Question

Consider the class \(\mathcal{K}\) of all structures of type \(\langle 1 ;-; 0\rangle\) with a denumerable unary relation. Show that any \(\mathfrak{A}\) and \(\mathfrak{B}\) in \(\mathcal{K}\) of the same cardinality \(\kappa>\aleph_{0}\) are isomorphic. Show that \(T=T h(\mathcal{K})\) is not \(\kappa\)-categorical for any \(\kappa \geq N_{0}\).

Short answer

Structures in \( \mathcal{K} \) of same cardinality \( \kappa > \aleph_0 \) are isomorphic; \( T \) is not \( \kappa \)-categorical for \( \kappa \geq N_0 \).

Step-by-step solution

Understand the Structure Type

The structure type is given as \( \langle 1 ;-; 0 \rangle \), meaning it consists of a single unary relation. The class \( \mathcal{K} \) includes all structures of this type, each with a countably infinite number of elements.

Define the Condition for Isomorphism

For two structures \( \mathfrak{A} \) and \( \mathfrak{B} \) to be isomorphic, they must have a bijection between their domains that preserves the unary relation. Given \( \kappa > \aleph_0 \), both structures have an uncountably infinite cardinality.

Apply Löwenheim–Skolem Theorem

The Löwenheim–Skolem theorem suggests that for a theory with infinite models, if the sentence is satisfiable, then it has models of all infinite cardinalities. Therefore, since \( \mathfrak{A} \) and \( \mathfrak{B} \) have the same cardinality \( \kappa \), they can be constructed to be isomorphic by choosing the same set for the isomorphic mapping.

Examine \( \kappa \)-Categoricity

A theory is \( \kappa \)-categorical if all models of size \( \kappa \) are isomorphic. Since \( T = Th(\mathcal{K}) \) can have models with different cardinalities greater than \( \aleph_0 \), it's not \( \kappa \)-categorical for any \( \kappa \geq N_0 \), as there can exist non-isomorphic models with the same cardinality.

Key concepts

Isomorphism

In mathematical logic, an isomorphism between two structures is a special kind of mapping or bijection. For two structures, \( \mathfrak{A} \) and \( \mathfrak{B} \), to be isomorphic, there must be a one-to-one and onto function between their sets of elements. This function must also preserve the structure, meaning it respects the relations and operations of the original and target structures.

When it comes to unary relations, like in our class \( \mathcal{K} \), this bijection must map related elements in \( \mathfrak{A} \) to related elements in \( \mathfrak{B} \). For structures with the same infinite cardinality \( \kappa > \aleph_{0} \), it's possible to construct such an isomorphism.

• Bijection: a one-to-one, onto mapping.
• Preserves Structure: matching relations in both structures.
• Same Cardinality: ensures comparability for isomorphism.

"Learning how to identify and create these mappings is crucial in determining isomorphism between complex structures."

Löwenheim–Skolem Theorem

The Löwenheim–Skolem theorem is a fundamental result in model theory, a branch of mathematical logic dealing with the relationship between formal languages and their interpretations or models.

• **Downward Löwenheim–Skolem Theorem:** If a theory has an infinite model, it has models of every cardinality at least \( \aleph_{0} \).
• **Upward Löwenheim–Skolem Theorem:** If a theory has an infinite model of some uncountable cardinality, it has models of every larger cardinality.

For theory \( T \), if it includes infinite models, the theorem assures us that models can be adjusted or "downsized" or "upsized" to any desired larger infinite size. This plays a crucial role in our exercise, guaranteeing that structures \( \mathfrak{A} \) and \( \mathfrak{B} \) of the same uncountably infinite cardinality can indeed be isomorphic if we carefully construct them appropriately.

Understanding this theorem aids in grasping how logical theories adapt to different infinite sizes, providing insights into the flexible nature of mathematical structures.

Categoricity

Categoricity is a property of a theory in logic that describes how unique its models are at a certain cardinality.

A theory is called \( \kappa \)-categorical if all models of the same cardinality \( \kappa \) are isomorphic. This is a strong condition, essentially stating that the theory "predicts" only one type of interpretation for that size. However, not all infinite structures can adhere to this property.

In our problem, the theory \( T = Th(\mathcal{K}) \) displays non-\( \kappa \)-categoricity for \( \kappa \geq N_0 \), meaning models of the same infinite size but larger than countable numbers may not all be isomorphic. This results in variations or "non-uniformity" in possible models.

• Non-\( \kappa \)-categorical: more than one model possible of a given infinite size.
• Affects the consistency and predictability of model structure.
• Helps in understanding the diversity among models for advanced structures.

"By exploring \( \kappa \)-categoricity, students can appreciate the depth of logical frameworks and the complexity of structures they describe."