order formulas in ${\mathbb{L}}(\tau)$, the
models $M_{1},M_{2}$ are ${\mathscr{E}}^{1,\tau_{1}}_{\Gamma,\mu,\alpha}$.

##  1\. Characterizing equivalence by $\omega$-limit ultra-powers

In [Sh:797], a logic
${\mathbb{L}}^{1}_{<\kappa}=\bigcup\limits_{\mu<\kappa}{\mathbb{L}}^{1}_{\leq\mu}$
is introduced (here we consider $\kappa$ is strongly inaccessible for
transparency), and is proved to be stronger than
${\mathbb{L}}_{\kappa,\aleph_{0}}$ but weaker than
${\mathbb{L}}_{\kappa,\kappa}$, has interpolation and a characterization, well
ordering not definable in it and has an addition theorem. Also it is the
maximal logic with some such properties.

For $\kappa=\theta$, we give a characterization of when two models are
${\mathbb{L}}^{1}_{<\theta}$-equivalent giving an additional evidence for the
logic naturality.

######  Convention 1.1.

In this section every vocabulary $\tau$ has ${\rm arity}(\tau)=\aleph_{0}$.

Recall [Sh:797, 2.11=La18] which says

######  Claim 1.2.

Assume $|\tau|\leq\mu,M_{n}$ is a $\tau$-model and
$M_{n}\prec_{{\mathbb{L}}_{\mu^{+},\mu^{+}}}M_{n+1}$ for $n<\omega$ and
$M_{\omega}=\cup\\{M_{n}:n<\omega\\}$. Then $M_{0},M_{\omega}$ are
${\mathbb{L}}^{1}_{\leq\mu}$-equivalent.

We need two definitions before stating and proving the theorem below. The
first definition generalizes common concepts.

######  Definition 1.3.

We say that a pair of models $(M_{1},M_{2})$ has isomorphic $\theta$-complete
$\omega$-iterated ultrapowers iff one can find $D_{n}\in{\rm
uf}_{\theta}(I_{n})$ for every $n\in\omega$ such that $M^{1}_{\omega}\cong
M^{2}_{\omega}$, when
$M^{\ell}_{\omega}=\cup\\{M^{\ell}_{k}:k\in\omega\\},M^{\ell}_{0}=M_{\ell}$
and
$M^{\ell}_{n}\prec_{{\mathbb{L}}_{\theta,\theta}}(M^{\ell}_{n})^{I_{n}}/D_{n}=M^{\ell}_{n+1}$
for $\ell=1,2$ and $n\in\omega$.

For the second definition, let $\mathbb{x}$ be a l.u.f.t. and we define the
following:

######  Definition 1.4.

If $\mathbb{x}=(I,D,\bar{E})$ is an ${\rm l.u.f.t.}$ and $\bar{E}=\langle
E_{n}:n\in\omega\rangle$ then $\bar{w}$ is a niceness witness for
$(I,D,\bar{E})$ when :

  1. (a)

$\bar{w}=\langle w_{s,n},\gamma_{s,n}:s\in I,n\in\omega\rangle$

  2. (b)

$w_{s,n}\subseteq\lambda_{n}$ and $|w_{s,n}|<\theta$ and
$|w_{s,n}|\geq|w_{s,n+1}|$

  3. (c)

$\gamma_{s,n}<\theta$ and
$(\gamma_{s,n}>\gamma_{s,n+1})\vee(\gamma_{s,n+1}=0)$

  4. (d)

$\gamma_{s,n}=0\Rightarrow w_{s,n}=\emptyset$ but $w_{s,0}\neq\emptyset$ and
for simplicity $w_{s,0}$ is infinite for every $s\in I$

  5. (e)

if $n\in\omega,u\in[\lambda_{n}]^{<\theta}$ then $\\{s\in I:u\subseteq
w_{s,n}\\}\in D$

  6. (f)

$w_{s,n}=w_{t,n}$ and $\gamma_{s,n}=\gamma_{t,n}$ when $sE_{n}t$.

######  Theorem 1.5.

Let $\theta$ be a compact cardinal and $M_{1},M_{2}$ be $\tau$-models (and
${\rm arity}(\tau)=\aleph_{0}$).

The following conditions are equivalent:

  1. $(a)$

$M_{1},M_{2}$ are ${\mathbb{L}}^{1}_{\theta}$-equivalent

  2. $(b)$

there are $(\theta,\theta)-{\rm
l.u.f.t.}\,\mathbb{x}_{n}=(I,D,{\mathscr{E}}_{n}),{\mathscr{E}}_{n}\subseteq{\mathscr{E}}_{n+1}$
for $n<\omega$ and we let ${\mathscr{E}}=\cup\\{{\mathscr{E}}_{n}:n<\omega\\}$
such that $(M_{1})^{I}_{D}|{\mathscr{E}}$ is isomorphic to
$(M_{2})^{I}_{D}|{\mathscr{E}}$

  3. $(c)$

$(M_{1},M_{2})$ have isomorphic $\theta$-complete $\omega$-iterated
ultrapowers

  4. $(d)$

if $D_{n}\in{\rm ruf}_{\lambda_{n},\theta}(I_{n})$ so $|I_{n}|\geq\lambda_{n}$
and $\lambda_{n+1}\geq 2^{|I_{n}|}\lambda_{n}>\|M_{1}\|+\|M_{2}\|+|\tau|$ for
every $n$ then the sequence $\langle(I_{n},D_{n}):n<\omega\rangle$ is as
required in clause (c)

  5. $(e)$

if $\mathbb{x}=(I,D,{\mathscr{E}})$ is a ${\rm
l.u.f.t.},{\mathscr{E}}=\\{E_{n}:n<\omega\\}$, for $n<\omega$ we have
$E_{n+1}$ refines $E_{n},2^{|I/E_{n}|}\leq\lambda_{n+1},D/E_{n}$ is a
$(\lambda_{n},\theta)$-regular $\theta$-complete ultrafilter,
$\lambda_{0}\geq\|M_{1}\|+\|M_{2}\|+|\tau|,\bar{w}$ is a niceness witness,
then ${\rm l.u.p.}_{\mathbb{x}}(M_{1})\cong{\rm l.u.p.}_{\mathbb{x}}(M_{2})$.

