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Chapter 8: Problem 12

Let \(L\) be the language consisting of a single binary predicate symbol \(\leq\) and let \(T\) be the theory of dense linear orderings with no first or last element. Show that for every formula \(F\left[v_{0}, v_{1}^{\prime}, v_{2}, \ldots, v_{n}\right]\), there exists a quantifier-free formula \(H\left[v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right]\) such that $$ T \vdash \forall v_{0} \forall v_{1} \ldots \forall v_{n}\left(F\left[v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right] \Leftrightarrow H\left[v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right]\right) $$

Short Answer

Expert verified
The solution to this problem lies in constructing equivalent quantifier-free formulas and applying logical equivalences and substitutions, based on the properties of dense linear ordering.

Step by step solution

01

Understand Theory of Dense Linear Ordering

Dense linear ordering is an ordering \( \leq \) on a set L where every two elements have another element between them and there is no first or last element. This is the theory \( T \).
02

Interpret the Formula \( F[v_0, v_1', v_2, ...., v_n] \)

This is a formula which uses variables \( v_0, v_1', v_2, ...., v_n \) and may consist of prescriptions using the relation \( \leq \), logical connectives and possibly quantifiers.
03

Construct Quantifier-Free Formula \( H[v_0, v_1, v_2, ...., v_n] \)

A quantifier-free formula is a formula that does not use quantifiers. It is formed using only the relation \( \leq \) and the logical connectives and variables \( v_0, v_1, v_2 ...., v_n \)._
04

Establish the If and Only If Relationship

Assume \( H \) in order to prove \( F \), and assume \( F \) to prove \( H \). After assuming, apply the properties of the relation \( \leq \), use logical equivalences and connectives. Deduce the statements, using substitution and simplification. This needs to be done for all variables which means that the scenario applies for all possible members of the set on which the relation \( \leq \) is defined.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dense Linear Ordering
Understanding the concept of a dense linear ordering is crucial when studying order properties in structures like numbers or sets. Imagine a never-ending line where between any two points, no matter how close they are, you can always find another point. That鈥檚 exactly what a dense linear order is like. This ordering is denoted by the predicate symbol \( \leq \), and it means that for any two distinct elements, say \(a\) and \(b\), there exists another element \(c\) such that \(a \leq c \leq b\), providing there's no 'beginning' or 'end' to this sequence.

In the context of the exercise, the theory \(T\) of dense linear orderings applies to any set with such a property. This theory becomes a valuable tool for mathematicians and logicians in understanding and illustrating the behavior of densely ordered sets. It鈥檚 like having an infinite playground where each swing (element) always has another swing between it and the next, without ever seeing the start or the end of the entire set of swings.
Quantifier-Free Formula
Quantifiers in logic are expressions like 'for all' (universal quantifier) and 'there exists' (existential quantifier). They are used to indicate the extent to which a predicate is true over a range of elements. However, a quantifier-free formula, as the name suggests, does not contain any quantifiers. Such formulas are composed only of variables, constants, logical connectives, and non-logical symbols (like \( \leq \) in our example).

In simpler terms, think of a quantifier-free formula as a straightforward declaration that expresses a relationship without specifying whether it's true for 'all' or 'some' elements. For instance, if \( v_0, v_1, \ldots, v_n \) are variables, a quantifier-free formula might just state a relationship among these variables using \( \leq \) without needing to clarify whether this holds for all possible values. This aspect implies a more direct computation or evaluation, which is central to the exercise provided.
Logical Connectives
Just like in a language where we have conjunctions that link words or sentences, in logic, we have logical connectives that link propositions or statements. These include 'and' (\(\land\)), 'or' (\(\lor\)), 'not' (\(\eg\)), 'if...then...' (\(\rightarrow\)), and 'if and only if' (\(\leftrightarrow\)). They are the fundamental building blocks used to combine or negate predicates and formulas to create more complex expressions.

In the example at hand, logical connectives play a pivotal role in creating the quantifier-free formula \(H\). By skillfully applying connectives, one can navigate through the properties of a dense linear ordering to produce a statement \(H\) that holds the same truth conditions as the original quantifier-based formula \(F\), without using any quantifiers.
Binary Predicate Symbol
At the heart of the theory of dense linear orderings lies a specific kind of symbol known as a binary predicate symbol. In our case, this symbol is \(\leq\). A binary predicate is a way to compare two items, like a way of saying one thing is 'less than or equal to' another in this context. It's called binary because it always involves two entities. It's this fundamental comparison tool that enables mathematicians to frame and solve problems about order.

