数理解析研究所講究録 1213 巻 2001 年 39-49

39
NONSTANDARD UNIVERSE
MASAHIKO MURAKAMI
DEPARTMENT OF MATHEMATICSHOSEI UNIVERSITY

ABSTRACT. The nonstandard universes are frameworks of nonstandard analysis. We find sheaf representation for anonstandard universe, in Theorem
3.7.

1. NONSTANDARD UNIVERSE
Definitions 1.1 (superstructure, base set). Given aset X, we define the iterated
power set $V_{n}(X)$ over X recursively by$V_{0}(X)=X$ ,and$V_{n+1}(X)=Vn(X)\cup\varphi(V_{n}(X))$.

The superstructure $V(X)$ is the union. The set $X$ is said to be abase
set ifand each element of $X$ is disjoint from $V(X)$ .
$\bigcup_{n<\omega}V_{n}(X)$$\emptyset\not\in X$

Definition 1.2 (nonstandard universe). Anonstandard universe is atriple
such that:$\langle V(X), V(\mathrm{Y}), \star\rangle$

are infinite base sets.
(2) (Transfer Principle) The symbol(1) $X$ and$\mathrm{Y}$$V(X)\models\varphi(a_{1}, \ldots, a_{n})$.
(4) For every infinite subset ofis amap from $V(X)$ into

if and only ifholds for any bounded formula(3)$\star$$V(\mathrm{Y})$such that$V(\mathrm{Y})\models\varphi(^{\star}a_{1}, \ldots, \star a_{n})$$\varphi(x_{1}, \ldots, x_{n})$and$a_{1}$,$\ldots$,$a_{n}\in \mathrm{V}(\mathrm{X})$.

$\star X=\mathrm{Y}$$A$

of $X$ , $\{^{\star}a|a\in A\}$ is aproper subset of$\star A$.

Definitions 1.3 (standard, internal). For $a\in V(^{\star}X)$ , we call a standard if there
is an $x\in V(X)$ such that $a=x\star$ .
For $a\in V(^{\star}X)$ , we call intemal if there is an $x\in V(X)$ such that
. We$\starV(X)$denote by
the set of all internal elements in.
From now on, we denote anonstandard universe by single $\star V(X)$ .
$a\in\star x$$a$$V(^{\star}X)$

Definitions 1.4 (norm, radius). The nor$m$ (of standardness) of an internal element is acardinal defined by$a$$\mathrm{n}\mathrm{o}\mathrm{s}(a)=\min\{|x||a\in x\}\star$.

The radius of $\star V(X)$ is acardinal defined byrad(\star V(X))
Date: November 14, 2000$= \min\{\kappa|\forall y\in \mathrm{V}(\mathrm{X})\mathrm{n}\mathrm{o}\mathrm{s}(y)<\kappa\}$.

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MASAHIKO MURAKAMI

Definition 1.5 (covering number). Letpoetter at is defined by$a$be an internal element. The local ultra-$a${$V(X)[a]=$For asubset$E\subseteq\star V(X)$$(^{\star}w)(a)|w\in V(X)$and, we denote{$V(X)[E]=\cup$ $V(X)[s]|s$ is afinite subset of $E$

The covering number of}.

$a\in\star(\mathrm{d}\mathrm{o}\mathrm{m}(w))$$\star V(X)$$\mathrm{c}\mathrm{o}\mathrm{v}(\eta\nearrow(X))=\min${}.

is defined by$|E||E\subseteq \mathrm{V}(X)$and $V(X)[E]=V*(X)$ }.

2. LOCALLY ATOMIC COMpLETE ALGEBRA
beDefinition 2.1 (regular complete subalgebra). Let
is said be aregular complete subalgebra ofBoolean algebra. Asubset
is acomplete subalgebra of and the inclusion map is also complete.

a$\langle \mathfrak{B}, \wedge, \vee, \neg, 0_{\mathfrak{B}}, 1\mathfrak{B}\rangle$$C\subseteq \mathfrak{B}$$C$$\mathfrak{B}$if$\mathfrak{B}$Notation, Let$\mathfrak{B}$be aBoolean algebra. For asubset S$\mathrm{S}^{0}=${C$\in \mathrm{S}$$\subseteq\varphi(\mathfrak{B})$, we denote|C is aregular complete subalgebra of }.
$\mathfrak{B}$Definition 2.2 (LCA). Alocally complete algebra (LCA) is aset Aof subsets ofaBoolean algebra satisfying the conditions below.
$\mathfrak{B}$(1).
(2) If ,then(3) Ifthenand, there is a(4) For every$\cup\Lambda=\mathfrak{B}$$S_{1}$$S_{1}\cup S_{2}\in \mathrm{A}$$S_{2}\in \mathrm{A}$$T\in \mathrm{A}$$T\subseteq S$$S\in \mathrm{A}$.

$S\in\Lambda$.

$C\in\Lambda^{0}$containing$S$.

the Boolean algebra
For an LCA , we denote byalgebrathe base Boolean algebra of A.
$\Lambda$$\mathfrak{B}(\Lambda)$$\cup\Lambda$. We call the Boolean$\mathfrak{B}(\Lambda)$

Definition 2.3 (LACA). An LCA Ais alocally atomic complete algebra (LACA)
by Atom(C).
is atomic. We denote the set of atoms ofif every$C\in\Lambda^{0}$$C\in\Lambda^{0}$

Definition 2.4 (homomorphism). We introduce notation “‘ @={ “ $S|$ S\in @}.
is apseudOLCAs. ABoolean homomorphism
Let A. We denotes apseud0-homomorphism byhomomorphism of LCAs if “‘$f::\Lambdaarrow---$ .
LCAs is a(complete) homOApseudO-homomorphism. An embedding or monomorphism
morphism if $h$ “ $S=h(S)$ for all
an injective homomorphism.
$R$

$R$

$f:\mathfrak{B}(\Lambda)arrow \mathfrak{B}(_{-}^{-}-)$$\mathrm{a}\mathrm{n}\mathrm{d}---\mathrm{b}\mathrm{e}$$f$$\mathrm{A}\subseteq \mathrm{E}$$h::\mathrm{A}arrow---\mathrm{o}\mathrm{f}$$S\in \mathrm{A}$$j::\mathrm{A}arrow---\mathrm{i}\mathrm{s}$Definition 2.5 (subLCA). AsubLCA of an LCA Ais anonempty subset of A
which is itself an LCA and the inclusion map is an embedding.
Definition 2.6 (generator). Let Abe an LCA. Asubset
Aor generates Aif Ais the only subLCA of Acontaining .