###### Proof..

Clause $(b)\Rightarrow$ Clause $(a)$:

So let $I,D,{\mathscr{E}}_{n}(n<\omega)$ be as in clause (b) and
${\mathscr{E}}=\cup\\{{\mathscr{E}}_{n}:n<\omega\\}$. By the transitivity of
being ${\mathbb{L}}^{1}_{<\theta}$-equivalent, clearly clause (a) follows
from:

  1. $\boxplus_{1}$

for every model $N$ the models $N,N^{I}_{D}|{\mathscr{E}}$ are
${\mathbb{L}}^{1}_{\theta}$-equivalent.

[Why $\boxplus_{1}$ holds? Let $N_{n}=N^{I}_{D}|{\mathscr{E}}_{n}$ for
$n<\omega$ and $N_{\omega}=\cup\\{N_{n}:n<\omega\\}$. So by 0.11 we have
$N\equiv_{{\mathbb{L}}_{\theta,\theta}}N_{0}$ and moreover
$N_{n}\prec_{{\mathbb{L}}_{\theta,\theta}}N_{n+1}$. Hence by 1.2, that is the
“Crucial Claim” [Sh:797, 2.11=a18] we have
$N_{n}\equiv_{{\mathbb{L}}^{1}_{<\theta}}N_{\omega}$ hence, in particular,
$N\equiv_{{\mathbb{L}}^{1}_{<\theta}}N_{\omega}$.]

Clause $(c)\Rightarrow$ Clause $(b)$:

Let $I=\prod\limits_{n<\omega}I_{n},E_{n}=\\{(\eta,\nu):\eta,\nu\in I$ and
$\eta{\restriction}n=\nu{\restriction}n\\}$ and $D=\\{X\subseteq I$: for some
$n,(\forall^{D_{n}}i_{n}\in I_{n})(\forall^{D_{n-1}}i_{n-1}\in
I_{n-1})\ldots(\forall^{D_{0}}i_{0}\in I_{0})(\forall\eta)[\eta\in
I\wedge\bigwedge\limits_{\ell\leq n}\eta(\ell)=i_{\ell}\rightarrow\eta\in
X\\}$. Now let
$M^{\ell}_{\omega}\equiv(M_{\ell})^{I}_{D}|\\{E_{n}:n<\omega\\}$.

So it should be clear that $(M_{\ell})^{I}_{D}|\\{E_{n}:n<\omega\\}$ is
isomorphic to $M^{\ell}_{\omega}$ for $\ell=1,2$, so recalling
$M^{1}_{\omega}\cong M^{2}_{\omega}$ by the present assumption, the models
$(M_{\ell})^{I}_{D}|\\{E_{n}:n<\omega\\}$ for $\ell=1,2$ are isomorphic, so
letting ${\mathscr{E}}_{n}=\\{E_{0},\dotsc,E_{n}\\}$ easily
$(I,D,{\mathscr{E}}_{n})_{n<\omega}$ are as required in clause (b).

Clause $(d)\Rightarrow$ Clause $(c)$:

Clause (d) is obviously stronger. We can choose
$\lambda_{0}=(\|M_{1}\|+\|M_{2}\|+|\tau|+\theta)^{<\theta},\lambda_{n+1}=2^{\lambda_{n}}$
for $n<\omega$ then letting $I_{n}=\lambda_{n}$ there is $D_{n}\in{\rm
ruf}_{\lambda_{n},\theta}(I_{n})$ recalling $\theta$ is a compact cardinal,
noting $\lambda_{n}=\lambda_{n}^{<\theta}$. Now $\langle
I_{n},D_{n}:n<\omega\rangle$ is as required in clause (c), in particualr the
isomorphism holsd by clause (d) which we are assuming.

Clause $(e)\Rightarrow$ Clause $(d)$:

Let $\langle(I_{n},D_{n},\lambda_{n}):n<\omega\rangle$ be as in the assumption
of clause (d).

We define $I=\prod\limits_{n}I_{n},E_{n}=\\{(\eta,\nu):\eta,\nu\in
I,\eta{\restriction}(n+1)=\nu{\restriction}(n+1)\\}$ and define $D$ as in the
proof of $(c)\Rightarrow(b)$ above and we choose $\bar{w}=\langle
w_{\eta,n}:\eta\in I,n<\omega\rangle$ as follows.

First, choose $\bar{u}_{n}=\langle u^{n}_{s}:s\in I_{n}\rangle$ which witness
$D_{n}$ is $(\lambda_{n},\theta)$-regular, i.e.
$u^{n}_{s}\in[\lambda_{n}]^{<\theta}$ and $(\forall\alpha<\lambda_{n})[\\{s\in
I_{n}:\alpha\in u^{n}_{s}\\}\in D_{n}]$. For $\eta\in I$ and $n<\omega$ let
$w_{\eta,n}$ be $u^{n}_{\eta(n)}$ if $\langle{\rm
otp}(u_{\eta(\ell)}):\ell\leq n\rangle$ is decreasing and $\emptyset$
otherwise. Let $\gamma_{\eta,n}$ be ${\rm otp}(w_{\eta,n})$. Now we can check
that the assumptions of clause (e) hold (because of the choice of $D$).

The main point is that for every $n$

  1. $(*)_{n}$

for some $X_{n}\in D_{n}$, for every $s_{n}\in X_{n}$, for some $X_{n-1}\in
D_{n-1}$ … for some $X_{0}\in D_{0}$ for every $s_{0}\in X_{0}$, if $\langle
s_{0},\dotsc,s_{n}\rangle\trianglelefteq\eta\in I$ then

    1. (a)

$|W_{\eta,0}|>|W_{\eta,1}|>\ldots>|W_{\eta,0}|$

    2. (b)

$|u^{\ell}_{s_{\ell}}|>|u^{\ell+1}_{s_{\ell+1}}|$ for $\ell<n$.

Why $(*)_{n}$? Clause (a) holds by clause (b) and the choice of $W_{\eta,n}$
as $u^{n}_{\eta(n)}$. Clause (b) holds because $u^{\ell+1}_{s_{\ell+1}}$ is of
cardinality $<\theta$ and $\\{s\in
I_{\ell}:|u^{\ell+1}_{s_{\ell+1}}|^{+}\subseteq u^{\ell}_{s}\\}\in D_{\ell}$.

Hence the conclusion of clause (e) holds and we are done as in the proof of
$(c)\Rightarrow(b)$.

Clause $(a)\Rightarrow$ Clause $(e)$:

So assume that clause (a) holds, that is $M_{1},M_{2}$ are
${\mathbb{L}}^{1}_{\theta}$-equivalent and assume $I,D,{\mathscr{E}},\langle
E_{n}:n<\omega\rangle$ and $\bar{w}$ are as in the assumption of clause (e),
and we should prove that its conclusion holds, that is, ${\rm
l.u.p.}_{\mathbb{x}}(M_{1})\cong{\rm l.u.p.}_{\mathbb{x}}(M_{2})$.

