Löwenheim–Skolem Theorem

Overview

The Löwenheim–Skolem theorem is a cornerstone of model theory. It addresses the relationship between the cardinality of a formal theory and the cardinalities of its models. Essentially, it states that if a theory has an infinite model, it must have models of every infinite cardinality. This has profound implications for the expressive power of first-order logic.

Key Concepts

• Countable Theory: A theory whose axioms and logical symbols are from a countable alphabet.
• Infinite Model: A structure that satisfies the theory and is infinite.
• Cardinality: The size of a set (e.g., finite, countably infinite, uncountably infinite).

Deep Dive

The theorem implies that first-order logic cannot uniquely determine the size of an infinite model. For any countable first-order theory with an infinite model, there exist models of cardinality $\aleph_0$ (countably infinite), $\aleph_1$, $\aleph_2$, and so on, up to any infinite cardinality.

This leads to the existence of non-standard models. For example, in the theory of arithmetic, if there is a standard model (like the natural numbers), there must also be non-standard models of arithmetic that are elementarily equivalent but have different cardinalities.

Applications

The Löwenheim–Skolem theorem is crucial in understanding the limitations of formal systems. It shows that no first-order theory can fully capture the intended structure of infinite sets like the real numbers or the natural numbers by specifying their exact cardinality.

Challenges & Misconceptions

A common misconception is that the theorem implies all infinite models of a theory are isomorphic. This is false. While they share the same elementary properties, they can differ significantly in their structure and size.

FAQs

What does the theorem say about finite models?
The theorem specifically applies to theories with infinite models. It does not restrict the number or cardinality of finite models a theory might have.

Why is it important for first-order logic?
It highlights that first-order logic is not categorical in infinite domains. This means theories like Peano Arithmetic cannot have a unique model up to isomorphism.