$9\subseteq\Lambda^{0}$

$S$

$g$

is agenerator of

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NONSTANDARD UNIVERSE

Definitions 2.7 (radius, covering number, diameter). The radius of an LACA A
is acardinal defined byrad(A)$= \min\{\kappa|\forall C\in \mathrm{A}^{0}|\mathrm{A}\mathrm{t}\mathrm{o}\mathrm{m}(C)|<\kappa\}$

The covering number of an LCA Ais acardinal defined by$\mathrm{c}\mathrm{o}\mathrm{v}(\Lambda)=\min$

{

$|9|$

$|S$

is agenerator of$\Lambda$}.

The diameter of an LACA Ais acardinal defined bydiam(A)$= \min\{\sum_{c\in 9}|\mathrm{A}\mathrm{t}\mathrm{o}\mathrm{m}(C\ovalbox{\tt\small REJECT}$$|9$iiss aa ggeenneerraattoorr ooff$\Lambda\}$.

Definition 2.8 (direct product). Let I be an index set. The direct productthe LCA is defined by:$\Lambda^{[I]}$of$\Lambda$$\Lambda^{[I]}=\{S\subseteq \mathfrak{B}(\Lambda)^{I}|g\in S\cup \mathrm{r}\mathrm{n}\mathrm{g}g\in\Lambda\}$. Thenwith the pointwise Boolean operations onby the canonical embedding
an LCA. The LCA Ais embedded into$\mathfrak{B}(\Lambda^{[I]})=\cup\Lambda^{[I]}\subseteq \mathfrak{B}(\Lambda)^{I}$$\Lambda^{[I]}$$\Lambda^{[I]}$is$b\mapsto I\cross\{b\}$.

Definitions 2.9 (embedding system, direct limit). The embedding system of
LCAs is afamily of embeddings$\mathcal{E}=\{j_{d}^{d’}::\Lambda_{d}arrow\Lambda_{d’}\}_{d\leq d’,d,d’\in D}$

satisfying
direct limit oflimit offor all $d\leq d’\leq d’$ , where $D$ is an upper direct set. The
is the direct
is $\cup\{jd"‘\Lambda d|d\in D\}$ where
as Boolean algebras.

$j_{d}^{d’},\circ j_{d}^{d’}=j_{d}^{d’}$$\{j_{d}^{d’} :$\epsilon$$\{jd:\mathfrak{B}(\Lambda_{d})arrow \mathfrak{B}\}_{d\in D}$\mathfrak{B}(\Lambda_{d})arrow \mathfrak{B}(\Lambda_{d’})\}_{d\leq dd,d’\in D}$”

Definition 2.10 (ultrafilter). Let Abe an LCA. Asubset ofof an $LCA\Lambda$ if it is an ultrafilter of the base Boolean algebra$\mathrm{u}$is an ultrafilter$\mathfrak{B}(\Lambda)$$\mathfrak{B}(\Lambda)$.

3. ULTRALIMIT
Definition 3.1 (LACA-valued model). Let Abe an LACA and let $M$ be amodel-valued universe of $M$ is defined byfor alanguage . The$L$$\mathfrak{B}(\Lambda)$’$M^{\langle\langle\Lambda\rangle\rangle}=\{u:Marrow \mathfrak{B}(\Lambda)|u(x)\wedge u(y)=0\mathrm{f}\mathrm{o}\mathrm{r}x\neq y\mathrm{r}\mathrm{n}\mathrm{g}u\in\Lambda,\mathrm{r}\mathrm{n}\mathrm{g}u=1$

For$u\in M^{\langle\langle\Lambda\rangle\rangle}$, the suppor$n$ of

$u$

$\}$

.

is asubset of $M$ defined by$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}u=\{x\in M|u(x)\neq 0\}$To each function $F$ of $L(M)$ and eachby:$u_{1}$, ... ,.

$u_{n}\in M^{\langle(\Lambda\rangle\rangle}$, we assign a$\check{F}(u_{1}, \ldots, u_{n})\in M^{\langle\langle\Lambda\rangle\rangle}$$\check{F}(u_{1}, \ldots, u_{n})(y)=\vee\{\bigwedge_{i=1}^{n}u_{i}(x_{i})|M\models y=F(x_{1}, \ldots,x_{n})\}$

for $y\in M$ .

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MASAHIKO MURAKAMI

We regard aconstant of $L(M)$ as afunction without any variables. Note that
if $n=0$ . To each sentence of
we assign atruth value[
by following recursive rules:$\bigwedge_{i=1}^{n}u_{i}(x_{i})=1$$\varphi \mathrm{J}$$L(M^{\langle\langle\Lambda\rangle\rangle})$$\varphi$$\in\overline{\mathfrak{B}(\Lambda})$$[u=v\mathrm{J}$[$=\vee\{u(x)\wedge v(x)|x\in M\}$ ,$\mathrm{R}(u_{1}, \ldots, u_{m})\mathrm{J}=\vee\{_{i=1}^{m}\wedge u:(x_{i})|\mathfrak{M}$$[\neg\varphi \mathrm{I}=\neg[\varphi \mathrm{J}$$\models \mathrm{v}(\mathrm{x})\ldots$

is any predicate in

$L$

) ,
$\}$

,

$[\exists x\varphi(x)\mathrm{J}=\vee\{[\varphi(u)\mathrm{J}|u\in M^{(\langle\Lambda\rangle\rangle}\}$$\mathrm{R}$$x_{m}$,$[\varphi_{1}\vee\varphi_{2}\mathrm{I}=[\varphi_{1}\mathrm{J}\vee[\varphi_{2}\mathrm{J}$

where

,

,

.

Definition 3.2 (LAC -valued superstructure). Let Abe an LACA. The -value
superstructure of $V(X)$ is defined by$\mathrm{A}$$\mathrm{A}$$\hat{V}(X)^{((\Lambda\rangle\rangle}=\{u\in V(X)^{((\Lambda\rangle\rangle}|\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}u\in V(X)\}$

While the truth values range over$\mathfrak{B}(\Lambda)$.

on this definition, we shall see$[\varphi \mathrm{I}_{\Lambda}\in$.