For every $\tau_{*}\subseteq\tau$ of cardinality $<\theta$ and $\mu<\theta$,
by 0.19 we know that $M_{1}{\restriction}\tau_{*},M_{2}{\restriction}\tau_{*}$
are ${\mathbb{L}}^{1}_{\leq\mu}$-equivalent, hence for every $\alpha<\mu^{+}$
there is a finite sequence $\langle
N_{\tau_{*},\mu,\alpha,k}:k\leq\mathbb{k}(\tau_{*},\mu,\alpha)\rangle$ such
that (see [Sh:797, 2.1=La8]):

  1. $(*)_{1}$

$(a)\quad N_{\tau_{*},\mu,\alpha,0}=M_{1}{\restriction}\tau_{*}$

  2. $(b)\quad N_{\tau_{*},\mu,\alpha,\mathbb{k}(\tau_{*},\mu,\alpha)}=M_{2}{\restriction}\tau_{*}$

  3. $(c)\quad$ in the game $\Game_{\tau_{*},\mu,\alpha}[N_{\tau_{*},\mu,\alpha,k},N_{\tau_{*},\mu,\alpha,k+1}]$ the ISO player has a

winning strategy for each $k<\mathbb{k}(\tau_{*},\mu,\alpha)$, but we
stipulate

a play to have $\omega$ moves, by deciding they continue to choose the

moves even when one side already wins

  4. $(*)_{2}$

without loss of generality $\|N_{\tau_{*},\mu,\alpha,k}\|\leq\lambda_{0}$ for
$k\in\\{1,\dotsc,\mathbb{k}(\tau_{*},\mu,\alpha)-1\\}$ (even $<\theta$).

[Why? By [Sh:797] quoted in 0.17 and what comes after.]

By monotonicity in $\tau^{*},\mu$ and in $\alpha$ of
$M_{1}{\mathscr{E}}^{1,\tau^{*}}_{{\rm qf}(\tau_{*}),\mu,\alpha}M_{2}$ we can
(without loss of generality ) assume:

  1. $(*)_{3}$

$(a)\quad$ above $\mathbb{k}(\tau_{*},\mu,\alpha)=\mathbb{k}$

  2. $(b)\quad\tau$ have only predicates

  3. $(*)_{4}$

$(a)\quad\langle P_{\alpha}:\alpha<|\tau|\rangle$ list the predicates of
$\tau$, recall that $|\tau|\leq\mu<\lambda_{0}$

  4. $(b)\quad$ for $t\in I$ let $\tau_{t}=\\{P_{\alpha}:\alpha\in w_{t,0}\cap|\tau|\\}$

  5. $(*)_{5}$

let $N_{s,k}:=N_{\tau_{s},|w_{s,0}|,\gamma_{s,0}+1,k}$ for $s\in I$ and
$k\leq\mathbb{k}$.

For $k\leq\mathbb{k}$, let $\bar{f}_{k,n}=\langle
f_{k,n,\alpha}:\alpha<2^{\lambda_{n}}\rangle$ list the members $f$ of
$\prod\limits_{s\in I}N_{s,k}$ such that $E_{n}$ refines ${\rm eq}(f)$, so
$f_{k,n,\alpha}=\langle f_{k,n,\alpha}(\eta):\eta\in I\rangle$ but $\eta\in
I\wedge\nu\in I\wedge\eta E_{n}\nu\Rightarrow
f_{k,n,\alpha}(\eta)=f_{k,n,\alpha}(\nu)$.

Now

  1. $(*)_{6}$

    1. (a)

for $t\in I$ and $k<\mathbb{k}$ let $\Game_{t,k}$ be the game
$\Game_{\tau_{t},|w_{t,0}|,\gamma_{t,0}+1}[N_{t,k},N_{t,k+1}]$

    2. (b)

let stt,k be a winning strategy for the ISO player in $\Game_{t,k}$

    3. (c)

if $t_{1}E_{0}t_{2}$ then $\langle N_{t_{\iota},k}:k\leq\mathbb{k}\rangle$ are
the same for $\iota=1,2$, moreover ($\Game_{t_{1},k}=\Game_{t_{2},k}$ and)
st${}_{t_{1},k}=$ st${}_{t_{2},k}$ for $k<\mathbb{k}$.

[Why clause (c)? Because by $(*)_{5}$,
$N_{s,k},N_{\tau_{s},|w_{s,0}|,\gamma_{s,0}+1,k}$ and $\tau_{s}$ depend on
$w_{s,0}$ only, so $N_{s,k}$ is determined by $(w_{s,0},k)$ hence by clause
(e), $\oplus(f)$ from Definition 1.5 depend just on $(s/E_{0},k)$.]

Now for each $k$ by induction on $n$ we choose $\langle\mathbb{s}_{t,k,n}:t\in
I\rangle$ such that:

  1. $(*)_{7}$

    1. (a)

$\mathbb{s}_{t,k,n}$ is a state of the game $\Game_{t,k}$

    2. (b)

$\langle\mathbb{s}_{t,k,m}:m\leq n\rangle$ is an initial segment of a play of
$\Game_{t,k}$ in which the ISO player uses the strategy stt,k

    3. (c)

if $t_{1}E_{n}t_{2}$ then $\mathbb{s}_{t_{1},k,n}=\mathbb{s}_{t_{2},k,n}$

    4. (d)

$\beta_{\mathbb{s}_{t,k,n}}=\gamma_{t,n}$, see Definition 0.17

    5. (e)

if $t\in I,n=\iota\mod 2$ and $\iota\in\\{0,1\\}$ then
$A^{\iota}_{\mathbb{s}_{t,k,n}}\supseteq\\{f_{k+\iota,m,\alpha}(t):m<n$ and
$\alpha\in w_{t,m}\\}$, see Definition LABEL:w35(yy)

  2. $(*)_{8}$

we can carry the induction on $n$.

[Why? Straightforward.]

  1. $(*)_{9}$

for each $k<\mathbb{k},n<\omega,t\in I$ we define $h_{s,k,n}$, a partial
function from $N_{s,k}$ to $N_{s,k+1}$ by $h_{s,k,n}(a_{1})=a_{2}$ iff for
some $m\leq n,w_{s,m}\neq\emptyset$ and $g_{\mathbb{s}_{t,k,m}}(a_{1})=a_{2}$,
see LABEL:w35(zz).

