A Simple Combinatorial Proof of Szemerédi’s Theorem via Three Levels of Infinities^†††Mathematics Subject Classification 2020: Primary 11B25, Secondary 03H05

This article is strongly influenced by Terence Tao’s notes [9].

Szemerédi’s theorem confirms a conjecture of P. Erdős and P. Turán made in 1936, which implies van der Waerden’s theorem.

Nonstandard versions of Furstenberg’s ergodic proof and Gowers’s harmonic proof of Szemerédi’s theorem have been tried by T. Tao (see Tao’s blog post [10]). In the workshop Nonstandard methods in combinatorial number theory sponsored by American Institute of Mathematics in San Jose, CA, August 2017, Tao gave a series of lectures to explain Szemerédi’s original combinatorial proof and hope to simplify it so that the proof can be better understood. He believed that Szemerédi’s combinatorial method should have a greater impact on combinatorics.

During these lectures Tao challenged the audience to produce a nonstandard proof of Szemerédi’s theorem which is noticeably simpler and more transparent than Szemerédi’s original proof. The current article is the result of Tao’s challenge and inspiration. However, in his later blog post [11], Tao commented that “in fact there are now signs that perhaps nonstandard analysis is not the optimal framework in which to place this argument.” We disagree. The current article is our effort to show that with the help of a nonstandard universe with three levels of infinities, Szemerédi’s original argument can be made simpler and more transparent.

The main simplification in our proof of Szemerédi’s theorem compared to the standard proof in [8, 9] is that a Tower of Hanoi type induction in [9, Theorem 6.6] and in [8, Lemma 5, Lemma 6, and Fact 12] is replaced by a straightforward induction (see Lemma 5.1 below), which makes Szemerédi’s idea more transparent. To achieve this, we work within a chain of nonstandard extensions ${\mathbb{V}}_{0}\prec{\mathbb{V}}_{1}\prec{\mathbb{V}}_{2}\prec{\mathbb{V}}_{3}$ which supply three levels of infinities, plus various bounded elementary embeddings from ${\mathbb{V}}_{j}$ to ${\mathbb{V}}_{j^{\prime}}$ for some $0\leq j<j^{\prime}\leq 3$.

The paper is organized in the following sections. §2 is a brief introduction of logic foundation for constructing the nonstandard extensions ${\mathbb{V}}_{0}\prec{\mathbb{V}}_{1}\prec{\mathbb{V}}_{2}\prec{\mathbb{V}}_{3}$ and bounded elementary embeddings including $i_{0}$, $i_{*}$, $i_{1}$, and $i_{2}$ to the reader who does not have logic background. The reader who is only interested in applications can familiarize with the notation, Property 2.7, Proposition 2.14, and Proposition 2.15, safely skip the proof, and return to it at a later time. In §3 we translate the density along arithmetic progressions in standard setting to strong upper Banach density in nonstandard setting as well as translate some consequences of the double counting argument in standard setting to nonstandard setting. The reader who is only interested in applications can familiarize with the notation, Lemma 3.3, Lemma 3.4, and Lemma 3.5, safely skip the proof, and return to it at a later time. In §4 we re-write, in a nonstandard setting, the proof of a so–called mixing lemma in [9] based on a weak regularity lemma. This section does not offer new idea but is included only for self-containment. §5 is the main part of the paper where we present the proof of Szemerédi’s theorem. In §6 we pose a question whether the presented proof of Szmerédi’s theorem can be carried out without the axiom of choice.

The notation we use here should be consistent with some standard textbooks. Consult, for example, [1, 4, 7] for more details. If $f:A\to B$ is a function, then $f(a)$ denotes the image of $a$ as an element in $B$ and $f[C]:=\{f(a)\mid a\in C\}$ for some $C\subseteq A$.

Let ${\mathbb{N}}_{0}$ be the set of all standard positive integers and ${\mathbb{R}}_{0}$ be the set of all standard real numbers. By the standard universe we mean the superstructure ${\mathbb{V}}_{0}:={\mathbb{V}}({\mathbb{R}}_{0})$ on $X={\mathbb{R}}_{0}$. Notice that all standard mathematical objects mentioned in this paper have ranks below, say, $100$. For example, an ordered pair $(a,b)$ of standard real numbers can be viewed as the set $\{\{a\},\{a,b\}\}\in{\mathbb{V}}({\mathbb{R}}_{0},2)$ and a function $f:{\mathbb{N}}_{0}\to{\mathbb{R}}_{0}$ can be viewed as a set of ordered pairs in ${\mathbb{V}}({\mathbb{R}}_{0},2)$. Hence $f\in{\mathbb{V}}({\mathbb{R}}_{0},3)$.