In the exercise's language \(L\), the sole symbol is the binary predicate that expresses the order between any two elements of a set. Understanding this symbol is key because it鈥檚 used in every step of defining the dense linear ordering. It shapes both the formulae in the theory and the way mathematicians think about the order within a given set.

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Most popular questions from this chapter

The language \(L\) consists of a binary predicate symbol \(R\) and a denumerably infinite set of constant symbols \(\left\\{c_{0}, c_{1}, \ldots, c_{n}, \ldots\right\\}\). Let \(A\) be a closed formula of \(L\) expressing that the interpretation of \(R\) is a strict, dense, total ordering without endpoints. For every \(n \in \mathbb{N}, F_{n}\) is the formula \(R c_{n} c_{n+1}\). Consider the theory \(\\{A\\} \cup\left\\{F_{n}: n \in \mathbb{N}\right\\} .\) We let \(\mathcal{A}, \mathcal{B}\), and \(\mathcal{C}\) denote the three \(L\)-structures whose base set is \(\mathbb{Q}\), in which the interpretation of \(R\) is the usual strict ordering, and in which the sequence of constant symbols \(\left(c_{n}\right)_{n \in \mathbb{N}}\) is interpreted, respectively, by the following sequences of rationals: $$ \alpha=\left(\alpha_{n}\right)_{n \in \mathbb{N}}, \quad \beta=\left(\beta_{n}\right)_{n \in \mathbb{N}}, \quad \gamma=\left(\gamma_{n}\right)_{n \in \mathbb{N}} $$ where $$ \alpha_{n}=n, \quad \beta_{n}=-\frac{1}{n+1}, \quad \gamma_{n}=\sum_{k=0}^{n} \frac{1}{k !} . $$ (a) Show that \(T\) is complete in \(L\). (b) Show that every denumerable model of \(T\) is isomorphic to one of the three structures \(\mathcal{A}, \mathcal{B}\), or \(\mathcal{C}\). (c) Show that the theory \(T\) is model-complete (see Exercise 8). (d) Show that every denumerable model of \(T\) has an elementary extension that is isomorphic to \(\mathcal{B}\) and an elementary extension that 'is isomorphic to \(\mathcal{C}\).

Let \(L\) be the first-order language consisting of a unary function symbol \(f\) and a binary relation symbol \(R\). Let \(A\) denote the conjunction of the following seven formulas: $$ \begin{aligned} &\forall v_{0} R v_{0} v_{0} \\ &\forall v_{0} \forall v_{1}\left(\left(R v_{0} v_{1} \Leftrightarrow R v_{1} v_{0}\right) \Rightarrow v_{0} \simeq v_{1}\right) \\ &\forall v_{0} \forall v_{1} \forall v_{2}\left(\left(R v_{0} v_{1} \wedge R v_{1} v_{2}\right) \Rightarrow R v_{0} v_{2}\right) \\ &\forall v_{0} \exists v_{1}\left(f v_{1} \simeq v_{0} \wedge \forall v_{2}\left(f v_{2} \simeq v_{0} \Rightarrow v_{2} \simeq v_{1}\right)\right) \\\ &\forall v_{0} \forall v_{\mathrm{I}}\left(R v_{0} v_{1} \Leftrightarrow R f v_{0} f v_{1}\right) ; \\ &\forall v_{0}\left(R v_{0} f v_{0} \wedge \neg v_{0} \simeq f v_{0}\right) ; \\\ &\forall v_{0} \forall v_{1}\left(\left(\neg v_{0} \simeq v_{1} \wedge R v_{0} v_{1}\right) \Rightarrow R f v_{0} v_{1}\right) \end{aligned} $$ (a) Show that in every model of the formula \(A\), the interpretation of the symbol \(R\) is a total ordering of the base set of the model, with no least or greatest element, such that every element has a successor, i.e. a strict least upper bound. (b) Show that \(\mathbb{Z}\) with its usual ordering and the successor function is a model of \(A\). Let \(X=\langle B, \leq\rangle\) be an arbitrary totally ordered set. Consider the following \(L\)-structure \(\mathcal{M}_{X}\) : \- the base set of \(\mathcal{M}_{X}\) is the set \(B \times \mathbb{Z}\); \- the interpretation of \(R\) in \(\mathcal{M}_{X}\) is the set $$ \left\\{((x, n),(y, m)) \in(B \times \mathbb{Z})^{2}: x

