{"id":31,"date":"2024-08-29T23:44:18","date_gmt":"2024-08-29T23:44:18","guid":{"rendered":"https:\/\/wp.math.berkeley.edu\/model-theory\/?p=31"},"modified":"2024-11-02T05:47:50","modified_gmt":"2024-11-02T05:47:50","slug":"introduction-to-o-minimality","status":"publish","type":"post","link":"https:\/\/wp.math.berkeley.edu\/model-theory\/2024\/08\/29\/introduction-to-o-minimality\/","title":{"rendered":"Introduction to o-minimality"},"content":{"rendered":"<p>For the most part, we will be following Lou van den Dries&#8217; book <a href=\"https:\/\/search.library.berkeley.edu\/discovery\/fulldisplay?docid=alma991031701629706532&amp;context=L&amp;vid=01UCS_BER:UCB&amp;lang=en&amp;search_scope=DN_and_CI&amp;adaptor=Local%20Search%20Engine&amp;tab=Default_UCLibrarySearch&amp;query=any,contains,tame%20topology&amp;offset=0\">Tame Topology and O-minimal Structures<\/a> for this seminar.\u00a0 We will supplement with material from reseach papers.\u00a0 If there are specific theorems about o-minimality you would like to see this term, let me know.\u00a0 Some possible topics include<\/p>\n<ul>\n<li>Speissegger&#8217;s<a href=\"https:\/\/arxiv.org\/abs\/math\/9710220\"> theorem on Pfaffian closures of o-minimal structures<\/a><\/li>\n<li>The Peterzil-Starchenko<a href=\"https:\/\/people.maths.ox.ac.uk\/bays\/ominGruppen\/peterzilStarchenko-trichotomy.pdf\"> o-minimal trichotomy<\/a><\/li>\n<li>The Pila-Wilkie <a href=\"https:\/\/projecteuclid.org\/journals\/duke-mathematical-journal\/volume-133\/issue-3\/The-rational-points-of-a-definable-set\/10.1215\/S0012-7094-06-13336-7.full\">counting theorem<\/a><\/li>\n<li>Proofs of o-minimality of specific structures (e.g. $\\mathbb{R}_{an,\\exp}$ or the expansion by <a href=\"https:\/\/arxiv.org\/pdf\/2203.15047\">Gevrey functions<\/a>)<\/li>\n<li>The Peterzil-Starchenko <a href=\"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/CRELLE.2009.002\/html\">o-minimal Chow theorem<\/a><\/li>\n<\/ul>\n<p><!--more--><\/p>\n<p>Generally, we will work with structures in the usual logical sense.\u00a0 However, van den Dries offers an alternative definition closer in spirit to other mathematical constructions and borrowing from cylindrical algebra.\u00a0 \u00a0Even though we will not be using his convention, it is useful to see how it relates to ours, especially as many mathematicians working in o-minimality use a van den Dries style definition of structure.<\/p>\n<p><strong>Definition:<\/strong>\u00a0 A structure $(R,\\mathcal{S})$ consists of a nonempty set $R$ and a sequence $\\mathcal{S} = (\\mathcal{S}_n)_{n=0}^\\infty$ such that<\/p>\n<ol>\n<li>each $\\mathcal{S}_n$ is a Boolean subalgebra of the power set $\\mathcal{P}(R^n)$,<\/li>\n<li>if $A \\in \\mathcal{S}_n$, then $R \\times A \\in \\mathcal{S}_{n+1}$ and $A \\times R \\in \\mathcal{S}_{n+1}$,<\/li>\n<li>for each $n$ and pair of numbers $i$ and $j$ with $1 \\leq i, j \\leq n$ the multidiagonal $\\Delta_{i,j}^n := \\{ (x_1, \\ldots, x_n) \\in R^n : x_i = x_j \\}$ belongs to $\\mathcal{S}_n$, and<\/li>\n<li>if $\\varpi_{n}:R^{n+1} \\to R^n$ is the projection $(x_1, \\ldots, x_{n+1}) \\mapsto (x_1, \\ldots, x_n)$ and $A \\in \\mathcal{S}_{n+1}$, then $\\varpi_{n}(A) \\in \\mathcal{S}_n$.<\/li>\n<\/ol>\n<p><strong>Proposition:<\/strong>\u00a0 For a structure $(R,\\mathcal{S})$ there is a first-order language $\\mathcal{L}$ and an $\\mathcal{L}$-structure $\\mathfrak{R}$ with universe $R$ so that the definable sets in $\\mathfrak{R}$ are precisely the sets in $\\mathcal{S}$.<\/p>\n<p><strong>Proof:<\/strong>\u00a0 \u00a0Let $\\mathcal{L}$ be the relational language that has an $n$-ary relation symbol $R_X$ for each $X \\in \\mathcal{S}_n$.\u00a0 Interpret $R_X^{\\mathfrak{R}} := X \\subseteq R^n$.\u00a0 \u00a0Clearly, every set in $\\mathcal{S}$ is definable in $\\mathfrak{R}$.\u00a0 For the other inclusion we need the following claim.<\/p>\n<p><strong>Claim:<\/strong> For natural numbers $m$ and $n$, sequences $i_1, \\ldots, i_n$ with $1 \\leq i_1, \\ldots, i_n \\leq m$, and $X \\in \\mathcal{S}_n$, if we set $Y := \\{ (a_1, \\ldots, a_m) \\in R^m : (a_{i_1}, \\ldots, a_{i_n}) \\in X \\}$, then $Y \\in \\mathcal{S}_m$.