Theorem 3.1. Let,,$u_{1}$$\overline{\mathfrak{B}(\Lambda)}$$d$$\varphi(x_{1},$$x_{\mathrm{r}})$\ldots ,be a formula of L with only,

$x_{1}$\ldots ,$x_{f}$

free. For$u_{\mathrm{r}}\in M^{((\Lambda\rangle\rangle}$\ldots ,$(^{*})$$\mathbb{I}\varphi(u_{1}, \ldots, u_{r})\mathrm{I}_{\Lambda}=\vee$$\{_{i=1}f\wedge u:(x:)|\mathfrak{M}$$\models\varphi(x_{1}, \ldots, x_{f})\}$.

either $” x_{1}=x_{2}$ ”or ,holds by definition. If
holds for anthen, by simple calculus of Boolean algebra,
atomic formula
holds for. Thus, by induction,holds for atomic. Suppose
holds
for ,
and . Since there is an atomic
containing all the ranges of, , and every range of :is apartition of unity except for 0,

Proof. For$\mathrm{R}$$\varphi$$(^{*})$$(^{*})$$\varphi(x)$$(^{*})$$\varphi(\mathrm{F}(x_{1}, \ldots, x_{n}))$$\varphi_{1}$$\varphi$$u_{1}$$(^{*})$$C\in\Lambda^{0}$$\varphi_{2}$$u$$u_{f}$$\ldots$$(^{*})$$\varphi$$[\neg\varphi \mathrm{J}_{\Lambda}=\vee\{_{\dot{l}=1}\wedge^{r}u:(x_{\dot{1}})|\mathfrak{M}$.

$\models\neg\varphi(x_{1}, \ldots, x_{\mathrm{r}})\}$

It is easy to see:$[\varphi_{1}\vee\varphi_{2}\mathrm{J}_{\Lambda}=\vee$Since [
Therefore$\{_{\dot{l}=}\bigwedge_{1}^{f}u:(x:)|\mathfrak{M}$$\models\varphi_{1}(x_{1}, \ldots, x_{f})\vee\varphi_{2}(x_{1}, \ldots, x_{f})\}$$\varphi(u)\mathrm{J}_{\Lambda}=\mathrm{V}_{x\in M}(u(x)\wedge[\varphi(\check{x})\mathrm{I}_{\Lambda})$$(^{*})$

holds for$\exists x\varphi(x)$, we have.

$[\exists x\varphi(x)\mathrm{J}_{\Lambda}=\mathrm{V}_{x\in M}[\varphi(\check{x})\mathrm{I}_{\Lambda}\cdot$.

U3

Similarly, we shall obtain the superstructure version.
Corollary 3.2. Let,$u_{1}$$x_{f})$\ldots ,$\varphi(x_{1},$beformu la of$L_{\in}with$only$x_{1}$,\ldots ,$x_{f}$$u_{f}\in\hat{V}(X)^{((\Lambda\rangle\rangle}f$\ldots ,$[\varphi(u_{1},$\ldots ,$u_{f}) \mathrm{J}_{\Lambda}=\vee\{\bigwedge_{\dot{l}=1}^{f}u_{i}(x:)|V(X)\models\varphi(x_{1},$\ldots ,$\mathrm{x}\mathrm{r})\}$.

free. For43

NONSTANDARD UNIVERSE

By the theorem and the corollary above, we have afundamental property-valued model. We have just introduced[[. We say that-valued superstructure
andand that asentence ofif [holds inasentence of. Theorem 3.1 and Corollary 3.2
if [holds in
. For $E\subseteq M$ , we may
follow that we consider the values $u(x)$ only for
to $M$ .
by extending the domain of
regardas asubset of$u=v\mathrm{J}$$\wedge[\varphi(u)\mathrm{J}$$\leq$$\mathfrak{B}(\Lambda)$$\varphi(v)\mathrm{I}$$\mathfrak{M}^{\langle\langle\Lambda\rangle\rangle}=\langle M^{\langle\langle\Lambda\rangle\rangle},\check{R},\check{F},\check{c}\rangle$$\hat{V}(X)^{\langle(\Lambda)\rangle}$$\mathfrak{B}(\Lambda)$$\mathfrak{M}^{\langle(\Lambda)\rangle}$$L(M^{(\langle\Lambda\rangle\rangle})$$\hat{V}(X)^{(\langle\Lambda\}\rangle}$$L_{\in}(\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle})$$\psi$$\varphi \mathrm{J}_{\Lambda}=1$$\varphi$$\psi \mathrm{J}_{\Lambda}=1$$x\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}u$$u\in E^{\langle(\Lambda)\rangle}$$M^{\langle\langle\Lambda\rangle)}$$E^{\langle\langle\Lambda\rangle\rangle}$

This means that we define for$u\in E^{(\langle\Lambda\rangle\rangle}$

if x$u(x)=0$$\not\in E$.

In the superstructure version, if E is aset relative to $V(X)$ then we may assume$E^{\langle\langle\Lambda)\rangle}=${u$\in\hat{V}(X)^{\langle\langle\Lambda)\rangle}|u\in\check{E}$

Theorem 3.3 (Maximum principle). Let
such that
free. Then there is$u\in M^{\langle\langle\Lambda\rangle\rangle}$$x$$\varphi(x)$holds in$\hat{V}(X)^{\langle(\Lambda\rangle\rangle}$

be a formula of}.

$L(M^{((\Lambda\rangle\rangle})$$[\varphi(u)\mathrm{I}_{\Lambda}--[\exists x\varphi(x)\mathrm{J}_{\Lambda}$with only.

be awell-0rdering for $M$ . By theorem 3.1, there is, we have. Putting. Sinceis apairwise disjoint family, we can pick
. Sincefor any. Then [, we have

Proof Let$\{a_{\zeta}\}_{\zeta<\alpha}$

containing$\{[\varphi(\check{x})\mathrm{J}|x\in \mathrm{A}\#\}$$C\in\Lambda^{0}$$b_{\zeta}=[\varphi(\check{a}\zeta)\mathrm{I}\wedge\neg _{\xi<\zeta}[\varphi(\check{a}\xi)\mathrm{I}$$u\in M^{(C\rangle}$$\{b_{\zeta}\}_{\zeta<\alpha}\subseteq C$

with[$\{b_{\zeta}\}_{\zeta<\alpha}$

Corollary 3.4. Let
be an element of$\varphi(x)$$\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$

be a formula of. Then there is$x$free and let$v$$[u\in v\wedge\varphi(u)\mathrm{I}_{\Lambda}=$.
$n$$\check{x}\in v\mathrm{I}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v\subseteq V_{n+1}(X)$,for $x\not\in V_{n}(X)$ .

$=\vee\{v(y)|x\in y\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v\}=0$Therefore we can choose$u$whose support is asubset of $V_{n}(X)$ .