Now clearly:

  1. $\boxplus_{1}$

for each $t\in I,k<\mathbb{k}$ and $n<\omega,h_{s,k,n}$ is a partial one-to-
one function and even a partial isomorphism from $N_{s,k}$ to $N_{s,k+1}$,
non-empty when $n>0$ and increasing with $n$.

[Why? By the choice of stt,k and $(*)_{7}(a)$.]

  1. $\boxplus_{2}$

let $Y_{k,n}=\\{(f_{1},f_{2}):f_{\ell}\in\prod\limits_{s\in I}{\rm
Dom}(h_{s,k,n})$ for $\ell=1,2$ and $s\in I\Rightarrow
f_{2}(s)=h_{s,k,n}(f_{1}(s))\\}$

  2. $\boxplus_{3}$

$\mathbb{f}_{k,n}=\\{(f_{1}/D,f_{2}/D):(f_{1},f_{2})\in Y_{k,n}\\}$ is a
partial isomorphism from
$M^{I}_{1}{\restriction}\\{f/D:f\in\prod\limits_{s}N_{s,k}$ and $f$ respects
$E_{n}\\}$ to $M^{I}_{2}{\restriction}\\{f/D:f\in\prod\limits_{s}N_{s,k+1}$
and $f$ respects $E_{n}\\}$

  3. $\boxplus_{4}$

$\mathbb{f}_{k,n}\subseteq\mathbb{f}_{k,n+1}$

  4. $\boxplus_{5}$

    1. (a)

if $f_{1}\in\prod\limits_{s}N_{s,k}$ and ${\rm eq}(f_{1})$ is refined by
$E_{n}$ then for some $n_{1}>n$ and $f_{2}\in\prod\limits_{s}N_{s,k+1}$ the
pair $(f_{1}/D,f_{2}/D)$ belongs to $\mathbb{f}_{k,n_{1}}$

    2. (b)

if $f_{2}\in\prod\limits_{s}N_{s,k+1}$ and ${\rm eq}(f_{2})$ is refined by
$E_{n}$ then for some $n_{1}>n$ and $f_{1}\in\prod\limits_{s}N_{s,k}$ the pair
$(f_{1}/D,f_{2}/D)$ belongs to $\mathbb{f}_{k,n_{1}}$.

[Why? By symmetry it suffices to deal with clause (a). For some
$\alpha,f_{1}=f_{k,n,\alpha}$, hence for every $t\in{\rm
Dom}(f_{1}),f_{1}(t)\in A^{1}_{\mathbb{s}_{t,k,n}}$. We use the “delaying
function”, $h_{\mathbb{s}_{t,k,n}}(f_{1}(t))<\omega$ so for some $m$ the set
$\\{t\in I:h_{\mathbb{s}_{t,k,n}}(f_{1}(t))\leq m\\}$ which respects $E_{n}$
belongs to $D$. In particular $\\{s:\gamma_{s,k,n}>m\\}\in D$, the rest should
be clear recalling the regularity of each $D/E_{m}$.]

Letting ${\mathscr{E}}=\\{E_{n}:n<\omega\\}$, putting together

  1. $(*)_{10}$

$\mathbb{f}_{k}=\bigcup\limits_{n}\mathbb{f}_{k,n}$ is an isomorphism from
$(\prod\limits_{s}N_{k,s})_{D}|{\mathscr{E}}$ onto
$(\prod\limits_{s}N_{k+1,s})_{D}|{\mathscr{E}}$.

Hence

  1. $(*)_{11}$

$\mathbb{f}_{\mathbb{k}-1}\circ\ldots\circ\mathbb{f}_{0}$ is an isomorphism
from $(M_{1})^{I}_{D}|{\mathscr{E}}$ onto $(M_{2})^{I}_{D}|{\mathscr{E}}$.

So we are done. ∎

######  Discussion 1.6.

1) So for our $\theta$, we get another characterization of
${\mathbb{L}}^{1}_{\theta}$.

2) We may deal with universal homogeneous $(\theta,\sigma)-{\rm
l.u.p.}\,\mathbb{x}$, at least for $\sigma=\aleph_{0}$, using Definition 0.8.

######  Claim 1.7.

In Theorem 1.5 if $\kappa=\kappa^{<\theta}\geq\|M_{1}\|+\|M_{2}\|$ we can add:

  1. $(b)^{+}$

like clause (b) of 1.5 but $|I|\leq 2^{\kappa}$.

######  Remark 1.8.

Note we do not restrict $\tau=\tau(M_{\ell})$. See proof of $(*)_{9}$ below.

###### Proof..

Clearly $(b)^{+}\Rightarrow(b)$, so it is enough to prove
$(b)\Rightarrow(b)^{+}$; we shall assume
$M_{1},M_{2},\kappa,\mathbb{x}_{n},D,{\mathscr{E}}_{n},{\mathscr{E}}$ are as
in (b) and let $g$ be an isomorphism from $(M_{1})^{I}_{D}/{\mathscr{E}}$ onto
$(M_{2})^{I}_{D}/{\mathscr{E}}$.

Let

  1. $(*)_{1}$

$(a)\quad{\mathscr{E}}^{\prime}_{n}=\\{E:E$ is an equivalence relation on $I$
with $\leq\kappa$ equivalence

classes such that some $E^{\prime}\in{\mathscr{E}}_{n}$ refines $E\\}$

  2. $(b)\quad$ let ${\mathscr{E}}^{\prime}=\cup\\{{\mathscr{E}}^{\prime}_{n}:n\in{\mathbb{N}}\\}$.

Clearly

  1. $(*)_{2}$

$(M_{\ell})^{I}_{D}|{\mathscr{E}}=(M_{\ell})^{I}_{D}|{\mathscr{E}}^{\prime}$
for $\ell=1,2$.

Let $\chi$ be large enough such that
$M_{1},M_{2},\kappa,D,I,{\mathscr{E}},\bar{{\mathscr{E}}}^{\prime}=\langle{\mathscr{E}}^{\prime}_{n}:n\in{\mathbb{N}}\rangle,g$
and $(M_{\ell})^{I}_{D}|{\mathscr{E}}$ for $\ell=1,2$ belong to
${\mathscr{H}}(\chi)$. We can choose
${\mathfrak{B}}\prec_{{\mathbb{L}}_{\kappa^{+},\kappa^{+}}}({\mathscr{H}}(\chi),\in)$
of cardinality $2^{\kappa}$ to which all the members of ${\mathscr{H}}(\chi)$
mentioned above belong and such that $2^{\kappa}+1\subseteq{\mathfrak{B}}$. So
as $\tau=\tau(M_{1})\in{\mathfrak{B}}$ and without loss of generality
$|\tau|\leq 2^{\kappa}$ necessarily $\tau\subseteq{\mathfrak{B}}$.