<\/p>\n<p>Proof of claim:\u00a0 We may express $Y$ as $Y = \\varpi_m \\varpi_{m+1} \\cdots \\varpi_{m+n-1} \\left( (R^m \\times X) \\cap \\bigcap_{j=1}^n \\Delta_{i_j,m+j}^{n+m}\u00a0 \\right)$\u00a0 $\\maltese$<\/p>\n<p>With the claim in place, we may argue induction on the complexity of $\\phi(x_1, \\ldots, x_m) \\in \\mathcal{L}(x_1, \\ldots, x_m)$ that $\\phi(\\mathfrak{R}) \\in \\mathcal{S}_m$.\u00a0 For $\\phi$ atomic, $\\phi$ is given by either $x_i = x_j$, which defines $\\Delta_{i,j}^m$, or $R_X(x_{i_1}, \\ldots, x_{i_n})$ fo some $X \\in \\mathcal{S}_n$ and sequence $i_1, \\ldots, i_n$, in which $\\phi$ defines the set $Y$ of the claim, which belongs to $\\mathcal{S}_m$.\u00a0 \u00a0 In the inductive case that $\\phi$ is a Boolean combination of simpler formulae we use the condition that $\\mathcal{S}_m$ is a Boolean algebra.\u00a0 In the case that $\\phi = \\exists x_j \\psi$ where $\\psi$ is simpler, we first use the claim to reduce to the case that $j = n+1$, and then apply the final closure condion on $\\mathcal{S}$ to see that the set defined\u00a0 by $\\phi$, which is the projection of the set defined by $\\psi$, belongs to $\\mathcal{S}$.\u00a0 $\\Box$.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Definition:<\/strong>\u00a0 An o-minimal structure is an $\\mathcal{L}$-structure $(R,&lt;,\\ldots)$ for which $(R,&lt;)$ is a dense linear order without endpoints and for which every $\\mathcal{L}_R$-definable subset of $R$ is a finite union of points and intervals.\u00a0 Here an\u00a0 interval has the form $(a,b) := \\{ x \\in R : a &lt; x &lt; b \\}$ where $a \\in R \\cup \\{ -\\infty \\}$ and $b \\in R \\cup \\{ \\infty \\}$.\u00a0 We give $R \\cup \\{ \\pm \\infty \\}$ an order extending $&lt;$ on $R$ by defining $-\\infty &lt; R &lt; \\infty$.<\/p>\n<p><strong>Remark:\u00a0<\/strong> In the definition of o-minimality, it is important that $a$ and $b$ come from $R \\cup \\{ -\\infty, \\infty \\}$.\u00a0 The structure $\\mathfrak{Q} := (\\mathfrak{Q},&lt;, P)$ where\u00a0 $P^\\mathfrak{Q} := (-\\infty,\\sqrt{2}) \\cap \\mathbb{Q}$ is not o-minimal.<\/p>\n<p><strong>Remark:\u00a0<\/strong> Notice that we allow parameters when defining subsets of $R$.\u00a0 From now on, when we use the phrase &#8220;definable&#8221; without qualification we mean &#8220;definable with parameters&#8221;.<\/p>\n<p>We will consider examples of o-minimal structures later in the term.\u00a0 From the quantifier elimination theorem for real closed fields, we know that if $(R,&lt;,+,\\cdot,0,1)$ is a real closed field, then it is an o-minimal structure.\u00a0 It follows that any reduct of a real closed field which still has the ordering is also o-minimal.\u00a0 We will offer a different proof of the o-minimality of real closed fields next week.<\/p>\n<p><strong>Lemma 1:<\/strong>\u00a0 O-minimal structures are definably complete: if $(R,&lt;,\\ldots)$ is o-minimal and $A \\subseteq R$ is a nonempty definable subset, then $A$ has a supremum in $R \\cup \\{ \\infty \\}$.<\/p>\n<p>Proof:\u00a0 By o-minimality, we may write $A = \\bigcup_{i=1}^n \\{ a_i \\} \\cup \\bigcup_{j=1}^m (b_j,c_j)$ for some $a_i \\in R$, $b_j \\in R \\cup \\{ -\\infty \\}$, and $c_j \\in R \\cup \\{ \\infty \\}$ with $b_j &lt; c_j$ for $1 \\leq i \\leq n$ and $1 \\leq j \\leq m$.\u00a0 Set $s := \\max (\\{ a_i : i \\leq n \\} \\cup \\{c_j : j \\leq m \\})$, which exists as every finite nonempty subset of a totally ordered set has a maximum.\u00a0 It is easy to check that $s = \\sup A$. $\\Box$<\/p>\n<p><strong>Theorem:<\/strong>\u00a0 If $(G,&lt;,\\cdot,\\ldots)$ is an o-minimal ordered group, then it is a divisible, ordered abelian group.<\/p>\n<p>For the next few results, we fix $(G,&lt;,\\cdot,\\ldots)$ an o-minimal ordered group.<\/p>\n<p><strong>Lemma 2:<\/strong>\u00a0 If $H \\leq G$ is a definable subgroup, then $H$ is convex.<\/p>\n<p><strong>Proof:\u00a0<\/strong> If $H$ is not convex, then there are elements $g \\in G \\smallsetminus H$ and $h \\in H$ with $1 &lt; g &lt; h$.