Understanding Models and Axiomatic Systems in Set Theory

Mathematics, often thought of as a repository of abstract truths, is deeply intertwined with various foundational theories, notably set theory. While set theory is one possible foundation for mathematics, it is not the only one. Through the lens of axiomatic systems and models, mathematicians have found ways to represent and understand complex mathematical objects. This article delves into the intricacies of axiomatic systems, models, and their relationship in the context of set theory, particularly focusing on ZFC (Zermelo-Fraenkel set theory with the axiom of choice).

The Role of Set Theory in Mathematics

Set theory, as the name suggests, deals with collections of objects and their properties. It has been implemented as a foundational system through which mathematical objects (such as numbers, functions, and geometric shapes) can be described using set-theoretic language. This approach can be likened to a compiler translating high-level code into efficient machine code. For instance, in the 19th century, mathematicians showed that the real numbers could be modeled using sets of rational numbers, and rational numbers could be described using equivalence classes of pairs of integers, with integers further defined as equivalence classes of pairs of natural numbers.

Axial Systems and Their Representations

Formalizing mathematical arguments often involves the use of first-order logic, where a collection of sentences can be expressed. Within this framework, a model is a set equipped with relations and functions corresponding to the primitive terms in the sentences. A model is said to satisfy a sentence if, after substituting the functions and relations in the model for the primitive ones, the sentence becomes true. This concept allows for the rigorous examination of mathematical statements by translating them into a more concrete, set-theoretic form.

The Difference Between Sets and Models

It is a common misconception that a model of ZFC (Zermelo-Fraenkel set theory with the axiom of choice) should be considered the universe of all sets. In reality, such models are not equivalent to the full universe of sets. The Lwenheim–Skolem theorem, a key result in model theory, illustrates this disparity. According to this theorem, any first-order theory with an infinite model has models of every infinite cardinality. This means that there are always other models of ZFC that are not the full universe of sets but are still satisfying subsets of it.

Lwenheim-Skolem Theorem and Its Implications

The Lwenheim–Skolem theorem is a cornerstone in understanding the relationship between axiomatic systems and their models. It implies that if a theory has an infinite model, it can be extended or reduced to models of any infinite cardinality. Even if the set of axioms is finite, the models can still be infinitely varied, as long as the number of constants, relations, or functions is infinite. This theorem also has the converse implication: if the set of axioms has a countably infinite model, then it also has a countably infinite model, as in the case of ZFC.

Conclusion and Further Reading

In summary, the relationship between models and axiomatic systems in set theory is a rich field of study. Through the Lwenheim–Skolem theorem, we see that while ZFC can be represented by models of different sizes, none of these can truly encompass the full universe of sets. This underscores the importance of understanding both the flexibility and limitations of these foundational systems. For further exploration, the theorem and its implications in the broader context of model theory are compelling topics for advanced study in mathematical logic.