We can define (first-order) $\mathscr{L}$–formulas recursively starting from atomic formulas. If $x$ and $x^{\prime}$ are variables, then $x=x^{\prime}$ is an atomic formula. If $P_{i}\in\mathscr{L}$ and $\overline{x}:=(x_{1},x_{2},\ldots,x_{m_{i}})$ is an $m_{i}$-tuple of variables, then $P_{i}(\overline{x})$ is an atomic formula. The word “first-order” means that these variables are intended to take only elements of some $\mathscr{L}$–model as their values. All formulas mentioned in this paper are first-order. We will use the symbol $\overline{a}$ to represent an $m$-tuple of elements with some suitable generic number $m\in\omega$. When $\overline{x}$ is intended to be substituted by $\overline{a}$, we assume implicitly that they have the same length.

If $\varphi$ and $\psi$ are $\mathscr{L}$–formulas and $x$ is a variable, then $\neg\varphi$, $\varphi\wedge\psi$, $\varphi\vee\psi$, $\varphi\to\psi$, $\varphi\leftrightarrow\psi$, $\forall x\varphi$, and $\exists x\varphi$ are also $\mathscr{L}$–formulas. The symbols $\neg$, $\wedge$, $\vee$, $\to$, and $\leftrightarrow$ are called logic connectives, and $\forall$ and $\exists$ are called universal and existential quantifiers. An occurrence of $x$ in $\varphi$ is called bounded if it occurs in a sub-formula of the form $\forall x\psi$ or $\exists x\psi$ in $\varphi$. An occurrence of $x$ in $\varphi$ is called free if it is not bounded. A formula without free variable is called a sentence.

Let $\mathcal{M}$ be an $\mathscr{L}$–model. For any $\mathscr{L}$–formula $\varphi(\overline{x})$, where $\overline{x}$ is an $m$-tuple of variables containing all free variables in $\varphi$, and any $\overline{a}\in\mathcal{M}^{m}$, called parameters, we can define $\mathcal{M}\models\varphi(\overline{a})$, meaning $\varphi(\overline{a})$ is true in $\mathcal{M}$, recursively by (a) $\mathcal{M}\models a=a^{\prime}$ iff $a$ and $a^{\prime}$ are identical elements in $\mathcal{M}$ and $\mathcal{M}\models P_{i}(\overline{a})$ iff $\overline{a}\in P_{i}^{\mathcal{M}}$; (b) for any $\mathscr{L}$–formulas $\varphi$ and $\psi$ with all free variables $\overline{x}$ being substituted by $\overline{a}$ in $\mathcal{M}$, $\mathcal{M}\models\neg\varphi$ iff $\mathcal{M}\not\models\varphi$ ($\neg$ means “not”), $\mathcal{M}\models\varphi\vee\psi$ iff $\mathcal{M}\models\varphi$ or $\mathcal{M}\models\psi$ ($\vee$ means “or”), $\mathcal{M}\models\varphi\wedge\psi$ iff $\mathcal{M}\models\varphi$ and $\mathcal{M}\models\psi$, ($\wedge$ means “and”), $\mathcal{M}\models\varphi\to\psi$ iff $\mathcal{M}\models\neg\varphi\vee\psi$ ($\to$ means “imply”), and $\mathcal{M}\models\varphi\leftrightarrow\psi$ iff $\mathcal{M}\models(\varphi\to\psi)\wedge(\psi\to\varphi)$ ($\leftrightarrow$ means “if and only if”); (c) for any $\mathscr{L}$–formulas $\varphi(y,\overline{x})$, $\mathcal{M}\models\forall y\,\varphi(y,\overline{a})$ iff $\mathcal{M}\models\varphi(b,\overline{a})$ for every $b\in\mathcal{M}$ ($\forall$ means “for all”) and $\mathcal{M}\models\exists y\varphi(y,\overline{a})$ iff $\mathcal{M}\models\varphi(b,\overline{a})$ for some $b\in\mathcal{M}$ ($\exists$ means “there exist”). From the truth definition above, for every $\mathscr{L}$–formula $\varphi(\overline{x})$ there is another $\mathscr{L}$–formula $\psi(\overline{x})$ using only logic connectives $\neg$ and $\vee$, and only quantifier $\exists$, such that $\mathcal{M}\models\varphi(\overline{a})$ iff $\mathcal{M}\models\psi(\overline{a})$ for any $\mathscr{L}$–model $\mathcal{M}$ and any $\overline{a}\in\mathcal{M}^{m}$.