\u00a0 Multiplying by $h^n &gt; 1$ for $n \\in \\mathbb{Z}_+$, we obtain a sequence of inequalities $1 &lt; g &lt; h &lt; gh &lt; h^2 &lt; gh^2 &lt; h^3 &lt; \\cdots &lt; gh^n &lt; h^{n+1} &lt; \\cdots$.\u00a0 For each $n \\in \\mathbb{N}$ we have $h^n \\in H$ and $g h^n \\in gH \\subseteq G \\smallsetminus H$.\u00a0 Hence, this sequence alternates between lying in $H$ and in its complemement.\u00a0 Such a situation violates o-minimality: as $H$ is definable, it is a finite union of points of intervals.\u00a0 There would have to be one such interval $(a,b)$ in this presentation for which $h^n \\in (a,b)$ for all sufficiently large $n$, but $a &lt;h^n &lt; g h^n &lt; h^{n+1} &lt; b$ would yield $g h^n \\in (a,b) \\subseteq H$ which is not true. $\\Box$<\/p>\n<p><strong>Lemma 3:<\/strong> If $H \\leq G$ is a definable subgroup, then $H = \\{ 1 \\}$ or $H = G$.<\/p>\n<p>Proof:\u00a0 By Lemma 1, $H$ is convex. By Lemma 2, $s := \\sup H \\in G \\cup \\{ \\infty \\}$.\u00a0 \u00a0If $s = 1$, then $H = \\{ 1 \\}$ and if $s = \\infty$, then $H = G$.\u00a0 Suppose that $1 &lt; s &lt; \\infty$.\u00a0 \u00a0We have that $s \\notin H$ for if $s \\in H$, then $s^2 \\in H$ and $s &lt; s^2$ showing that $s$ is not an upper bound of $H$.\u00a0 Note that because $1 &lt; s$, we have that\u00a0 $H \\neq \\{ 1 \\}$.\u00a0 So, there is some $h \\in H$ with $h &lt; 1$.\u00a0 \u00a0Then $hs &lt; s$ and both $s$ and $hs$ belong to $Hs$ which is disjoint from $H$.\u00a0 As $H$ is convex, there can be no element of $H$ between $hs$ and $s$.\u00a0 But then $hs$ is also an upper bound for $H$, showing that $s$ is not the supremum. $\\Box$<\/p>\n<p>&nbsp;<\/p>\n<p>Let us return to the proof of the theorem.<\/p>\n<p><strong>Proof of theorem:<\/strong>\u00a0 For any $g \\in G$, the centralizer $C_G(g) := \\{ x \\in G : gx = xg \\}$ is a definable subgroup of $G$.\u00a0 When $g = 1$, as is the case in every group, $C_G(g) = G$.\u00a0 When $g \\neq 1$, then $g \\in C_G(g) \\smallsetminus \\{ 1 \\}$.\u00a0 Thus, by Lemma 3, $C_G(g) = G$.\u00a0 That is, every element of $G$ belongs to its center so that $G$ is abelian.<\/p>\n<p>We will now write the group structure on $G$ additively.\u00a0 Fix some positive element $g \\in G$.\u00a0 For each positive integers $n$, the set $nG = \\{ nx : x \\in G \\} = \\{ y \\in G : (\\exists x) nx = y \\}$ is a definable subgroup of $G$ containing the nontrivial element $ng$.\u00a0 Thus, by Lemma 3 again, $nG = G$.\u00a0 As this identity holds for every $n$, we see that $G$ is divisible.\u00a0 $\\Box$<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>For the most part, we will be following Lou van den Dries&#8217; book Tame Topology and O-minimal Structures for this seminar.\u00a0 We will supplement with material from reseach papers.\u00a0 If there are specific theorems about o-minimality you would like to &hellip; <a href=\"https:\/\/wp.math.berkeley.edu\/model-theory\/2024\/08\/29\/introduction-to-o-minimality\/\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":99,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16],"tags":[],"class_list":["post-31","post","type-post","status-publish","format-standard","hentry","category-o-minimality-learning-seminar"],"_links":{"self":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts\/31","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/users\/99"}],"replies":[{"embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/comments?post=31"}],"version-history":[{"count":3,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts\/31\/revisions"}],"predecessor-version":[{"id":161,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/posts\/31\/revisions\/161"}],"wp:attachment":[{"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/media?parent=31"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/categories?post=31"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wp.math.berkeley.edu\/model-theory\/wp-json\/wp\/v2\/tags?post=31"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}