Question

Let \(\mathbf{D}=\mathbf{R}[X] / X^{2}\) be the ring of dual numbers. D has a unique maximal ideal, generated by \(X\). Consider a Kripke model with two nodes \(k_{0}, k_{1} ; k_{0}

Short answer

The model is an intuitionistic field because \(\mathbf{D}\) behaves like a field in \(\mathbf{R}\) under the ordering, and apartness is defined by differing real parts.

Step-by-step solution

Understand the Kripke Model Structure

The Kripke model has two nodes, \(k_0\) and \(k_1\), with \(k_0 < k_1\), which means information at \(k_0\) can be accessed at \(k_1\) but not vice versa. At \(k_0\), the structure is \(\mathbf{D} = \mathbf{R}[X] / X^2\), the ring of dual numbers, while at \(k_1\), the structure is \(\mathbf{R}\), the field of real numbers.

Identify the Structure at Each Node

For \(k_0\), the structure \(\mathfrak{A}(k_0)\) is the ring of dual numbers \(\mathbf{D}\), which consists of elements of the form \(a + bX\), where \(a, b \in \mathbf{R}\) and \(X^2 = 0\). For \(k_1\), \(\mathscr{A}(k_1) = \mathbf{R}\), the structure is simply the real numbers.

Determine the Canonical Map

The canonical map \(f : \mathbf{D} \to \mathbf{R}\) is defined by \(f(a + bX) = a\). This map essentially extracts the real part of a dual number, ignoring the part that involves \(X\).

Check the Model Conditions for an Intuitionistic Field

In an intuitionistic field within a Kripke model, each node must be equipped with a local field structure that is consistent with the ordering \(k_0 < k_1\). At \(k_0\), the ring \(\mathbf{D}\) behaves like a field because the only maximal ideal is generated by \(X\). At \(k_1\), the real numbers already form a field, meeting the requirements.

Define the Apartness Relation

Apartness of two elements is a constructive way of saying they are distinct. In this Kripke model, for two elements \(a + bX\) and \(c + dX\) in \(\mathbf{D}\), they are apart if their real parts are different, i.e., \(a eq c\). This relation is consistent in both \(k_0\) and \(k_1\) through the map \(f\), which ensures that apartness is preserved as \(k_0 < k_1\).

Key concepts

Kripke Model

In the realm of intuitionistic logic, a Kripke model is a fundamental concept. It is used to evaluate the truth of a proposition across various possible worlds or nodes. In a Kripke model:

• Nodes represent different states or worlds.
• There is a relation of accessibility between nodes, typically written as \(k_0 < k_1\) to denote that \(k_1\) accesses information from \(k_0\).
• At each node, specific structures, such as algebraic or logical constructs, are defined.

In this exercise, the Kripke model consists of two nodes, \(k_0\) and \(k_1\), with \(k_0 < k_1\). This notion of accessibility is crucial. It implies that notions or values defined at \(k_0\) can influence or be accessed at \(k_1\), but not vice versa. At node \(k_0\), we have the ring of dual numbers \(\mathbf{D}\), while at node \(k_1\), the simpler structure of real numbers \(\mathbf{R}\) is used.

Dual Numbers

Dual numbers are a fascinating mathematical construct. They extend the real numbers much like complex numbers but with a twist. A dual number has the form \(a + bX\), where \(a\) and \(b\) are real numbers and \(X\) is a special element such that \(X^2 = 0\). Here’s what makes them particularly interesting:

• The term \(bX\) behaves like an infinitesimal part since its square is zero, highlighting jumps or small perturbations when applied in various contexts like calculus.
• They form a ring, not precisely a field, since not all non-zero elements have a multiplicative inverse.

In the exercise, the structure at node \(k_0\) is the ring of dual numbers \(\mathbf{D} = \mathbf{R}[X] / X^2\). This represents a localized field-like behavior at this node as it is augmented by the maximal ideal generated by \(X\).

Apartness Relation

In intuitionistic logic, rather than merely stating two elements are different, a stronger and more constructive approach is to define an apartness relation. This relation asserts a verifiable distinction between two entities. Consider elements \(a + bX\) and \(c + dX\) in the ring of dual numbers:

• They are apart if the real parts differ, i.e., \(a eq c\).
• This ensures a constructive method to distinguish between elements, crucial in many logic and computational applications.

In our Kripke model, even though we move from \(k_0\) to \(k_1\), this apartness relation is maintained. It is consistent because of the canonical map that essentially preserves the real part of any dual number element.

Maximal Ideal

A maximal ideal in a ring is an ideal that is as large as possible without being the whole ring. It is fundamental in understanding the structure of rings and fields. Here's how it works:

• If you take the quotient of a ring by a maximal ideal, what remains is a field.
• In the dual number ring \(\mathbf{D}\), the ideal generated by \(X\) (all multiples of \(X\), such as \(bX\)) is maximal.

Thus, in structural terms, \(\mathbf{D}/(X)\) behaves as a field. This ensures that \(k_0\) maintains a field-like behavior in the Kripke model, even as it interacts with the real numbers in \(k_1\). Understanding this concept emphasizes the flexibility and utility of the Kripke model in representing different mathematical structures with both ring and field properties.