We sometimes call a formula with all free variables being substituted by parameters a sentence. Clearly, the truth value of a sentence with parameters from $\mathcal{M}$ is determined in $\mathcal{M}$. When we write $\varphi(\overline{x},\overline{a})$, we mean implicitly that all free variables in the formula are among $\overline{x}$ and all parameters from model $\mathcal{M}$ are among $\overline{a}$.

Suppose $\mathcal{M}$ and $\mathcal{N}$ are two $\mathscr{L}$–models. A function $i:\mathcal{M}\to\mathcal{N}$ is called an elementary embedding from $\mathcal{M}$ to $\mathcal{N}$ if for any $\mathscr{L}$–formula $\varphi(\overline{x})$ and any $m$-tuple $\overline{a}=(a_{1},a_{2},\ldots,a_{m})\in\mathcal{M}^{m}$ we have

where $i(\overline{a}):=(i(a_{1}),i(a_{2}),\ldots,i(a_{m}))\in\mathcal{N}^{m}$. An elementary embedding is necessarily injective. If there exists an elementary embedding $i:\mathcal{M}\to\mathcal{N}$, we can view $\mathcal{M}$ as an elementary submodel of $\mathcal{N}$ and call $\mathcal{N}$ an elementary extension of $\mathcal{M}$, denoted by $\mathcal{M}\preceq\mathcal{N}$. We sometimes write $\mathcal{M}\prec\mathcal{N}$ to emphasize that $i$ is not surjective.

If $\mathcal{M}$ is an $\mathscr{L}$–model, $\mathscr{L}^{\prime}:=\mathscr{L}\cup\{P_{n+1},\ldots,P_{k}\}$, and $P_{i}^{\mathcal{M}}\subseteq\mathcal{M}^{m_{i}}$ for $n<i\leq k$, the $\mathscr{L}^{\prime}$–model $\mathcal{M^{\prime}}$ by adding the relations $P_{i}^{\mathcal{M}}$ to $\mathcal{M}$ is denoted by $(\mathcal{M};P^{\mathcal{M}}_{n+1},\ldots,P^{\mathcal{M}}_{k})$. We call $\mathcal{M}^{\prime}$ a model expansion of $\mathcal{M}$.

It is well known that the existence of non-principal ultrafilters on $X$ follows from $\mathsf{ZFC}$.

For an element $c\in\mathcal{M}$ denote $\phi_{c}:X\to\mathcal{M}$ for the constant function with value $c$. Let $i:\mathcal{M}\to\mathcal{M}^{X}/{\mathcal{F}}$ be the natural embedding, i.e., $i(c)=[\phi_{c}]_{{\mathcal{F}}}$. The following is often called Łoś’s theorem. (cf. [1, Theorem 4.1.9, Corollary 4.1.13].)

The proof of Proposition 2.3 is done by induction on the complexity of $\varphi$.

So, the model $\mathcal{M}$ can be viewed as an elementary submodel of $\mathcal{M}^{X}/{\mathcal{F}}$ via the natural embedding $i$ and $\mathcal{M}^{X}/{\mathcal{F}}$ is an elementary extension of $\mathcal{M}$.

$\blacklozenge$: From now on let $\mathscr{L}=\{\in\}$. The ultrafilters which will be used are ${\mathcal{F}}_{0}$, the tensor product of ${\mathcal{F}}_{0}$’s, and nonstandard versions of them.