  1. $(*)_{3}$

let

    1. $(a)$

$I^{*}=I\cap{\mathfrak{B}}$

    2. $(b)$

${\mathscr{E}}^{*}_{n}=\\{E{\restriction}I^{*}:E\in{\mathscr{E}}^{\prime}_{n}\cap{\mathfrak{B}}\\}$

    3. $(c)$

${\mathscr{E}}^{*}=\cup\\{{\mathscr{E}}^{*}_{n}:n\in{\mathbb{N}}\\}$

    4. $(d)$

let $D^{*}$ be any ultrafilter on $I^{*}$ which includes $\\{I\cap I^{*}:I\in
D\cap{\mathfrak{B}}\\}$.

It is enough to check the following points:

  1. $(*)_{4}$

$\mathbb{x}^{*}_{n}:=(I^{*},D^{*},{\mathscr{E}}^{*}_{n})$ is a
$(\theta,\theta)-{\rm l.u.f.t.}$ for every $n\in\omega$.

Why? E.g. note that if $E\in{\mathscr{E}}^{*}_{n}$ then for some
$E^{\prime}\in{\mathscr{E}}^{\prime}_{n}\cap{\mathfrak{B}}$ we have
$E^{\prime}{\restriction}I^{*}=E$ hence $E$ has $\leq\kappa$ equivalence
classes. Now for any such $E^{\prime}$, as $E^{\prime}$ has
$\leq\kappa$-equivalence classes and belongs to ${\mathfrak{B}}$ clearly every
$E^{\prime}$-equivalence class is not disjoint to $I^{*}$ and every
$A\subseteq I^{*}$ respecting $E$ is $A^{\prime}\cap I^{*}$ for some
$A^{\prime}\in{\mathfrak{B}}$ respecting $E^{\prime}$. So
$D/E^{\prime}_{n},D^{*}/E$ are essentially equal, etc., that is, let
$\pi_{n}:{\mathscr{E}}^{*}_{n}\rightarrow{\mathscr{E}}^{\prime}_{n}$ be such
that $E\in{\mathscr{E}}^{*}_{n}\Rightarrow\pi_{n}(E){\restriction}I^{*}=E$ and
let $\pi_{n,E}:\\{A:A\subseteq I^{*}$ respects $E\\}\rightarrow\\{A\subseteq
I:A$ respects $\pi_{n}(E)\\}$ be such that $\pi_{n,E}(A)=B\Rightarrow B\cap
I^{*}=A$; in fact, those functions are uniquely determined.

So clearly $(*)_{4}$ follows by

  1. $(*)_{5}$

$(a)\quad\pi_{n}$ is a one-to-one function from ${\mathscr{E}}^{*}_{n}$ onto
${\mathscr{E}}^{\prime}_{n}\cap{\mathfrak{B}}$

  2. $(b)\quad\pi_{n}$ preserves “$E^{1}$ refines $E^{2}$” and its negation

  3. $(c)\quad{\mathscr{E}}^{*}_{n}$ is ($<\theta)$-directed

  4. $(d)\quad$ if $n=m+1$ then ${\mathscr{E}}^{*}_{m}\subseteq{\mathscr{E}}^{*}_{n}$ and $\pi_{m}\subseteq\pi_{n}$.

Moreover

  1. $(*)_{6}$

$(a)\quad$ if $E\in{\mathscr{E}}^{*}_{n}$, then ${\rm
Dom}(\pi_{n,E})\subseteq{\mathfrak{B}}$ (because
$2^{\kappa}\subseteq{\mathfrak{B}}$ is assumed)

  2. $(b)\quad\pi_{n,E}$ is an isomorphism from the Boolean Algebra ${\rm Dom}(\pi_{n,E})$ onto

$\\{A\subseteq I:A\,\,{\rm respects}\,\,\pi_{n}(E)\\}$ which is canonically
isomorphic to

the Boolean Algebra ${\mathscr{P}}(I/\pi_{n}(E))$ and also to
${\mathscr{P}}(I^{*}/E)$

  3. $(c)\quad D^{*}\cap{\rm Dom}(\pi_{n,E})$ is an ultrafilter which $\pi_{n,E}$ maps onto the

$D\cap{\rm Rang}(\pi_{n,E})$ which is an ultrafilter; those ultrafilters are

$\theta$-complete

  4. $(*)_{7}$

$I^{*}$ has cardinality $\leq 2^{\kappa}$.

[Why? Because ${\mathfrak{B}}$ has cardinality $\leq 2^{\kappa}$.]

  1. $(*)_{8}$

$(M_{\ell})^{I^{*}}_{D^{*}}|{\mathscr{E}}^{*}$ is isomorphic to
$((M_{\ell})^{I}_{D}|{\mathscr{E}}^{\prime}){\restriction}{\mathfrak{B}}$ for
$\ell=1,2$.

[Why? Let $\varkappa$ be the following function:

  1. $(*)_{8.1}$

$(a)\quad{\rm Dom}(\varkappa)=(M_{1})^{I_{*}}|{\mathscr{E}}^{*}$

  2. $(b)\quad$ if $f_{1}\in(M_{1})^{I_{*}}$ and $E\in{\mathscr{E}}^{*}$ refines ${\rm eq}(f_{1})$ then $f_{2}:=\varkappa(f_{1})$ is

the unique function with domain $I$ such that
$(\bigcup\limits_{n}\pi_{n})(E)\in{\mathscr{E}}^{\prime}$

refines ${\rm eq}(f_{2})$ and $f_{2}{\restriction}I^{*}=f_{1}$.

Now easily $\varkappa$ induces an isomorphism as promised in $(*)_{8}$.]

  1. $(*)_{9}$

$((M_{1})^{I}_{D}|{\mathscr{E}}^{\prime}){\restriction}{\mathfrak{B}}$ is
isomorphic to
$(M_{2})^{I}_{D}|{\mathscr{E}}^{\prime}){\restriction}{\mathfrak{B}}$.

