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True History of She-Male Cock 3 2014 Evil Angel 1 DRO We Like Tranny 2014 Pulse Distribution In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x1, x2,, xn that are true of a sequence of elements of an L-structure {\displaystyle {\mathcal {M}}}{\mathcal {M}}. Depending on the context, types can be complete or partial and they may use a fixed set of constants, A, from the structure {\displaystyle {\mathcal {M}}}{\mathcal {M}}. The question of which types represent actual elements of {\displaystyle {\mathcal {M}}}{\mathcal {M}} leads to the ideas of saturated models and omitting types. Contents 1 Formal definition 2 Examples of types 3 Stone spaces 4 The omitting types theorem 5 References Formal definition Consider a structure {\displaystyle {\mathcal {M}}}{\mathcal {M}} for a language L. Let M be the universe of the structure. For every A ? M, let L(A) be the language obtained from L by adding a constant ca for every a ? A. In other words,



{\displaystyle L(A)=L\cup \{c_{a}:a\in A\}.}L(A)=L\cup \{c_{a}:a\in A\}. A 1-type (of {\displaystyle {\mathcal {M}}}{\mathcal {M}}) over A is a set p(x) of formulas in L(A) with at most one free variable x (therefore 1-type) such that for every finite subset p0(x) ? p(x) there is some b ? M, depending on p0(x), with {\displaystyle {\mathcal {M}}\models p_{0}(b)}{\mathcal {M}}\models p_{0}(b) (i.e. all formulas in p0(x) are true in {\displaystyle {\mathcal {M}}}{\mathcal {M}} when x is replaced by b). Similarly an n-type (of {\displaystyle {\mathcal {M}}}{\mathcal {M}}) over A is defined to be a set p(x1,,xn) = p(x) of formulas in L(A), each having its free variables occurring only among the given n free variables x1,,xn, such that for every finite subset p0(x) ? p(x) there are some elements b1,,bn ? M with {\displaystyle {\mathcal {M}}\models p_{0}(b_{1},\ldots ,b_{n})}{\mathcal {M}}\models p_{0}(b_{1},\ldots ,b_{n}). A complete type of {\displaystyle {\mathcal {M}}}{\mathcal {M}} over A is one that is maximal with respect to inclusion. Equivalently, for every {\displaystyle \phi ({\boldsymbol {x}})\in L(A,{\boldsymbol {x}})}\phi ({\boldsymbol {x}})\in L(A,{\boldsymbol {x}}) either {\displaystyle \phi ({\boldsymbol {x}})\in p({\boldsymbol {x}})}\phi ({\boldsymbol {x}})\in p({\boldsymbol {x}}) or {\displaystyle \lnot \phi ({\boldsymbol {x}})\in p({\boldsymbol {x}})}\lnot \phi ({\boldsymbol {x}})\in p({\boldsymbol {x}}). Any non-complete type is called a partial type. So, the word type in general refers to any n-type, partial or complete, over any chosen set of parameters (possibly the empty set). An n-type p(x) is said to be realized in {\displaystyle {\mathcal {M}}}{\mathcal {M}} if there is an element b ? Mn such that {\displaystyle {\mathcal {M}}\models p({\boldsymbol {b}})}{\mathcal {M}}\models p({\boldsymbol {b}}). The existence of such a realization is guaranteed for any type by the compactness theorem, although the realization might take place in some elementary extension of {\displaystyle {\mathcal {M}}}{\mathcal {M}}, rather than in {\displaystyle {\mathcal {M}}}{\mathcal {M}} itself. If a complete type is realized by b in {\displaystyle {\mathcal {M}}}{\mathcal {M}}, then the type is typically denoted {\displaystyle tp_{n}^{\mathcal {M}}({\boldsymbol {b}}/A)}tp_{n}^{{{\mathcal {M}}}}({\boldsymbol {b}}/A) and referred to as the complete type of b over A. A type p(x) is said to be isolated by {\displaystyle \varphi }\varphi , for {\displaystyle \varphi \in p(x)}{\displaystyle \varphi \in p(x)}, if {\displaystyle \forall \psi ({\boldsymbol {x}})\in p({\boldsymbol {x}}),\operatorname {Th} ({\mathcal {M}})\models \varphi ({\boldsymbol {x}})\rightarrow \psi ({\boldsymbol {x}})}{\displaystyle \forall \psi ({\boldsymbol {x}})\in p({\boldsymbol {x}}),\operatorname {Th} ({\mathcal {M}})\models \varphi ({\boldsymbol {x}})\rightarrow \psi ({\boldsymbol {x}})}. Since finite subsets of a type are always realized in {\displaystyle {\mathcal {M}}}{\mathcal {M}}, there is always an element b ? Mn such that f(b) is true in {\displaystyle {\mathcal {M}}}{\mathcal {M}}; i.e. {\displaystyle {\mathcal {M}}\models \varphi ({\boldsymbol {b}})}{\mathcal {M}}\models \varphi ({\boldsymbol {b}}), thus b realizes the entire isolated type. So isolated types will be realized in every elementary substructure or extension. Because of this, isolated types can never be omitted (see below). A