Definition 3.3 (ultralimit). We denote byby the equivalence relationx
The ultralimit

with only
such that$L_{\in}(\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle})$$u\in\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$

Proof Since there is such that[$\square$$[\varphi(u)\mathrm{J}\geq[\exists x\varphi(x)\mathrm{I}\cdot$$\exists x\varphi(x)\mathrm{I}=_{\zeta<\alpha}\varphi(\check{a}_{\zeta})=_{\zeta<\alpha}b_{\zeta}$$[\exists x\in v\varphi(x)\mathrm{I}_{\Lambda}$$\zeta<\alpha$$\varphi(u)\mathrm{I}\geq u(a_{\zeta})\wedge[\varphi(\check{a}_{\zeta})\mathbb{I}\geq b_{\zeta}$$u(a_{\zeta})\geq b_{\zeta}$$\mathfrak{M}^{\langle\langle\Lambda\rangle\rangle}/\mathrm{u}$of$\sim \mathrm{u}$$\mathfrak{M}$y$\equiv$

modulo$\mathrm{u}$[x$u/\mathrm{u}$$\square$the equivalence class of$=y\mathrm{I}_{\Lambda}\in \mathrm{u}$.

of Ais defined by:$M^{\langle\langle A\rangle\rangle}/\mathfrak{U}=\{u/\mathrm{U}|u\in M^{\langle\langle\Lambda\rangle\rangle}\}$$\check{F}/\mathrm{U}(u_{1}/\mathrm{u}, \ldots, u_{n}/\mathfrak{U})=(\check{F}(u_{1}, \ldots, u_{n}))/\mathrm{u}$$\mathfrak{M}^{\langle\langle\Lambda\rangle\rangle}/\mathrm{u}\models \mathrm{R}(u_{1}/\mathrm{u}, \ldots, u_{m}/\mathrm{U})$iff$u/\mathrm{u}$$u\in\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$$\sim \mathrm{u}$y$\equiv$[x.

.

$[\mathrm{R}(u_{1}, \ldots, u_{m})\mathrm{J}\in \mathrm{u}$Definition 3.4 (bounded ultralimit). We denote byby the equivalence relationx$u\in M^{\langle(\Lambda\rangle\rangle}$$=y\mathrm{J}_{\Lambda}\in \mathfrak{U}$.

.

the equivalence class of

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MASAHIKO MURAKAMI

The bounded ultralimit$\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}/\mathrm{u}$of $V(X)$ modulo$\mathrm{u}$of Ais defined by:$\hat{V}(X)^{\langle(\Lambda\rangle\rangle}/\mathrm{u}=\{u/\mathrm{U}|u\in\hat{V}(X)^{\langle(\Lambda\rangle\rangle}\}$$\hat{V}(X)^{\langle(\Lambda\rangle\rangle}/\mathfrak{U}\models u/\mathrm{u}\in v/\mathfrak{U}$Theorem 3.5 (,with only ,$\mathrm{L}\mathrm{o}\acute{\mathrm{s}}$$x_{1}$$\ldots$$x_{f}$

iff$[u\in v\mathrm{J}$Principle of Ultralimits). Let,
free. For , ,$\in \mathrm{u}$,.
be a formula of$\varphi(x_{1}, \ldots, x_{f})$$L$$u_{f}\in M^{\langle\langle\Lambda\rangle\rangle}$$u_{1}$$\ldots$iff$\mathfrak{M}^{\langle(\Lambda\rangle\rangle}/\mathrm{U}\models\varphi(u_{1}/\mathrm{U}, \ldots,u_{r}/\mathrm{U})$$[\varphi(u_{1}, \ldots, u_{f})\mathrm{J}\in \mathrm{u}$.

Proof. The proof proceeds by induction on the complexity of formulae. The only
nontrivial step is the case where is of the form
. Suppose.
$\exists x\psi(x)$$\varphi$$[\exists x\psi(x)\mathrm{J}\in \mathrm{u}$By the maximal principle (Theorem 3.3), there is satisfying
Thenby the induction assumption. We have thus
. Conversely, suppose. Then there is some such that
. By the induction assumption,$[\psi(u)\mathrm{I}=[\exists x\psi(x)\mathrm{I}\cdot$$u$$\mathfrak{M}^{\langle(\Lambda\rangle\rangle}\models\psi(u/\mathrm{U})$$\mathfrak{M}^{\langle\langle\Lambda\rangle\rangle}\models$$\exists x\psi(x)$$\mathfrak{M}^{\langle\langle\Lambda\rangle)}\models\exists x\psi(x)$$\mathfrak{M}^{((\Lambda\rangle\rangle}\models\psi(u/\mathrm{U})$$u$$[\exists x\psi(x)\mathrm{J}\geq[\psi(u/\mathrm{U})\mathrm{J}\in \mathrm{u}.$Corollary 3.6 ( -Mostowski Principle of Bounded Ultralimits).
Let
be a, free. For
only ,
formula of,$\square$$\mathrm{L}\mathrm{o}\acute{\mathrm{s}}$$\varphi(x_{1}, \ldots,x_{r})$$\iota_{\in}with$$\Delta_{0}$$x_{1}$$\ldots$$x_{f}$$u_{1}$,$\ldots$,$u_{r}\in$$\hat{V}(X)^{(\langle\Lambda\rangle\rangle}$$\hat{V}(X)^{\langle(\Lambda\rangle)}/\mathfrak{U}\models\varphi(u_{1}/\mathfrak{U}, \ldots, u_{f}/\mathrm{U})$iff$[\varphi(u_{1}, \ldots, u_{r})\mathrm{I}\in \mathrm{u}$.

Proof. The proof is similar to that of Theorem 3.5. The only different part is the
if-part of the case where is of the form
. Suppose.
$\exists x\in y\psi(x)$$\varphi$

It follows ffom Corollary 3.4 that there is$[\exists x\in u_{k}\psi(x)\mathrm{J}$$[\exists x\in u_{k}\psi(x)\mathrm{I}\in \mathrm{u}$$u\in\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$satisfying$[u\in u_{k}\wedge\psi(u)\mathrm{J}=$.

$\square$Abounded ultralimit is apre nonstandard universe: that satisfies (1),(2) and(3) of Definition 1.2 with Mostowski collapsing.
Definition 3.5 (atlas). An atlas is apairof Asuch thatand$\langle\Lambda,\mathfrak{U}\rangle$$\mathrm{r}\mathrm{a}\mathrm{d}(\hat{V}(X)^{((\Lambda\rangle\rangle}/\mathrm{u})=\mathrm{r}\mathrm{a}\mathrm{d}(\mathrm{A})$of an LACA Aand an ultrafilter$\mathrm{c}\mathrm{o}\mathrm{v}(\hat{V}(X)^{((\Lambda))}/\mathrm{u})=\mathrm{c}\mathrm{o}\mathrm{v}(\Lambda)$.

Theorem 3.7 (Sheaf representation Theorem for Nonstandard Universes). For
any nonstanda $rd$ universe, there is an atlas (such thatisomorphic to.
$\star V(X)$$\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}/\mathfrak{U}$$\Lambda,\mathrm{u}\rangle$$\star V(X)$

We prove the theorem in the next section.

