Question

Show that well-ordering is not a first-order notion. Suppose that \(\Gamma\) axiomatises the class of well-orderings. Add countably many constants \(c_{i}\) and show that \(\Gamma \cup\left\\{c_{i+1}

Short answer

Well-ordering is not first-order as it allows a model with an infinite descending sequence.

Step-by-step solution

Understanding the Context

The problem involves showing that well-ordering is not expressible in first-order logic. We are given that \( \Gamma \) axiomatizes the class of well-orderings and have added countably many constants \( c_i \), where each constant has a successor that is lesser, i.e., \( c_{i+1} < c_i \). This setting does not align with well-ordering which states every non-empty subset must have a least element.

Analyzing the Given Condition

The condition \( \Gamma \cup \{c_{i+1} < c_i \mid i \in \mathcal{N}\}\) suggests an infinite descending sequence, as each constant is defined to be less than the previous one. In a well-order, there should be no infinite descending sequence, which implies that this setup violates the well-ordering principle.

Constructing a Model

Despite violating well-ordering, in first-order logic, a model can be constructed for \( \Gamma \cup \{c_{i+1} < c_i \mid i \in \mathcal{N}\}\) by interpreting the countably infinite sequence of constants \( c_i \) to form a structure resembling the integers with reversed order, e.g., \( \.\.\. , c_3, c_2, c_1 \). This model satisfies each clause \( c_{i+1} < c_i \), demonstrating the constructible model.

Drawing the Conclusion

Since a model can be constructed for \( \Gamma \cup \{c_{i+1} < c_i \mid i \in \mathcal{N}\}\), which demonstrates an infinite descending sequence, it indicates that the first-order theory \( \Gamma \) cannot adequately express the concept of well-ordering. Thus, well-ordering is not a first-order expressible notion.

Key concepts

first-order logic

First-order logic is a powerful and widely used system in mathematical logic. It is targeted at expressing statements about objects and their relationships. In essence, it allows for the construction of logical sentences using quantifiers, like "for all" (denoted as \( \forall \)) and "there exists" (denoted as \( \exists \)). First-order logic is often involved when trying to define mathematical structures, such as groups, rings, or fields. A critical constraint, however, is that it cannot express properties involving infinitely many elements in a single statement. For example, expressing the well-ordering of the natural numbers is problematic for first-order logic because it cannot encapsulate the idea of an infinite descending sequence. In this context, our exercise finds that while we can build a first-order logical model with certain properties, it fails to uphold the essence of well-ordering due to this limitation.

axiomatization

Axiomatization refers to the process of defining a system entirely based on a set of axioms or base principles. This set forms the foundation of logical reasoning within a specific domain. In mathematics and logic, axiomatizing means taking fundamental truths or propositions, and deducing further truths from them.In our exercise, the set \( \Gamma \) is said to axiomatize the notion of well-orderings. This means \( \Gamma \) contains foundational rules intended to define and encapsulate the properties of well-orderings. However, despite this axiomatic basis, when additional constructs are introduced—such as countably many constants \( c_i \) in an infinite descending order—this axiomatization fails to sustain the definition of well-ordering, which is critical for it being considered as such.Axiomatization is crucial in logic and mathematics for defining new structures and systems. But as seen in our exercise, axiom sets in first-order logic may encounter limitations in expressing complex infinite properties.

model theory

Model theory is an area of mathematical logic focused on studying structures (models) that satisfy the axioms of a given theory. It looks at the relationship between formal languages and their interpretations, or models. Essentially, model theory asks what kinds of structures satisfy specific sets of sentences.In the exercise given, we engage with model theory by demonstrating a possible structure where the statements \( c_{i+1} < c_i \) hold true for all \( i \). By doing so, we've created a model resembling a reverse ordering of the integers. This satisfies the requirements of our logical sentences without capturing the true nature of a well-order.This practical construction highlights an essential point in model theory: even though a model satisfies specific sentences of a logical theory, it does not guarantee that the model captures all intended properties—such as the nonexistence of an infinite descending sequence in a well-order.

infinite descending sequence

An infinite descending sequence is a chain of elements arranged such that each subsequent element is less than the previous one. In the context of well-orderings, such a sequence should not exist. Well-ordering demands that every non-empty set should have a least element.In our mathematical exploration, we construct a set of constants \( c_i \) such that \( c_{i+1} < c_i \), forming an infinite descending sequence. This directly conflicts with the property of well-ordering because a well-order cannot have any infinite descending chain. The formation of this sequence in the model demonstrates the inadequacy of first-order logic to describe well-ordering completely. By showcasing this sequence, it becomes clear why well-ordering extends beyond the expressive capacity of first-order logic, as it inherently disallows these sequences.