Recall that ${\mathbb{V}}_{0}$ represents the standard universe, which is an $\mathscr{L}$–model. Let ${\mathbb{V}}_{0}^{{\mathbb{N}}_{0}}/{\mathcal{F}}_{0}$ be the ultrapower of ${\mathbb{V}}_{0}$ modulo ${\mathcal{F}}_{0}$ and $i^{0}:{\mathbb{V}}_{0}\to{\mathbb{V}}_{0}^{{\mathbb{N}}_{0}}/{\mathcal{F}}_{0}$ be the natural embedding. Denote ${}^{*}\!\!\in$ for the interpretation of $\in$ in ${\mathbb{V}}_{0}^{{\mathbb{N}}_{0}}/{\mathcal{F}}_{0}$. Notice that $a\in b$ iff $i^{0}(a)\,^{*}\!\!\in i^{0}(b)$ for any $a,b\in{\mathbb{V}}_{0}$. One can define the rank function $\mbox{rank}(b)$ for every $b\in{\mathbb{V}}_{0}^{{\mathbb{N}}_{0}}/{\mathcal{F}}_{0}$ according to (1) with $\in$ being replaced by ${}^{*}\!\!\in$. Notice that some elements $[f]_{{\mathcal{F}}}$ in ${\mathbb{V}}_{0}^{{\mathbb{N}}_{0}}/{\mathcal{F}}_{0}$ may not have a finite ${}^{*}\!\!\in$-rank. For example, if $f(1)=0$ and $f(n+1)=\{f(n)\}$ for every $n\in{\mathbb{N}}_{0}$, then $[f]_{{\mathcal{F}}_{0}}\in{\mathbb{V}}_{0}^{{\mathbb{N}}_{0}}/{\mathcal{F}}_{0}$ does not have a finite ${}^{*}\!\!\in$-rank. Let

The set $V_{1}$ is just the ultrapower of ${\mathbb{V}}_{0}$ modulo ${\mathcal{F}}_{0}$ truncated at ${}^{*}\!\!\in$-rank $\omega$. Notice that $\in\!\mbox{-rank}(a)=\,^{*}\!\!\in\!\mbox{-rank}(i^{0}(a))$ for every $a\in{\mathbb{V}}_{0}$ and $V_{1}=\bigcup_{n\in\omega}i^{0}(V({\mathbb{R}}_{0},n))$.

Let ${\mathbb{R}}_{1}:=i^{0}({\mathbb{R}}_{0})$ and ${\mathbb{N}}_{1}:=i^{0}({\mathbb{N}}_{0})$. Assume that every element in ${\mathbb{R}}_{1}$ is an urelement and identify $i^{0}(r)$ by $r$ for every $r\in{\mathbb{R}}_{0}$. Then ${\mathbb{R}}_{0}\subseteq{\mathbb{R}}_{1}$. Since the natural order $\leq$ on ${\mathbb{R}}_{0}$, addition $+$ and multiplication $\times$ on ${\mathbb{R}}_{0}$ can be viewed as elements in ${\mathbb{V}}_{0}$, we have that $i^{0}(\leq)$ is a linear order on ${\mathbb{R}}_{1}$ extending $\leq$, $i^{0}(+)$ is the addition on ${\mathbb{R}}_{1}$ extending $+$, and $i^{0}(\times)$ is the multiplication on ${\mathbb{R}}_{1}$ extending $\times$. For notational convenience we write $\leq$, $+$, and $\times$ for $i^{0}(\leq)$, $i^{0}(+)$, and $i^{0}(\times)$, respectively. By the elementality of $i^{0}$ the structure $({\mathbb{R}}_{1};+,\times,\leq,0,1)$ is an ordered field containing the standard real field $({\mathbb{R}}_{0};+,\times,\leq,0,1)$ as its subfield. Notice that if $\mbox{Id}(n)=n$ for every $n\in{\mathbb{N}}_{0}$, then $[\mbox{Id}]_{{\mathcal{F}}_{0}}\in i^{0}({\mathbb{N}}_{0})={\mathbb{N}}_{1}$ and $[\mbox{Id}]_{{\mathcal{F}}_{0}}\geq r$ for every $r\in{\mathbb{R}}_{0}$, i.e., ${\mathbb{N}}_{1}$ contains natural numbers such as $[\mbox{Id}]_{{\mathcal{F}}_{0}}$ which are infinitely large relative to real numbers in ${\mathbb{R}}_{0}$.