model that realizes the maximum possible variety of types is called a saturated model, and the ultrapower construction provides one way of producing saturated models. Examples of types Consider the language with one binary connective, which we denote as {\displaystyle \in }\in . Let {\displaystyle {\mathcal {M}}}{\mathcal {M}} be the structure {\displaystyle \langle \omega ,\in _{\omega }\rangle }\langle \omega ,\in _{{\omega }}\rangle for this language, which is the ordinal {\displaystyle \omega }\omega with its standard well-ordering. Let {\displaystyle {\mathcal {T}}}{\mathcal {T}} denote the theory of {\displaystyle {\mathcal {M}}}{\mathcal {M}}. Consider the set of formulas {\displaystyle p(x):=\{n\in _{\omega }x\mid n\in \omega \}}{\displaystyle p(x):=\{n\in _{\omega }x\mid n\in \omega \}}. First, we claim this is a type. Let {\displaystyle p_{0}(x)\subseteq p(x)}p_{0}(x)\subseteq p(x) be a finite subset of {\displaystyle p(x)}p(x). We need to find a {\displaystyle b\in \omega }{\displaystyle b\in \omega } that satisfies all the formulas in {\displaystyle p_{0}}p_{0}. Well, we can just take the successor of the largest ordinal mentioned in the set of formulas {\displaystyle p_{0}(x)}p_{0}(x). Then this will clearly contain all the ordinals mentioned in {\displaystyle p_{0}(x)}p_{0}(x). Thus we have that {\displaystyle p(x)}p(x) is a type. Next, note that {\displaystyle p(x)}p(x) is not realized in {\displaystyle {\mathcal {M}}}{\mathcal {M}}. For, if it were there would be some {\displaystyle n\in \omega }n\in \omega that contains every element of {\displaystyle \omega }\omega . If we wanted to realize the type, we might be tempted to consider the model {\displaystyle \langle \omega +1,\in _{\omega +1}\rangle }\langle \omega +1,\in _{{\omega +1}}\rangle , which is indeed a supermodel of {\displaystyle {\mathcal {M}}}{\mathcal {M}} that realizes the type. Unfortunately, this extension is not elementary, that is, this model does not have to satisfy {\displaystyle {\mathcal {T}}}{\mathcal {T}}. In particular, the sentence {\displaystyle \exists x\forall y(y\in x\lor y=x)}{\displaystyle \exists x\forall y(y\in x\lor y=x)} is satisfied by this model and not by {\displaystyle {\mathcal {M}}}{\mathcal {M}}. So, we wish to realize the type in an elementary extension. We can do this by defining a new structure in the language, which we will denote {\displaystyle {\mathcal {M}}'}{\mathcal {M}}'. The domain of the structure will be {\displaystyle \omega \cup \mathbb {Z} '}\omega \cup {\mathbb {Z}}' where {\displaystyle \mathbb {Z} '}{\mathbb {Z}}' is the set of integers adorned in such a way that {\displaystyle \mathbb {Z} '\cap \omega =\emptyset }{\mathbb {Z}}'\cap \omega =\emptyset . Let {\displaystyle <}< denote the usual order of {\displaystyle \mathbb {Z} '}{\mathbb {Z}}'. We interpret the symbol {\displaystyle \in }\in in our new structure by {\displaystyle \in _{{\mathcal {M}}'}=\in _{\omega }\cup <\cup \,(\omega \times \mathbb {Z} ')}\in _{{{\mathcal {M}}'}}=\in _{{\omega }}\cup <\cup \,(\omega \times {\mathbb {Z}}'). The idea being that we are adding a "{\displaystyle \mathbb {Z} }\mathbb {Z} -chain", or copy of the integers, above all the finite ordinals. Clearly any element of {\displaystyle \mathbb {Z} '}{\mathbb {Z}}' realizes the type {\displaystyle p(x)}p(x). Moreover, one can verify that this extension is elementary. Another example: the complete type of the number 2 over the empty set, considered as a member of the natural numbers, would be the set of all first-order statements, describing a variable x, that are true when x = 2. This set would include formulas such as {\displaystyle \,\!x\neq 1+1+1}\,\!x\neq 1+1+1, {\displaystyle x\leq 1+1+1+1+1}x\leq 1+1+1+1+1, and {\displaystyle \exists y(y0\land y^{2}>2)\implies y>x)}{\displaystyle \forall y((y>0\land y^{2}>2)\implies y>x)} describing the square root of 2 are consistent with the axioms of ordered fields, and can be extended to a complete type. This type is not realized in the ordered field of rational numbers, but is realized in the ordered field of reals. Similarly, the infinite set of formulas (over the empty set) {x>1, x>1+1, x>1+1+1, ...} is not realized in the ordered field of real numbers, but is realized in the ordered field of hyperreals. If we allow parameters, for instance all of the reals, we can specify a type {\displaystyle \{0

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