4. Local ULTRALIMITSWe shall see that ahomomorphism of LACAs induces an elementary embeddingof ultralimits and abounded elementary embedding of bounded ultralimits. Let
ahomomorphism. The induced map
is definedby $h_{*}(u)=h\circ u$ . Then we have the lemma below.
$h::\Lambdaarrow---\mathrm{b}\mathrm{e}$$h_{*}:$

Lemma 4.1. Let,,$u_{1}$$\varphi(x_{1}, \ldots, x_{r})$be a formula of$L$$\mathfrak{M}^{(\langle\Lambda\rangle\rangle}arrow \mathfrak{M}^{((\overline{-}\rangle\rangle}-$with only$x_{1}$$u_{f}\in M^{((\Lambda\rangle)}$$\ldots$$[\varphi(h_{*}(u_{1}), \ldots, h_{*}(u_{r}))\mathrm{J}_{-}--=h([\varphi(u_{1}, \ldots, u_{f})\mathrm{I}_{\Lambda})$.

,$\ldots$,$x_{r}$

free. For45

NONSTANDARD UNIVERSE

Proof. There is$C\in\Lambda^{0}$containing all the ranges of$u_{k}$. Sinceis complete, we$h\lceil C$

have from Theorem 3.1$\vee\{i=\wedge^{r}h(u_{i}(x_{i}))1|\mathfrak{M}$a$\varphi(x_{1}, \ldots, x_{r})\}=h(\vee\{i=\wedge u_{i}(x_{i})1r|\mathfrak{M}$ $\models\varphi(x_{1}, \ldots, x_{r})\}$$[\varphi(h_{*}(u_{1}), \ldots, h_{*}(u_{r}))\mathrm{I}_{-}--=h([\varphi(u_{1}, \ldots, u_{r})\mathrm{J}_{\Lambda})$.

We have thus proved the lemma.
For$h_{*}:$$u\in\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$$\square$, sincesimilarly.

$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(/\mathrm{i}\circ u)\subseteq \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}$it, we can define the induced map$\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}arrow\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$

Corollary 4.2. Let,,$\varphi(x_{1}, \ldots, x_{r})$be a formula of $L_{\in}with$ only

$x_{1}$,$\ldots$,$x_{r}$

free. For$u_{r}\in\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$$u_{1}$$\ldots$$[\varphi(h_{*}(u_{1}), \ldots, h_{*}(u_{r}))\mathrm{I}_{-}\overline{-}=h([\varphi(u_{1}, \ldots, u_{r})\mathrm{I}_{\Lambda})$.

Proof. Using Corollary 3.2, we see the proof is similar to that of Lemma 4.1.
. Then
Let and be ultrafilters of Aand —, respectively. Suppose$\square$$h^{-1}$ “$\mathrm{V}$$\mathrm{u}$$\mathrm{V}$$=\mathrm{U}$

we have from Lemma 4.1 or from Corollary 4.2[$u=u’\mathrm{J}_{\Lambda}\in \mathrm{u}$iff$[h_{*}(u)=h_{*}(u’)\mathrm{I}_{-}--\in \mathrm{V}$Therefore we can define the injection, by. Sincesimilarly.

$h_{*}:$$h_{*}(u/\mathrm{U})=h_{*}(u)/\mathrm{V}$$h_{*}$$h_{*}:$.

by same, we can define the injection$M^{\langle\langle\Lambda\rangle\rangle}/\mathrm{u}arrow M^{\langle\langle_{-}^{-}\rangle\rangle}-/\mathrm{V},\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{d}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}h_{*}(u)\subseteq \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}u$$\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}/\mathrm{u}arrow\hat{V}(X)^{\langle\langle\rangle\rangle}\underline{=}/\mathrm{V}$Lemma 4.3. The injection.

is an elementary embedding of$h_{*}$$M^{\langle\langle\Lambda\rangle\rangle}/\mathrm{u}$into$M^{\langle\langle_{-}^{-}\rangle\rangle}-/\mathrm{V}$

Proof. Let

be aformula of,,$\varphi(x_{1}, \ldots, x_{r})$rem 3.5 we have for$L$with only$x_{1}$,$\ldots$,$x_{r}$

free. From Theo$u_{r}\in M^{\langle\langle\Lambda\rangle\rangle}$$u_{1}$$\ldots$$h([\varphi(u_{1}, \ldots, u_{f})\mathrm{J}_{\Lambda})\in \mathrm{V}$iff[$\varphi(h_{*}(u_{1}), \ldots, h_{*}(u_{r}))\mathrm{I}_{-}--\in \mathrm{V}$iff$[\varphi(u_{1}, \ldots, u_{r})\mathrm{J}_{\Lambda}\in \mathrm{u}$$\mathfrak{M}^{\langle\langle_{-}^{-}\rangle\rangle}-/\mathrm{V}\models\varphi(h_{*}(u_{1}/\mathfrak{U}), \ldots, h_{*}(u_{r}/\mathrm{U}))$iff$\mathfrak{M}^{(\langle\Lambda\rangle\rangle}/\mathrm{u}\models\varphi(u_{1}, \ldots, u_{r})$[$\varphi(u_{1}, \ldots , u_{r})\mathrm{I}_{\Lambda}\in h^{-1}$“$\mathrm{V}$.
$\square$Corollary 4.4. The injection.
into$h_{*}$

ofis a bounded elementary embedding$\hat{V}(X)^{\langle\langle_{-}^{-}\rangle\rangle}-/\mathrm{V}$$\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}/\mathfrak{U}$Proof. Using Corollary 3.6, we see the proof is similar to that of Lemma 4.3.

$\square$

Let I be aset relative to $V(X)$ . We shall find aone-t0-0ne correspondence. Note thatis the set of “the subsets of in
betweenand. Define $g:Iarrow$, there is”. Forsuch that. Conversely,
andbyThen we have
. Define, there isbysuch thatfor$\varphi(I)^{\langle\langle\Lambda\rangle\rangle}$$\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$$\mathrm{S}(\mathrm{A})$$\check{I}$$\mathfrak{B}(\Lambda^{[I]})$$\varphi(I)^{\langle\langle\Lambda\rangle\rangle}$$C\in\Lambda^{0}$$A\in\varphi(I)^{\langle\langle\Lambda\rangle\rangle}$$g(i)=[\check{i}\in A\mathrm{I}_{\Lambda}\cdot$$g\in \mathfrak{B}(\Lambda^{[I]})$$\mathrm{r}\mathrm{n}\mathrm{g}A\subseteq C$$g\in \mathfrak{B}(\Lambda^{[I]})$$\mathrm{r}\mathrm{n}\mathrm{g}g\subseteq C$$C\in\Lambda^{0}$$A:\mathrm{A}(\mathrm{x})arrow C$$\mathrm{r}\mathrm{n}\mathrm{g}g\subseteq C$$b$$A(x)=\wedge \mathrm{s}\mathrm{g}_{x}(i, g(i))i\in I$

’

where

$\mathrm{s}\mathrm{g}_{x}(i, b)=\{$$\neg b$if $i\in x$ ,if

$i\in I\backslash x$.