Let $\mathscr{M}$ be the Mostowski collapsing map on $V_{1}$, i.e., $\mathscr{M}(a)=a$ for every $a\in{\mathbb{R}}_{1}$ and

for every $b\in V_{1}\setminus{\mathbb{R}}_{1}$. Then $\mathscr{M}$ is an injection and $a\,^{*}\!\!\in b$ iff $\mathscr{M}(a)\in\mathscr{M}(b)$. If one identifies $V_{1}$ with the image of $V_{1}$ under $\mathscr{M}$, one can pretend that ${}^{*}\!\!\in$ is the true membership relation and consider $V_{1}$ as a subset of the superstructure ${\mathbb{V}}({\mathbb{R}}_{1})$. Hence, we can pretend that ${}^{*}\!\!\in$ is the true membership relation $\in$ and drop ^∗ for notational convenience. The purpose of the truncation of ${\mathbb{V}}_{0}^{{\mathbb{N}}_{0}}/{\mathcal{F}}_{0}$ at ${}^{*}\!\!\in$-rank $\omega$ is to make sure $\mathscr{M}$ is well defined in the standard sense.

Let ${\mathbb{V}}_{1}:=(V_{1};\in)$. We call ${\mathbb{V}}_{1}$ a nonstandard universe extending ${\mathbb{V}}_{0}$. We will extend ${\mathbb{V}}_{1}$ further later. Notice that due to the truncation, ${\mathbb{V}}_{1}$ is no longer an elementary extension of ${\mathbb{V}}_{0}$ from the model theoretic point of view. However, ${\mathbb{V}}_{1}$ is a so-called bounded elementary extension of ${\mathbb{V}}_{0}$.

An $\mathscr{L}$–formula $\theta$ has bounded quantifiers if every occurrence of quantifiers $\forall$ and $\exists$ in $\theta$ has the form $\forall x\!\in\!y$ and $\exists x\!\in\!y$. Notice that $\forall x\!\in\!y\,\varphi$ is the abbreviation of $\forall x(x\!\in\!y\to\varphi)$ and $\exists x\!\in\!y\,\varphi$ is the abbreviation of $\exists x(x\!\in\!y\wedge\varphi)$. Similar to (2), it is easy to show that for any $\mathscr{L}$–formula $\varphi(\overline{x})$ with bounded quantifiers and any $\overline{a}\in{\mathbb{V}}_{0}$ we have

So, the map $i^{0}$ is called a bounded elementary embedding from ${\mathbb{V}}_{0}$ to ${\mathbb{V}}_{1}$. It is a common abuse of notation to write ${\mathbb{V}}_{0}\preceq{\mathbb{V}}_{1}$ to indicate the existence of the bounded elementary embedding (instead of just elementary embedding) $i^{0}$ and ${\mathbb{V}}_{0}\prec{\mathbb{V}}_{1}$ to emphasize that $i^{0}$ is not surjective. The property (3) is sometimes called the transfer principle in nonstandard analysis. Notice that if $A\in{\mathbb{V}}_{0}$, then $i^{0}(A)$ can be viewed as the ultrapower of $A$ modulo ${\mathcal{F}}_{0}$, i.e., $i^{0}(A)=\{[f]_{{\mathcal{F}}_{0}}\mid f\in A^{{\mathbb{N}}_{0}}\}$.

Each $i^{0}(a)$ for $a\in{\mathbb{V}}_{0}$ is called ${\mathbb{V}}_{0}$–internal (or “standard” in some literature) and each $b\in{\mathbb{V}}_{1}$ is called ${\mathbb{V}}_{1}$–internal. Hence ${\mathbb{V}}_{0}$–internal set is also a ${\mathbb{V}}_{1}$–internal set. For example, ${\mathbb{R}}_{1}=i^{0}({\mathbb{R}}_{0})$ and ${\mathbb{N}}_{1}=i^{0}({\mathbb{N}}_{0})$ are ${\mathbb{V}}_{0}$–internal sets. Some ${\mathbb{V}}_{1}$–internal sets are not ${\mathbb{V}}_{0}$–internal. For example, the set $\{1,2,\ldots,[\mbox{Id}]_{{\mathcal{F}}_{0}}\}$ is ${\mathbb{V}}_{1}$–internal subset of ${\mathbb{N}}_{1}$ but not ${\mathbb{V}}_{0}$–internal. Some subsets of a ${\mathbb{V}}_{1}$–internal set are not ${\mathbb{V}}_{1}$–internal. For example, ${\mathbb{N}}_{0}$ as a subset of ${\mathbb{N}}_{1}$ is not ${\mathbb{V}}_{1}$–internal because it is bounded above in ${\mathbb{N}}_{1}$ and has no largest element. Notice that $i^{0}[{\mathbb{N}}_{0}]={\mathbb{N}}_{0}$ and $i^{0}({\mathbb{N}}_{0})={\mathbb{N}}_{1}$. The following proposition says that a subset of $B\in{\mathbb{V}}_{1}$ defined by an $\mathscr{L}$–formula with parameters from ${\mathbb{V}}_{1}$ is ${\mathbb{V}}_{1}$–internal. The proposition is an easy consequence of Proposition 2.3.