[Why? By $(*)_{2}$ and the choices of $g$ (in the beginning) and of
${\mathfrak{B}}$ after $(*)_{2}$ this is obvious when $\tau=\tau(M_{1})$ is
included in ${\mathfrak{B}}$, which is equivalent to $|\tau|\leq 2^{\kappa}$.
By recalling that the ${\rm arity}(\tau)\leq\aleph_{0}$, i.e. every predicate
and function symbol of $\tau$ has finitely many places (see 1.5), without loss
of generality this holds. That is, let $\tau^{\prime}\subseteq\tau$ be such
that for every predicate $P\in\tau$ there is one and only one
$P^{\prime}\in\tau^{\prime}$ such that $\ell\in\\{1,2\\}\Rightarrow
P^{M_{\ell}}=(P^{\prime})^{M_{\ell}}$ and similarly for every function symbol;
clearly it suffices to deal with
$M_{1}{\restriction}\tau^{\prime},M_{2}{\restriction}\tau^{\prime}$ and
$|\tau^{\prime}|\leq 2^{\|M_{1}\|}\leq 2^{\kappa}$.]

Together we are done. ∎

Note that the proof of 1.7 really uses $\kappa=\kappa^{<\theta}$, as otherwise
${\mathscr{E}}^{\prime}_{n}$ is not $(<\theta)$-directed. How much is the
assumption $\kappa=\kappa^{<\theta}$ needed in 1.7? We can say something in
1.9.

######  Claim 1.9.

Assume that $\kappa\geq 2^{\theta}$ but $\kappa^{<\theta}>\kappa$ hence for
some regular $\sigma<\theta$ we have $\kappa^{<\sigma}=\kappa<\kappa^{\sigma}$
and ${\rm cf}(\kappa)=\sigma$ and by [Sol74] we have
$(\forall\mu<\kappa)(\mu^{\theta}<\kappa)$; recall ${\rm
arity}(\tau)=\aleph_{0}$.

1) If $\langle{\mathfrak{B}}_{i}:i\leq\sigma\rangle$ is a
$\subseteq$-increasing continuous sequence of $\tau$-models and $\mathbb{x}$
is a $(\theta,\theta)-{\rm l.u.f.t.}$ then ${\rm
l.u.p.}_{\mathbb{x}}({\mathfrak{B}}_{\sigma})=\cup\\{{\rm
l.u.p.}_{\mathbb{x}}({\mathfrak{B}}_{i}):i<\sigma\\}$ and $i<j\Rightarrow{\rm
l.u.p.}_{\mathbb{x}}({\mathfrak{B}}_{i})\subseteq{\rm
l.u.p.}_{\mathbb{x}}({\mathfrak{B}}_{j})$.

2) If $J$ is a directed partial order of cardinality $\leq\sigma$ $(<\theta)$
and $\mathbb{x}_{s}=(I,D,{\mathscr{E}}_{s})$ is a $(\theta,\theta)-{\rm
l.u.f.t.}$ for $s\in J$ such that
$s<_{J}t\Rightarrow{\mathscr{E}}_{s}\subseteq{\mathscr{E}}_{t}$ and $M$ is a
$\tau$-model then ${\rm l.u.p.}_{\mathbb{x}}({\mathfrak{B}})=\cup\\{{\rm
l.u.p.}_{\mathbb{x}_{s}}({\mathfrak{B}}):s\in J\\}$ and
$s<_{J}t\Rightarrow{\rm
l.u.f.t.}_{\mathbb{x}_{s}}({\mathfrak{B}})\subseteq{\rm
l.u.p.}_{\mathbb{x}_{t}}({\mathfrak{B}})$ under the natural identification.

3) In 1.7, $|I^{*}|\leq\Sigma\\{2^{\partial}:\partial<\kappa\\}$ is enough.

###### Proof..

Straightforward. ∎

##  2\. Special Models

######  Definition 2.1.

Assume $\lambda>\theta$ is strong limit of cofinality $\aleph_{0}$.

We say a model $M$ is $\lambda$-special when there are $\bar{\lambda},\bar{M}$
such that (we also may say $\bar{M}$ is a $\lambda$-special sequence):

  1. $(a)$

$M$ is a model of cardinality $\lambda$ with $|\tau(M)|<\lambda$

  2. $(b)$

    1. $(\alpha)$

$\bar{\lambda}=\langle\lambda_{n}:n\in{\mathbb{N}}\rangle$

    2. $(\beta)$

$\lambda_{n}\leq\lambda_{n+1}$

    3. $(\gamma)$

$\theta\leq\lambda_{n}<\lambda_{n+1}<\lambda=\sum\limits_{k}\lambda_{k}$ and
stipulate $\lambda_{-1}=\theta$

  3. $(c)$

    1. $(\alpha)$

$\bar{M}=\langle M_{n}:n<\omega\rangle$

    2. $(\beta)$

$M_{n}\prec_{{\mathbb{L}}_{\theta,\theta}}M_{n+1}$

    3. $(\gamma)$

$M=\bigcup\limits_{n}M_{n}$

    4. $(\delta)$

$\lambda_{n}=\|M_{n}\|$

  4. (d)

    1. $(\alpha)$

$\bar{D}=\langle D_{n}:n\in{\mathbb{N}}\rangle$ and $\|M_{n}\|\leq\lambda_{n}$

    2. $(\beta)$

$D_{n}\in{\rm ruf}_{\lambda_{n-1},\theta}(\lambda_{n})$

    3. $(\gamma)$

$M^{\lambda_{n}}_{n}/D_{n}\prec_{{\mathbb{L}}_{\theta,\theta}}M_{n+1}$ under
the canonical identification (so hence $2^{\lambda_{n}}\leq\lambda_{n+1}$)

or just (instead clause (d))

  1. (d)′

if $\Gamma$ is an ${\mathbb{L}}_{\theta,\theta}$-type on $M_{n}$ of
cardinality $\leq\lambda_{n}$ with $\leq\lambda_{n}$ free variables then
$\Gamma$ is realized in $M_{n+1}$.

######  Claim 2.2.

1) If for every $n<\omega$ we have $D_{n}$ is a $(\lambda_{n},\theta)$-regular
$\theta$-complete ultra-filter on $I_{n},M_{n+1}=(M_{n})^{I_{n}}/D_{n}$
identifying $M_{n}$ with its image under the canonical embedding into
$M_{n+1}$ so $M_{n}\prec_{{\mathbb{L}}_{\theta,\theta}}M_{n+1}$ and
$\lambda_{n}\geq\|M_{n}\|,\lambda=\sum\limits_{n}\lambda_{n}\geq\theta$ then
$\langle M_{n}:n\in{\mathbb{N}}\rangle$ is a $\lambda$-special sequence, so
$\bigcup\limits_{n}M_{n}$ is a $\lambda$-special model.

2) In Definition 2.1, clause (d) indeed implies clause $(d)^{\prime}$.