46

MASAHIKO MURAKAMI

Since is completely distributive, we have
Then we seeand. In the context above, the relation[
one correspondence between
andon, we identify.
with
We shall define the special element$C$$A\in\varphi(I)^{\langle\langle\Lambda\rangle\rangle}$. Suppose$g(i)=[\check{i}\in A\mathrm{J}_{\Lambda}$

and $(\neg g)(i)=$sets up aone-t0as Boolean algebras. Prom now$(g\wedge g’)(i)=[\check{i}\in A\cap A’\mathrm{I}_{\Lambda}$$g’(i)=[\check{i}\in A’\mathrm{I}_{\Lambda}\cdot$$i\vee\in\check{I}\backslash A\mathrm{J}_{\Lambda}$$g(i)=[\check{i}\in A\mathrm{J}_{\Lambda}$$\mathfrak{B}(\Lambda^{[I]})$$\varphi(I)^{\langle\langle\Lambda\rangle\rangle}$$\mathfrak{B}(\Lambda^{[I]})$$\varphi(I)^{\langle(\Lambda\rangle\rangle}$by$\delta\in I^{\langle(\Lambda^{[l]}\rangle\rangle}\subseteq\hat{V}(X)^{\langle\langle\Lambda^{[J]}\rangle\rangle}$$\delta(x)(i)=\{$

1if $x=i$ ,0if $x\neq i$ .

diagonal element of I on A. Let
We call theembedding. Then is also aBoolean monomorphism of
diagonal element ahas following properties.

be the canonical
. The
into

$j::\mathrm{A}arrow\Lambda^{[I]}$$\delta$$j$$\mathfrak{B}(\Lambda)$$\varphi(I)^{(\langle\Lambda\rangle\rangle}$

Lemma 4.5. The following statements hold.
(1) [$j(b)=\check{I}\mathrm{J}_{\Lambda}=b$(2) [$b\in \mathfrak{B}(\Lambda)$$\delta\in j_{*}(g)\mathrm{J}_{\Lambda^{[l]}}=g$

Prvof. Since [for every
for every.

$g\in\varphi(I)^{((\Lambda\rangle)}$for all $i\in I$ ,. Proro the definition of , it is clear that
Then we have$i\vee\in j(b)\mathrm{J}_{\Lambda}=j(b)(i)=b$$[j(b)=\check{I}\mathrm{I}_{\Lambda}=[j(b)\supseteq\check{I}\mathrm{J}_{\Lambda}=$$\delta$$\bigwedge_{:\in I}\mathbb{I}\check{i}\in j(b)\mathrm{J}=b$$6(\mathrm{i})(\mathrm{i})=1$$g(i)$.

.

$=\check{i}\mathrm{I}_{\Lambda^{[’]}}(i)=$$[\delta\in j_{*}(g)\mathrm{I}_{\Lambda^{[’]}}(i)=\mathbb{I}\check{i}\in j_{*}(g)\mathrm{J}_{\Lambda^{[l]}}(i)=[\check{i}\in g\mathrm{J}_{\Lambda}=$.

$\square$, there is a map
Theorem 4.6. For any v.
such that v $=j_{*}(w)(\delta)$ holds in$\in\hat{V}(X)^{((\Lambda^{[l]}\rangle\rangle}$$\hat{V}(X)^{(\langle\Lambda\rangle)}$$w:\check{I}arrow(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v)^{\sqrt}$

in$\hat{V}(X)^{(\langle\Lambda\rangle\rangle}$

Proof. Since rngtz
w:$\mathrm{I}\delta$$\in$$(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v)^{I}arrow \mathfrak{B}(\Lambda)$$\Lambda^{[I]}$,$\bigcup_{g\in \mathrm{r}\mathrm{n}\mathrm{g}v}$

rngg

$\in$

$\Lambda$.

Therefore we can defineby$\mathrm{w}(\mathrm{s})$$=\wedge$$v(s(i))(i)$ .

$:\in I$. If $s\neq s’$ , then there isThen we get as required. First, we show
is pairwise disjoint, we havesuch that $s(i_{0})\neq s’(i_{0})$ . Since$w\in\hat{V}(X)^{((\Lambda\rangle\rangle}$$w$$i_{0}\in I$$\mathrm{r}\mathrm{n}\mathrm{g}v$.

$\mathrm{w}(\mathrm{s})\wedge w(s’)\leq v(s(i_{0}))(i_{0})\wedge v(s’(i_{0}))(i_{0})=0$

There is$C\in\Lambda^{0}$such that. Then we have$\bigcup_{g\in \mathrm{r}\mathrm{n}\mathrm{g}v}\mathrm{r}\mathrm{n}\mathrm{g}g\subseteq C$$\mathrm{r}\mathrm{n}\mathrm{g}w\subseteq C$

and then.

$s\in(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v)^{I}s\in(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v)^{I}i\in Ii\in Iy\in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v\vee w(s)=\vee\wedge v(s(i))(i)=\wedge\vee v(y)(i)=1$

47

NONSTANDARD UNIVERSE

We have thus shown, we havefor every$w\in\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$. For each $i\in I$ , since $w(s)\leq v(s(i\})(i)$ holds$s\in(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v)^{I}$$[v=j_{*}(w)(\delta)\mathrm{I}_{\Lambda^{[I]}}(i)=[v=j_{*}(w)(\check{i})\mathrm{I}_{\Lambda^{[I]}}(i)$$=(\vee\{v(y)\wedge j(w(s))\wedge\check{i}(x)|y=s(x)\})(i)$$=\vee(v(s(i))(i)\wedge w(s))s\in(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v)^{I}$$=\vee w(s)s\in(\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}v)^{I}=1$.

We have thus proved the theorem.

Cl

Let be an ultrafilter of an LACA A. Alocal ultralimit
is abounded elementary embedding satisfying$\mathrm{u}$$\rho:\hat{V}(X)^{((\Lambda\rangle\rangle}/\mathrm{u}arrow\star V(X)$

for every $x\in V(X)$ .