The following is called the overspill principle in nonstandard analysis. Notice that ${\mathbb{N}}_{0}$ is an infinite initial segment of ${\mathbb{N}}_{1}$.

The proof of Proposition 2.6 is easy. If $A\cap({\mathbb{N}}_{1}\setminus U)=\emptyset$, then $U$ can be defined by a formula with bounded quantifiers and parameter $A$. Hence $U$ is ${\mathbb{V}}_{1}$–internal.

We are now going to work towards establishing this property in this section.

$\blacklozenge$: From now on, let ${\mathbb{V}}_{j}^{{\mathbb{N}}_{j}}$ always represent, for notational convenience, the set of all functions $f$ from ${\mathbb{N}}_{j}$ to ${\mathbb{V}}_{j}$ such that $\{\mbox{rank}(f(n))\mid n\in{\mathbb{N}}_{j}\}$ is a bounded set in $\omega$.

Notice that ${\mathcal{F}}_{1}:=i^{0}({\mathcal{F}}_{0})\in{\mathbb{V}}_{1}$ satisfies Part 1–4 of Definition 2.1 with $X$, $\mathscr{P}_{<\omega}(X)$, and $\mathscr{P}(X)$ being replaced by ${\mathbb{N}}_{1}:=i^{0}({\mathbb{N}}_{0})$, $i^{0}(\mathscr{P}_{<{\mathbb{N}}_{0}}({\mathbb{N}}_{0}))=\left\{A\subseteq{\mathbb{N}}_{1}\mid A\in{\mathbb{V}}_{1}\,\mbox{ and }\,|A|\in{\mathbb{N}}_{1}\right\}$, and $i^{0}(\mathscr{P}({\mathbb{N}}_{0}))={\mathbb{V}}_{1}\cap\mathscr{P}({\mathbb{N}}_{1})$, respectively. We call ${\mathcal{F}}_{1}$ a ${\mathbb{V}}_{1}$–internal non-principal ultrafilter on ${\mathbb{N}}_{1}$.

We call ${\mathbb{V}}_{2}$ a ${\mathbb{V}}_{1}$–internal ultrapower of ${\mathbb{V}}_{1}$ modulo the ${\mathbb{V}}_{1}$–internal ultrafilter ${\mathcal{F}}_{1}$.

The proof of Proposition 2.9 is almost the same as the proof of Proposition 2.3 except one step that shows $\left\{n\in{\mathbb{N}}_{1}\mid{\mathbb{V}}_{1}\models\exists x\!\in\!B(n)\,\varphi(\overline{a(n)},x)\right\}\in{\mathcal{F}}_{1}$ implies ${\mathbb{V}}_{2}\models\exists x\!\in\![B]_{{\mathcal{F}}_{1}}\,\varphi(\overline{[a]}_{{\mathcal{F}}_{1}},x)$. Let $M_{z}=i^{0}({\mathbb{V}}({\mathbb{R}}_{0},z))\in{\mathbb{V}}_{1}$ with $z\in\omega$ such that $B,\overline{a}\in M_{z}$. By the axiom of choice there is a well-order $\lhd$ on ${\mathbb{V}}({\mathbb{R}}_{0},z)$. So, every nonempty set $A\in M_{z}$ has a $i^{0}(\lhd)$-least element by the transfer principle. Suppose that $\left\{n\in{\mathbb{N}}_{1}\mid{\mathbb{V}}_{1}\models\exists x\!\in\!B(n)\,\varphi(\overline{a(n)},x)\right\}=X\in{\mathcal{F}}_{1}$. For each $n\in{\mathbb{N}}_{1}$, if $n\not\in X$, let $f(n)=0$ and if $n\in X$, let $f(n)$ be the $i^{0}(\lhd)$-least element in the nonempty set $\left\{x\in B(n)\mid{\mathbb{V}}_{1}\models\varphi(a(n),x)\right\}$. Since $B,\overline{a},i^{0}(\lhd)$ are in ${\mathbb{V}}_{1}$, so does the function $f$ by Proposition 2.5. Hence $[f]_{{\mathcal{F}}_{1}}\in{\mathbb{V}}_{2}\cap[B]_{{\mathcal{F}}_{1}}$. By the induction hypothesis we have that

