Question

Let \(L\) have the binary predicate symbol \(P, \sigma:=\forall x \neg P(x, x) \wedge \forall x y z(P(x, y) \wedge\) \(P(y, z) \rightarrow P(x, z)) \wedge \forall x \exists y P(x, y) .\) Show that Mod \((\sigma)\) contains only infinite models.

Short answer

Mod(\(\sigma\)) contains only infinite models due to irreflexivity, transitivity, and requirement for perpetual extension.

Step-by-step solution

Understand the Predicate and Its Components

The given predicate \(\sigma\) is divided into three parts: \(\forall x eg P(x, x)\) which means no element is related to itself, \(\forall x y z(P(x, y) \wedge P(y, z) \rightarrow P(x, z))\) which establishes the transitivity of the relation, and \(\forall x \exists y P(x, y)\) which states that every element is related to some other element.

Explore the Meaning of Each Component

The first part, \(\forall x eg P(x, x)\), implies the relation \(P\) is irreflexive. The second part indicates that \(P\) is transitive, meaning that if an element \(a\) is related to \(b\) and \(b\) to \(c\), then \(a\) is directly related to \(c\). The third part assures that for every element \(x\), there is at least one other element \(y\) such that \(P(x, y)\).

Realize the Implications as a Set of Relationships

The irreflexive nature combined with transitivity suggests a strict partial order, which is a directed graph with no loops. The requirement that every element relates to some other ensures no endpoints in the graph. If the model were finite, this would force a loop, contradicting irreflexivity.

Examine the Consequences of the Components Together

In finite models, a directed acyclic graph with no endpoints must ultimately form a cycle to satisfy \(\forall x \exists y P(x, y)\), but this would violate \(\forall x eg P(x, x)\). Infinite models, on the other hand, can maintain a constant forward extension without looping back, satisfying all parts of the predicate.

Conclude the Solution

Due to the stipulated conditions, every finite directed path must eventually close to meet the relational requirement, but closure introduces reflexivity somewhere in the operation, clashing with \(\forall x eg P(x, x)\). As such, only infinite graphs can entirely satisfy all conditions within \(\sigma\), implying that \(\text{Mod}(\sigma)\) contains only infinite models.

Key concepts

Predicate Logic

Predicate logic is a branch of logic that deals with predicates and quantified variables. It's a powerful tool for expressing statements about objects and their relationships in a formal way. In predicate logic, we use symbols like \(P(x, y)\), where \(P\) can represent a particular relation or property between elements \(x\) and \(y\). The expression is further refined using quantifiers:

• \(\forall\) (for all): This quantifier signifies that the statement applies to every element in the domain.
• \(\exists\) (there exists): This implies there is at least one element in the domain for which the statement holds true.

In our given predicate \(\sigma\), we have several components involving both \(\forall\) and \(\exists\) quantifiers, describing properties and relationships of elements within a set. Understanding these relationships helps us determine the nature of the model they describe, be it finite or infinite.

Transitivity

Transitivity is a fundamental property of certain relations. A relation is transitive if whenever it relates an element \(a\) to \(b\), and \(b\) to \(c\), it also directly relates \(a\) to \(c\). In mathematical terms, a relation \(R\) is transitive if: \[ \forall x, y, z, (R(x, y) \wedge R(y, z) \rightarrow R(x, z)).\]In the given predicate, we have a condition \(\forall x y z(P(x, y) \wedge P(y, z) \rightarrow P(x, z))\). This means the relation \(P\) is transitive. Transitivity, combined with other properties like irreflexivity, helps in forming structures similar to directed acyclic graphs, indicating directionality while avoiding cycles.

Irreflexivity

Irreflexivity is another key property in logic, particularly when dealing with relational models. A relation is irreflexive if no element is related to itself. Formally, this can be stated as:\[\forall x eg P(x, x)\]This implies that within the model, no self-loop is allowed for any element. The given predicate specifies that \(\forall x eg P(x, x)\), making the relation \(P\) irreflexive.Irreflexivity plays a crucial role in ensuring that relationships described by the model remain acyclic, which is essential in avoiding contradictions when combined with other properties like transitivity.

Infinite Models

Infinite models are crucial to logic, as they describe structures that cannot be finitely contained. In the context of the predicate \(\sigma\) given, which includes transitivity, irreflexivity, and the existence of at least one outbound relation for every element, only infinite models satisfy all conditions without contradiction.Why must these models be infinite?

• Irreflexivity mandates the absence of any self-referential relationship, preventing loops or cycles.
• Transitivity requires that relations can extend in a directed manner across elements.
• The condition \(\forall x \exists y P(x, y)\) ensures there are no endpoints, every element must relate to another element distinct from itself.

When attempted in finite space, these conditions would logically result in cycles to meet all relational requirements, which would inherently conflict with irreflexivity. Thus, only infinite paths can fulfill all these predicate requirements, ensuring direction, relation, and extension without contradiction.