$\rho(\check{x}/\mathfrak{U})=x\star$

Theorem 4.7 (Local Ultralimit Theorem). Let
be a l0$cal$ ultralimit and let
be an inter$mal$ element of $\star V(X)$ . Then there is a local
ultralimit
such that the following conditions hold.
$\rho:\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}/\mathrm{U}arrow\star V(X)$$p$$\tau:\hat{V}(X)^{\langle\langle\Lambda^{[I]}\rangle\rangle}/\mathrm{V}arrow\star V(X)$(i) The index set I is a set relative to $V(X)$ and(ii) Letbe the canonical embedding. Then$|I|=\mathrm{n}\mathrm{o}\mathrm{s}(p)$$j::\Lambdaarrow\Lambda^{[I]}$(iii) The submodel$\star V(X)$$\mathrm{r}\mathrm{n}\mathrm{g}\mathrm{r}$that contains

and.
of $\star V(X)$ is the minimal bounded elementary submodel of$\{p\}\mathrm{U}\mathrm{r}\mathrm{n}\mathrm{g}\mathrm{r}$$\mathrm{u}$$=j^{-1}$ “

Proof. Let I be aset relative to $V(X)$ such that
identified

with$\varphi(I)^{\langle\langle\Lambda\rangle\rangle}$. Define

iff$g\in \mathrm{V}$$\rho=\tau\circ j_{*}$$\mathrm{V}$.

,$\mathfrak{B}(\Lambda^{[I]})$.

$p\in\star I$$\mathrm{V}\subseteq \mathfrak{B}(\Lambda^{[I]})$$p\in\rho(g/\mathfrak{U})$and$p\in \mathrm{r}\mathrm{n}\mathrm{g}\tau$.

$|I|=\mathrm{n}\mathrm{o}\mathrm{s}(p)$. We haveby.

. Let be an element of 1[. Prom (1) of Lemma 4.5,
is an ultrafilter ofcoincides *I. Then we havefrom the definition of V. Since“andare maximal filters, we obtain
U. Letbe,,a,,-formula ofwith only
free. Letbe elements of. By Theorem 4.6, there are maps , ,from insuchthat $v_{k}=j_{*}(w_{k})(\delta)$ hold, where is the diagonal element of I on A. Putting$g_{0}=\{i\in\check{I}|\varphi(w_{1}(i), \ldots, w_{r}(i))\}$ in, we have from (2) of Lemma 4.5
Then$\Lambda^{[I]}$$\mathrm{V}$$b$$j(b)\in \mathrm{V}$$\rho(j(b)/\mathrm{U})$$j^{-1}$$\mathrm{V}$$\mathrm{u}$$\Delta_{0}$$L_{\in}$$x_{1}$$\ldots$$\mathrm{V}=$$x_{r}$$\varphi(x_{1}, \ldots, x_{f})$$v_{1}$$\ldots$$\hat{V}(X)^{\langle\langle\Lambda^{[I]}\rangle\rangle}$$v_{r}$$\check{I}$$w_{1}$$\ldots$$w_{r}$$\delta$$\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}$$g_{0}=[\delta\in j_{*}(g\mathrm{o})\mathrm{I}_{\Lambda^{[I]}}$$=[\delta\in j_{*}(\{i\in\check{I}|\varphi(w_{1}(i), \ldots,w_{r}(i))\})\mathrm{I}_{\Lambda^{[I]}}$$=[\delta\in\{i\in\check{I}|\varphi(j_{*}(w_{1})(i), \ldots,j_{*}(w_{f})(i))\}\mathrm{I}_{\Lambda^{[I]}}$$=[\varphi(j_{*}(w_{1})(\delta), \ldots,j_{*}(w_{f})(\delta))\mathrm{J}_{\Lambda^{[I]}}$$=[\varphi(v_{1}, \ldots, v_{r})\mathrm{I}_{\Lambda^{[I]}}$$\hat{V}(X)^{\langle\langle\Lambda)\rangle}$

48

MASAHIKO MURAKAMI

and we have$\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}/\mathfrak{U}\models g\mathrm{o}/\mathfrak{U}$$=\{i\in\check{I}/\mathfrak{U}|\varphi((w_{1}/\mathfrak{U})(i), \ldots, (w_{r}/\mathfrak{U})(i))\}$.

$\rho(g_{0}/\mathrm{u})=\{i\in I\star|\varphi(\rho(w_{1}/\mathfrak{U})(i), \ldots, \rho(w_{r}/\mathfrak{U})(i))\}$

By the definition of , we obtain$\mathrm{V}$itr p$g0\in \mathrm{V}$[$\varphi(v_{1},$\ldots ,iff$v_{f})\mathrm{J}_{\Lambda^{[I]}}\in \mathrm{V}$$\in\rho(g_{0}/\mathfrak{U})$$\varphi(\rho(w_{1}/\mathrm{u})(p),$$\rho(w_{f}/\mathfrak{U})(p))$\ldots ,.

“The case$x_{1}=x_{2}$ ”enables us to define the operation
where $v=j_{*}(w)(\delta)$ holds in. Thus, defining
bywhere $v=j_{*}(w)(\delta)$ holds in, we getasrequired. In fact, it is clear in the preceding context that is an bounded elementaryembedding of
into. Let be the identity map on , then we
see
. For
, let be theconstant map ffom I onto
, then we havein
. Suppose abounded elementary submodel $W$ of
contains.
Prom the definition of ,for some.
Therefore rngr is the minimum. We have completed the proof of Theorem 4.7.
$\varphi(x_{1}, x_{2})\equiv$$v/\mathrm{V}\mapsto\rho(w/\mathrm{U})(p)$$\hat{V}(X)^{\langle\langle\Lambda^{[J]}\rangle\rangle}$$\tau:\hat{V}(X)^{\langle\langle\Lambda^{[J]}\rangle\rangle}/\mathrm{V}arrow\star V(X)$$\tau(v/\mathrm{V})=\rho(w/\mathrm{U})(p)$$\hat{V}(X)^{((\Lambda^{[\mathit{1}]}\rangle\rangle}$$\tau$$\tau$$\hat{V}(X)^{((\Lambda^{[I]}\rangle\rangle}/\mathrm{V}$$\star V(X)$$I$$\iota$$\tau(j_{*}(\check{\iota}/\mathrm{u})(\delta/\mathrm{V}))=\rho(\check{\iota}/\mathfrak{U})(p)=\iota(\star p)=p$$u/\mathfrak{U}\in\hat{V}(X)^{((\Lambda\rangle\rangle}/\mathrm{u}$$\tilde{u}$$\vee$$\hat{V}(X)^{((\Lambda\rangle\rangle}$$\{u\}$$\tau(j_{*}(u/\mathrm{U}))=\mathrm{p}(\mathrm{w}/\mathrm{U})(\mathrm{p})=$$\rho(u/\mathrm{U})$$\star V(X)$$\tau$$\tau(v/\mathrm{V})=\mathrm{p}(\mathrm{w}/\mathrm{U})(\mathrm{p})\in W$$\{p\}\cup \mathrm{r}\mathrm{n}\mathrm{g}\rho$$w/\mathfrak{U}\in\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}/\mathfrak{U}$$\square$

Let
limit
ultrafilter of$\{j_{d}^{d’}::,be an embedding system of LACAs with direct“
be an ultrafilter of , then eachis an\Lambda_{d}arrow\Lambda_{d’}\}_{d\leq d,d,d\in D},$$\{jd::\Lambda_{d}arrow\Lambda\}_{d\in D}$$\Lambda_{d}$

. Let

.