which implies ${\mathbb{V}}_{2}\models\exists x\!\in\![B]_{{\mathcal{F}}_{1}}\,\varphi(\overline{[a]}_{{\mathcal{F}}_{1}},x)$.

Let

Then $i_{0}$ is a bounded elementary embedding from ${\mathbb{V}}_{0}$ to ${\mathbb{V}}_{2}$, which will be used in Theorem 5.4.

Same as for ${\mathbb{V}}_{1}$ we can assume by Mostowski collapsing that ${\mathbb{V}}_{2}$ is a subset of the superstructure ${\mathbb{V}}({\mathbb{R}}_{2})$ and $\in_{2}$ is the true membership relation $\in$. Notice that $i^{1}\!\upharpoonright\!{\mathbb{R}}_{1}$ is an identity map. The element $a\in{\mathbb{V}}_{2}$ is called ${\mathbb{V}}_{2}$–internal, $i^{1}(b)$ is called ${\mathbb{V}}_{1}$–internal for any $b\in{\mathbb{V}}_{1}$, and $i_{0}(c)$ is called ${\mathbb{V}}_{0}$–internal for every $c\in{\mathbb{V}}_{0}$. Notice that ${\mathbb{N}}_{0}$ and ${\mathbb{N}}_{1}$ as subsets of ${\mathbb{N}}_{2}$ are not ${\mathbb{V}}_{2}$–internal.

If $f\in{\mathbb{V}}_{1}$ is a function from ${\mathbb{N}}_{1}$ to $[n]$ for some $n\in{\mathbb{N}}_{1}$, then $[f]_{{\mathcal{F}}_{1}}=m$ for some $m\in{\mathbb{N}}_{1}$ by (3) with ${\mathbb{V}}_{0},{\mathbb{V}}_{1},i^{0}$ being replaced by ${\mathbb{V}}_{1},{\mathbb{V}}_{2},i^{1}$. Hence ${\mathbb{N}}_{2}$ is a proper end-extension of ${\mathbb{N}}_{1}$.

By the same way, we can define ${\mathbb{V}}_{3}$ as a ${\mathbb{V}}_{2}$–internal ultrapower of ${\mathbb{V}}_{2}$ modulo $i^{1}({\mathcal{F}}_{1})$.

Generalizing the arguments above, we have that the map $i^{2}$ is a bounded elementary embedding from ${\mathbb{V}}_{2}$ to ${\mathbb{V}}_{3}$. We say that ${\mathbb{V}}_{3}$ is a nonstandard extension of ${\mathbb{V}}_{2}$. It is also easy to see that ${\mathbb{N}}_{3}:=i^{2}({\mathbb{N}}_{2})$ is an end-extension of ${\mathbb{N}}_{2}$.

We have completed the construction of ${\mathbb{V}}_{0}\prec{\mathbb{V}}_{1}\prec{\mathbb{V}}_{2}\prec{\mathbb{V}}_{3}$ and verified that Part 1 of Property 2.7 is true.

We view $a\mapsto[F_{a}]_{{\mathcal{F}}_{0}\otimes{\mathcal{F}}^{\prime}_{0}}$ as a relation between ${\mathbb{V}}_{2}$ and ${\mathbb{V}}^{\prime}_{2}$. Notice that $[X]_{{\mathcal{F}}^{\prime}_{0}}\in{\mathcal{F}}_{1}$ iff $\{n\in{\mathbb{N}}_{0}\mid X(n)\in{\mathcal{F}}_{0}\}\in{\mathcal{F}}^{\prime}_{0}$. Notice also that for any $a,b\in{\mathbb{V}}_{2}$, we have