Question

Consider the language of groups. \(T=\\{\sigma \mid \mathfrak{A} \models \sigma\\}\), where \(\mathfrak{A}\) is a fixed non-trivial group. Show that \(T\) is not a Henkin theory.

Short answer

\(T\) is not Henkin because it cannot provide witnesses for all existential statements.

Step-by-step solution

Understand the Definition of Henkin Theory

A theory is Henkin if every consistent set of sentences can be extended to a complete and consistent theory, and it contains witnesses for every existential quantifier.

Define the Language and Terms

In this context, the language of groups includes symbols for group operations (multiplication and inverse) and constants for the identity element. \(T\) is the set of all sentences \(\sigma\) satisfied by a specific non-trivial group \(\mathfrak{A}\).

Analyze the Non-trivial Group \(\mathfrak{A}\)

Let \(\mathfrak{A}\) be a non-trivial group. Non-trivial means \(\mathfrak{A}\) has elements other than the identity element, allowing for distinct elements to exist.

Construct a Consistent Set Involving Existential Statements

Consider a consistent set of sentences involving some existential statement that might not have a witness in the non-trivial group \(\mathfrak{A}\). For example, a sentence asserting the existence of an element of finite order other than the identity might lack such a witness in \(\mathfrak{A}\).

Show No Witness in \(\mathfrak{A}\)

If \(\mathfrak{A}\) lacks an element that satisfies those existential conditions, then not every existential statement can have a witness, violating a requirement of Henkin theories.

Conclude \(T\) is Not Henkin

As \(T\) fails to provide witnesses for all existentially quantified statements, it cannot be a Henkin theory.

Key concepts

Non-Trivial Groups

In mathematics, a group is a set equipped with an operation that combines two elements to form a third element while satisfying certain conditions, including the presence of an identity element, associativity, and the existence of inverses for every element. A non-trivial group is a group that contains elements other than just the identity element. This means that the group includes at least one other element which, through the group operation, interacts with other elements to produce distinct results.

For example, consider a group \( \mathfrak{A} \) that includes an identity element, say \( e \), and another element, \( g \), such that \( g eq e \). This already makes \( \mathfrak{A} \) non-trivial. Non-trivial groups are important because they allow for more complex structures and interactions, going beyond the simple outcomes found in trivial groups. This diversity in structure can help explore more extensive algebraic properties and theories.

Consistent Theories

A consistent theory in mathematics and logic is one where no contradictions exist within its set of statements. If you pick any two statements within the theory, you won't find them both true and false simultaneously. Consistency is crucial for ensuring that the theory holds logically and does not lead to any absurd conclusions.

In the context of Henkin theories and non-trivial groups, the focus is on whether the mathematical language used, referred to as a theory, can be consistently maintained when expanded to account for all possibilities. Specifically, when we talk about extending a set of statements to a complete theory, consistency ensures that we aren’t introducing contradictions as we move from a partial understanding to a fully-fledged theory.

Existential Quantifiers

Existential quantifiers are logical symbols that express the existence of a certain element meeting a given property. They are often denoted by the symbol \( \exists \) and are prevalent in mathematical statements and logical formulations.

For example, a statement might read: "There exists an element \( x \) such that \( x + 1 = 3 \)." Here, \( \exists \) indicates that within the context of our group or set, at least one element satisfies the given condition (here, \( x = 2 \)).

This concept becomes a pivotal point in examining whether a set of sentences forms a Henkin theory. For the theory to be Henkin, it is necessary that every existential statement has a witness within the group, meaning there is an actual element that fulfills the existential condition. When a group cannot provide such a witness, it reveals a limitation that disqualifies it from being a Henkin theory.

Language of Groups

The language of groups is a formal system the describes operations and elements within group theory. It encompasses the symbols that represent group operations, like multiplication, and specific elements like the identity element or constants. When discussing the language of groups, you're essentially talking about the syntax and structure that define and manipulate groups mathematically.

This language thus serves as the foundation for formulating theories and assertions about groups. In the given example within the original exercise, the symbolic language represents the collection of sentences satisfied by a specific group, denoted \( T=\{\sigma \mid \mathfrak{A} \models \sigma\}\).

Understanding the language allows us to develop theories that are internally consistent and explore the properties of groups in a structured and logical fashion. It is key to translating real-world mathematical concepts into a form that can be analyzed, studied, and expanded upon in theoretical mathematics.