$\mathrm{u}$$\mathrm{u}_{d}=j_{d}^{-1}$$\Lambda$$\mathrm{u}$Theorem 4,8 (Elementary Net Theorem of Ultralimits). Let
and be models
satisfying
for L. Suppose there are elementary embeddings :$d\leq d’$ .
the condition
Then there is an elementary embeddingfor$d\in D$ .
such that$\Re$$\mathfrak{M}$$\mathfrak{M}^{((\Lambda_{d}\rangle\rangle}/\mathrm{u}_{d}arrow\Re$$\tau_{d}$$\tau d=\tau d’\circ j_{d*}^{d’}$$\tau:\mathfrak{M}^{((\Lambda\rangle)}/\mathrm{u}arrow\Re$$\tau_{d}=\tau\circ j_{d*}for$. Let be an element of$Pro\mathrm{o}/$$v$$\mathfrak{M}^{\langle(\Lambda\rangle\rangle}$. Since rngr{$\in\Lambda=$$j_{d}$“$S|d\in D$ and $S\in\Lambda_{d}$}ffom the definition of direct limits, there issuch that $v=j_{d*}(u)$ .
Therefore defining
where $v=j_{d_{*}}(u)$ , we get as required.
Let
be aformula of with only , ,free and let , ,be elements. Then there are $d\in D$ and , ,such that$v_{k}=j_{d*}(u_{k})$ . We conclude as below.
$u\in \mathfrak{M}^{((\Lambda_{d}\rangle\rangle}$$\tau(v/\mathrm{U})=\tau d(u/\mathrm{U}_{d})$$\varphi(x_{1}, \ldots,x_{\mathrm{r}})$$\tau$$L$$x_{1}$$\ldots$$x_{f}$$M^{((\Lambda\rangle\rangle}$$u_{1}$$\mathfrak{M}^{\langle(\Lambda_{d}\rangle\rangle}/\mathrm{U}_{d}\models\varphi(u_{1}/\mathrm{U}_{d}, \ldots, u_{f}/\mathrm{U}_{d})$$\mathfrak{M}^{\langle(\Lambda\rangle\rangle}/\mathrm{U}\models\varphi(v_{1}/\mathrm{U}, \ldots, v_{f}/\mathrm{U})$iff$\mathfrak{R}$

iff

$\Re$

$v_{1}$$\ldots$$u_{f}\in M^{(\langle\Lambda_{d}\rangle)}$$\ldots$$\models\varphi(\tau_{d}(u_{1}/\mathrm{U}), \ldots, \tau_{d}(u_{f}/\mathfrak{U}))$$\models\varphi(\tau(v_{1}/\mathrm{U}), \ldots, \tau(v_{r}/\mathfrak{U}))$.

$v_{r}$

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NONSTANDARD UNIVERSE

Theorem 4.9 (Bounded Elementary Net Theorem of Bounded Ultralimits).
$\star V(X)$satisfying
Suppose there are local ultralimits:
the condition Td
for $d\leq d’$ . Then there is a local ultralimit$d\in D$ .
such that$\hat{V}(X)^{\langle(\Lambda_{d}\rangle\rangle}/\mathrm{u}_{d}$$\tau_{d}$$arrow$$=\tau d’\circ j_{d*}^{d’}$$\tau:\hat{V}(X)^{\langle\langle\Lambda\rangle\rangle}/\mathrm{u}arrow\star V(X)$$\tau_{d}=\tau\circ j_{d_{*}}for$

Proof. Similar to the proof of Theorem 4.8.
We call the pair.

Proof of Theorem 3.7Let
We define local ultralimits$\Lambda_{0=}\varphi(\{0,1\})$$\Lambda_{\zeta+1}=\mathrm{A}^{[I_{\zeta}]}$$\langle\Lambda_{\lambda}, \mathfrak{U}_{\lambda}\rangle$

in Theorem 4.8 or Theorem 4.7 the direct limit of$\langle\Lambda, \mathfrak{U}\rangle$$\{\langle\Lambda_{d}, \mathrm{u}_{d}\rangle\}_{d\leq d’,d,d’\in D}$$\square$,$\{p_{\zeta}\}_{\zeta<\kappa}$$\{\rho_{\zeta} :$\kappa=\mathrm{c}\mathrm{o}\mathrm{v}(^{\star}V(X))$.

$|I_{\zeta}|=\mathrm{n}\mathrm{o}\mathrm{s}(p_{\zeta})$

is the direct limit ofThen the direct limit$\langle\Lambda, \mathfrak{U}\rangle$of.

\hat{V}^{\langle\langle\Lambda_{\zeta}\rangle\rangle}/\mathfrak{U}_{\zeta}arrow\star V(X)\}_{\zeta<\kappa}$$\mathfrak{U}_{0}=\{1\}$, wherebe asequence in $\star V(X)$ with
of $\star V(X)$ by:,$p_{\zeta}\in \mathrm{r}\mathrm{n}\mathrm{g}\rho\zeta+1$$\{\langle\Lambda_{\zeta}, \mathrm{u}_{\zeta}\rangle_{\zeta<\lambda}\}$$\{\langle\Lambda\zeta, \mathfrak{U}_{\zeta}\rangle\}_{\zeta<\kappa}$

in Theorem 4.7.

in Theorem 4.9.

is an atlas of $\star V(X)$ .
口

REFERENCES[1] C. CHANG and J. KEISLER, Model Theory, 3rd ed, North-Holland, Amsterdam, (1990).

[2] R. MANSFIELD, The theorry of Boolean ultrapowers, Ann. Math. Logic, 2(1971) 297-323.
[3] M. Ozawa, Forcing in nonstandard analysis, Annals of Pure and Applied Logic, 68 (1994),

263-297.

[4] M. MURAKAMI, Standa rdization principle of Nonstandard universes, Joumal of Symbolic
Logic, 64, $4(1999)$ , 1645-1655.

$E$-mail address: muramasa(Dms. -tokyo. ac.jp$\mathrm{u}$