3) If $M$ is a $\lambda$-special model and $\tau\subseteq\tau_{M}$, then
$M{\restriction}\tau$ is also a $\lambda$-special model.

###### Proof..

1) If we assume clause (d) in Definition 2.1, just by the definition. If we
assume clause $(d)^{\prime}$ in Definition 2.1, use part (2).

2) It follows by the $(\lambda_{n},\theta)$-regularity of $D_{n}$. ∎

######  Claim 2.3.

1) If $\langle M^{\ell}_{n}:n\in{\mathbb{N}}\rangle$ is a $\lambda$-special
sequence with union $M_{\ell}$ for $\ell=1,2$ and ${\rm
Th}_{{\mathbb{L}}_{\theta,\theta}}(M^{1}_{0})={\rm
Th}_{{\mathbb{L}}_{\theta,\theta}}(M^{2}_{0})$ then $M_{1},M_{2}$ are
isomorphic.

2) Moreover, if $n<\omega$ and $f$ is a partial function from $M^{1}_{n}$ into
$M^{2}_{n}$ which is
$(M^{1}_{n},M^{2}_{n},{\mathbb{L}}_{\theta,\theta})$-elementary (i.e.
$\bar{a}\in{}^{\theta>}({\rm Dom}(f))\Rightarrow f({\rm
tp}_{{\mathbb{L}}_{\theta,\theta}}(\bar{a},\emptyset,M^{1}_{n}))={\rm
tp}_{{\mathbb{L}}_{\theta,\theta}}(f(\bar{a}),\emptyset,M^{2}_{n}))$ then $f$
can be extended to an isomorphism from $M_{1}$ onto $M_{2}$.

3) If we weaken clause $(d)^{\prime}$ of Definition 2.1 by weakening the
conclusion to: for some $k>n,\Gamma$ is realized in $M_{k}$ we get an
equivalent definition.

###### Proof..

1) As in part of the proof of 1.5 only much simpler; the hence and forth
argument.

2) Same proof.

3) Use suitable subsequences. ∎

Note that comparing definition 2.1 with the first order parallel, in Claim
2.3(1), a priori it is not given that ${\rm
Th}_{{\mathbb{L}}_{\theta,\theta}}(M_{1})={\rm
Th}_{{\mathbb{L}}_{\theta,\theta}}(M_{2})$ suffices. Also 2.3 does not say
that ${\rm Th}_{{\mathbb{L}}^{1}_{\theta}}(M)$ and $\lambda$ determines $M$ up
to isomorphism because we demand that $M^{1}_{0},M^{2}_{0}$ are
${\mathbb{L}}^{1}_{\theta}$-equivalent. However:

######  Claim 2.4.

1) Assume $\lambda>\theta$ is strong limit of cofinality $\aleph_{0}$ and $T$
is a complete theory in ${\mathbb{L}}^{1}_{\theta}(\tau_{T}),|T|<\lambda$
equivalently $|\tau_{T}|<\lambda$. Then $T$ has exactly one $\lambda$-special
model (up to isomorphism).

2) Similarly when $\lambda>\theta,{\rm cf}(\lambda)=\aleph_{0}$ only; so here
we have to use clause $(d)^{\prime}$ in Definition 2.1.

###### Proof..

1) Assume $N_{1},N_{2}$ are special models of $T$ of cardinality $\lambda$. By
Definition 2.1 for $\ell=1,2$ there is a triple
$(\bar{\lambda}_{\ell},\bar{M}_{\ell},\bar{D}_{\ell})$ witnessing $N_{\ell}$
is $\lambda$-special as there.

As
$M_{\ell,0}\prec_{{\mathbb{L}}_{\theta,\theta}}M_{\ell,n}\prec_{{\mathbb{L}}_{\theta,\theta}}M_{\ell,n+1}\prec_{{\mathbb{L}}^{1}_{\theta}}\bigcup\limits_{m}M_{\ell,m}=N_{\ell}$
for $n\in{\mathbb{N}}$, by 0.7 and 1.2, we know
$M_{\ell,0}\equiv_{{\mathbb{L}}^{1}_{\kappa}}N_{\ell}$, so we can conclude
that $M_{1,0}\equiv_{{\mathbb{L}}^{1}_{\kappa}}M_{2,0}$ and both are models of
$T$.

By 1.5 there is a sequence
$\langle(\lambda_{n},D_{n}):n\in{\mathbb{N}}\rangle$ with
$\Sigma\lambda_{n}=\lambda,2^{\lambda_{n}}\leq\lambda_{n+1}$ and $D_{n}$ a
$(\lambda_{n},\theta)$-regular ultrafilter on $\lambda_{n}$ such that
$M^{\prime}_{1}\cong M^{\prime}_{2}$ when:

  1. $(*)$

$M^{\prime}_{\ell,0}=M_{\ell,0},M^{\prime}_{\ell,n+1}=(M^{\prime}_{\ell,n})^{\lambda_{n}}/D_{n}$
and $M^{\prime}_{\ell}=\bigcup\limits_{n}M^{\prime}_{\ell,n}$.

So $M^{\prime}_{1}\cong M^{\prime}_{2}$ by 1.5 and $N_{1}\cong M^{\prime}_{1}$
by 2.3(1) and $N_{2}\cong M^{\prime}_{2}$ similarly. Together $N_{1}\cong
N_{2}$ is promised.

2) The proof is similar to part of the proof of 1.5, i.e. by the hence and
forth argument. ∎

Now we can generalize Robinson lemma (hence gives an alternative proof of the
interpolation theorem, recall though that in [Sh:797] we do not assume the
cardinal $\theta$ is compact).

######  Claim 2.5.

1) Assume $\tau_{1}\cap\tau_{2}=\tau_{0},T_{\ell}$ is a complete theory in
${\mathbb{L}}^{1}_{\theta}(\tau_{\ell})$ for $\ell=1,2$ and $T_{0}=T_{1}\cap
T_{2}$. Then $T_{1}\cup T_{1}$ has a model.

2) We can allow in (1) the vocabularies to have more than one sort.

3) The logic ${\mathbb{L}}^{1}_{\theta}$ satisfies the interpolation theorem.

