Upward Löwenheim–Skolem Theorem

Overview

The upward Löwenheim–Skolem theorem is a fundamental result in mathematical logic, specifically in the field of model theory. It addresses the existence of models of varying infinite sizes for a given theory.

Key Concepts

The theorem asserts that if a first-order theory possesses at least one infinite model, then for any infinite cardinality $\kappa$, the theory also possesses a model of cardinality $\kappa$. This means that once a theory can be satisfied by an infinite structure, it can be satisfied by infinitely many different sizes of infinite structures.

Deep Dive

The proof often involves the construction of new models using techniques like the method of diagrams or ultraproducts. It demonstrates a crucial limitation of first-order logic: its inability to characterize infinite structures up to isomorphism by specifying their cardinality. For instance, a theory that has a countably infinite model cannot have a model that is *only* uncountably infinite, and vice-versa, if it has an uncountably infinite model.

Implications

The upward Löwenheim–Skolem theorem has significant consequences:

• It shows that first-order logic is not categorical in infinite languages.
• It implies that no first-order theory can uniquely characterize the natural numbers or the real numbers up to isomorphism, as these have specific infinite cardinalities.

Challenges and Misconceptions

A common misconception is that the theorem implies all infinite models of a theory are isomorphic. This is false. The theorem guarantees the existence of models of different cardinalities, not that these models are structurally identical beyond what the theory dictates.

FAQs

What does it mean for a theory to have a model?
A theory has a model if there exists a structure (a set with relations and functions) in which all the sentences of the theory are true.

How does this differ from the downward Löwenheim–Skolem theorem?
The downward version states that if a theory has an infinite model, it also has models of all smaller infinite cardinalities. The upward version deals with arbitrarily large infinite cardinalities.