Consider the languages \(L_{1}=\\{f\\}\) and \(L_{2}=\\{f, P\\}\), where \(f\) is a unary function symbol and \(P\) is a unary relation symbol. Let \(T_{1}\) denote the following theory in \(L_{1}\) : $$ \begin{aligned} \\{\forall x \forall y(f x \simeq f y \Rightarrow x \simeq y)\\} & \cup\\{\forall x \exists y f y \simeq x\\} \\ & \cup\left\\{\forall x \neg f^{n} x \simeq x: n \in \mathbb{N}^{*}\right\\} \end{aligned} $$ [The term \(f^{n} x\) is defined as usual: \(f^{0} x=x\) and, for all \(n \in \mathbb{N}, f^{n+1} x=\) \(\left.f\left(f^{n} x\right) .\right]\) EXERCISES FOR CHAPTER 8 233 Let \(T_{2}\) denote the following theory in \(L_{2}\) : $$ T_{1} \cup\\{\exists x P x, \quad \exists x \neg P x, \quad \forall x(P x \Leftrightarrow P f x)\\} $$ (a) Show that \(T_{1}\) is a complete theory in \(L_{1} .\) (b) Show that \(T_{2}\) is not categorical in any infinite cardinal. (c) Use the results from the preceding exercise to show that \(T_{2}\) is a complete theory in \(L_{2}\).

Let \(L\) be a language. A class \(\mathcal{C}\) of \(L\)-structures is closed under ultraproducts if for every set \(I\), for every ultrafilter \(\mathcal{U}\) on \(I\), and for every sequence \(\left(M_{i}: i \in I\right)\) of structures from \(\mathcal{C}, \prod_{i \in I} \mathcal{M}_{i} / \mathcal{U}\) belongs to \(\mathcal{C}\). Let \(T\) be a theory in \(L\). Show that the class of \(L\)-structures that are not models of \(T\) is closed under ultraproducts if and only if \(T\) is finitely axiomatizable.

Let \(\langle G, \cdot e\rangle\) be a group. With this group, we associate a first-order language \(L_{G}\) that has a unary function symbol \(f_{\alpha}\) for every element \(\alpha \in G\). Let \(T\) denote the following theory of \(L_{G}\) : $$ \begin{gathered} \left\\{\forall v_{0} f_{e} v_{0} \simeq v_{0}\right\\} \\ \cup\left\\{\forall v_{0} f_{\alpha} f_{\beta} v_{0} \simeq f_{\alpha \beta} v_{0}: \alpha \in G \text { and } \beta \in G\right\\} \\ \cup\left\\{\forall v_{0} \neg f_{\alpha} v_{0} \simeq v_{0}: \alpha \in G \text { and } \alpha \neq e\right\\} . \end{gathered} $$ (a) Show that for every term \(t\) of \(L_{G}\), there exists an element \(\alpha \in G\) and a symbol \(x\) for a variable such that $$ T \vdash \forall x t \simeq f_{\alpha} x $$ (b) After observing that any atomic formula of \(L_{G}\) can involve at most two variables, show that, for every atomic formula \(F=F\left[v_{0}, v_{1}\right]\) of \(L_{G}\), one of the following three possibilities holds: \- \(T \vdash \forall v_{0} \forall v_{1} F\); \- \(T \vdash \forall v_{0} \forall v_{1} \neg F\); \- there exists an element \(\alpha \in G\) such that \(T \vdash \forall v_{0} \forall v_{1}\left(F \Leftrightarrow v_{0} \simeq f_{\alpha} v_{1}\right)\). (c) Let \(\mathcal{G}\) be the \(L\)-structure whose base set is \(G\) and in which, for every \(\alpha \in G\), the symbol \(f_{\alpha}\) is interpreted by the map \(\beta \mapsto \alpha \cdot \beta\) from \(G\) into \(G\) (i.e. left multiplication by \(\alpha\) ). Show that \(\mathcal{G}\) is a model of \(T\). (d) Let \(\mathcal{M}=\left\langle M,\left(\phi_{\alpha}\right)_{\alpha \in G}\right\rangle\) be a model of \(T\) and let \(a\) be an element of \(M\). Consider the set \(O(a)=\left\\{x \in M:\right.\) there exists \(\alpha \in G\) such that \(\left.x=\phi_{\alpha}(a)\right\\}\) Show that \(O(a)\) is a substructure of \(\mathcal{M}\) that is isomorphic to \(\mathcal{G}\). Show that \(X_{M}=\\{O(a): a \in M\\}\) is a partition of \(M\). Show that if \(\mathcal{M}\) and \(\mathcal{N}\) are two models of \(T\) and if \(X_{M}\) and \(X_{N}\) are equipotent, then \(\mathcal{M}\) and \(\mathcal{N}\) are isomorphic. (e) Show that if \(G\) is infinite, then the theory \(T\) is complete. (f) Suppose that \(G\) is finite. Does there exist an infinite cardinal \(\lambda\) such that \(T\) is \(\lambda\)-categorical? Is the theory \(T\) complete?
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