4) ${\mathbb{L}}^{1}_{\theta}$ has disjoint amalgamation, i.e. if
$M_{0}\prec_{{\mathbb{L}}^{1}_{\theta}}M_{\ell}$ for $\ell=1,2$ that is
$(M_{0},c)_{c\in M_{0}},(M_{\ell},c)_{c\in M_{0}}$ has the same
${\mathbb{L}}^{1}_{\theta}$-theory and $|M_{1}|\cap|M_{2}|=|M_{0}|$, then
there is $M_{3}$ such that $M_{\ell}\prec_{{\mathbb{L}}^{1}_{\theta}}M_{3}$
for $\ell=0,1,2$ (hence orbital type are well defined).

5) ${\mathbb{L}}^{1}_{\theta}$ has111But the disjoint version may fail, e.g.
if we have individual constants. the JEP.

###### Proof..

1) Let $\lambda>|\tau_{1}|+|\tau_{2}|+\theta$ be a strong limit cardinal of
cofinality $\aleph_{0}$. For $\ell=1,2$ there is a $\lambda$-special model
$M_{\ell}$ of $T_{\ell}$ by 2.2(1). Now
$N_{\ell}=M_{\ell}{\restriction}\tau_{0}$ is a $\lambda$-special model of $T$,
see the Definition 2.1.

By 2.4(1), $N_{1}\cong N_{2}$ so without loss of generality $N_{1}=N_{2}$, and
let $M$ be the expansion of $N_{1}=N_{2}$ by the predicates and functions of
$M_{1}$ and of $M_{2}$. Clearly $M$ is a model of $T_{1}\cup T_{2}$.

2) Similarly.

3) Follows as ${\mathbb{L}}^{1}_{\theta}$ being
$\subseteq{\mathbb{L}}_{\theta,\theta}$ satisfies $\theta$-compactness and
part (1).

4) Follows by (1), that is, let $\mathbb{x}$ be as in 1.5(c) for
$M_{1},M_{2}$. So for every $C\subseteq M_{0}$ of cardinality $<\theta$,
letting $M_{C,\ell}=(M_{\ell},c)_{c\in C}$ we have $N_{C,1}\cong N_{C,2}\cong
N_{C,0}$ where $N_{C,\ell}={\rm l.u.p.}_{\mathbb{x}}(M_{C,\ell})$. Hence
$N_{C,0}\prec_{{\mathbb{L}}_{\theta,\theta}}N_{C,\ell}$ for $\ell=1,2$ and we
use “${\mathbb{L}}_{\theta,\theta}$ has disjoint amalgamation”.

5) Follows by 1.5. ∎

######  Remark 2.6.

This proof implies the generalization of preservation theorems, see [CK73].

Recall that Eherenfuecht-Mostowski [EM56] aim was: every first order theory
$T$ with infinite models has models with many automorphisms. This fails for
${\mathbb{L}}_{\theta,\theta}$ and even ${\mathbb{L}}_{\aleph_{1},\aleph_{1}}$
as we can express “$<$ is a well ordering”. What about
${\mathbb{L}}^{1}_{\theta}$?

######  Claim 2.7.

Assume ($\lambda,T$ are as above in 2.4 and) $M$ is a special model of $T$ of
cardinality $\lambda$. Then $M$ has $2^{\lambda}$ automorphisms.

###### Proof..

Let $\langle M_{n}:n<\omega\rangle$ witness $M$ is special. The result follows
by the proof of 2.3(2) noting that

  1. $(*)$

if $f_{n}$ is an
$(M_{n},M_{n},{\mathbb{L}}_{\theta,\theta}(\tau_{\mu}))$-elementary mapping
then there are $a_{2}\in{}^{\lambda}(M_{n+1})$ and
$f_{\alpha},a_{2,\alpha}\in(M_{n+1})$ for $\alpha<\lambda_{n}$ such that

    1. $(a)$

$a_{2,\alpha}\neq a_{2,\beta}$ for $\alpha<\beta<\lambda_{n}$

    2. $(b)$

for $f_{\alpha}$ is an
$(M^{1}_{n+1},M^{2}_{n+1},{\mathbb{L}}_{\theta,\theta}(\tau_{M}))$-elementary
mapping

    3. $(c)$

$f_{\alpha}\supseteq f$ and maps $a$ to $a_{\alpha}$.

Why this is possible? Choose $a^{\prime}\in M_{n+2}\backslash M_{n+1}$ and
choose $a_{\alpha}\in M_{n+1}\backslash\\{a_{\beta}:\beta<\alpha\\}$ by
induction on $\alpha<\lambda_{n}$ realizing ${\rm
tp}_{{\mathbb{L}}_{\theta,\theta}(\tau_{T})}(a^{\prime},M_{n},M_{n+2})$.

Lastly, let $f_{\alpha}=f\cup\\{(a_{0},g(a_{\alpha}))\\}$.

Why this is enough? Should be clear. ∎

## References

  * [CK73] Chen C. Chang and H. Jerome Keisler, _Model Theory_ , Studies in Logic and the Foundation of Math., vol. 73, North–Holland Publishing Co., Amsterdam, 1973.
  * [EM56] Andrzej Ehrenfeucht and Andrzej Mostowski, _Models of axiomatic theories admitting automorphisms_ , Fundamenta Mathematicae 43 (1956), 50–68.
  * [Gai74] Haim Gaifman, _Elementary embeddings of models of set-theory and certain subtheories_ , Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), Amer. Math. Soc., Providence R.I., 1974, pp. 33–101.
  * [Kei61] H. Jerome Keisler, _Ultraproducts and elementary classes_ , Nederl. Akad. Wetensch. Proc. Ser. A 64 (1961), 477–495.
  * [Kei63] by same author, _Limit ultrapowers_ , Transactions of the American Mathematical Society 107 (1963), 382–408.
  * [Koc61] Simon Kochen, _Ultraproducts in the theory of models_ , Ann. of Math. (2) 74 (1961), 221–261.
  * [Sol74] Robert M. Solovay, _Strongly compact cardinals and the GCH_ , Proceedings of the Tarski Symposium, Berkeley 1971, Proceedings of Symposia in Pure Mathematics, vol. 25, A.M.S., 1974, pp. 365–372.
  * [Sh:13] Saharon Shelah, _Every two elementarily equivalent models have isomorphic ultrapowers_ , Israel Journal of Mathematics 10 (1971), 224–233.
  * [HoSh:109] Wilfrid Hodges and Saharon Shelah, _Infinite games and reduced products_ , Annals of Mathematical Logic 20 (1981), 77–108.
  * [Sh:797] Saharon Shelah, _Nice infinitary logics_ , Journal of the American Mathematical Society 25 (2012), 395–427, arxiv:1005.2806.
  * [Sh:1019] by same author, _Model theory for a compact cardinal_ , Israel Journal of Mathematics, arxiv:new.