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Maths › 03. Logic and Foundations ›

7.2 Cardinality and Countability

Cardinality is the part of set theory that compares the sizes of sets. For finite sets this agrees with ordinary counting, but for infinite sets the idea becomes more subtle, because two infinite sets may have the same size even when one appears to sit inside the other as a proper subset. Size by Correspondence The basic idea is that two sets have the same size when their elements can...
logic set theory cardinality countability infinity May 2, 2026
Maths › 03. Logic and Foundations ›

7. Basic Set Theory

Set theory provides a formal language for speaking about collections, membership, functions, relations, size, and order, and it serves as one of the standard foundations in which ordinary mathematical objects can be represented. The chapter begins with sets, relations, and functions, where the basic operations of mathematics are described in terms of membership and ordered pairs, giving a common framework for algebra, analysis, topology, and logic. Cardinality and countability are...
logic set theory cardinality ordinals zfc May 2, 2026
Maths › 03. Logic and Foundations ›

5.2 Lowenheim Skolem

The Lowenheim Skolem theorems describe how the size of models of a first order theory can be changed without affecting truth of sentences, and they show that if a theory has any infinite model, then it has models of many different infinite sizes, which reveals an important limitation of first order logic in controlling cardinality. Definition 5.10 (Cardinality of a Structure) Let $\mathcal{M}$ be a structure with underlying set $M$....
logic model theory lowenheim skolem cardinality May 2, 2026
Maths › 00. General Mathematics ›

01.4 Finite vs Infinite Objects

1.4 Finite vs Infinite Objects Mathematics distinguishes sharply between finite and infinite objects. Many definitions look similar in both settings, but their behavior diverges in essential ways. Understanding this boundary is necessary for reasoning, proof design, and algorithmic thinking. A set is finite if its elements can be listed completely and counted by a natural number. |A| = n for some n ∈ N A set is infinite if no...
mathematics foundations infinity cardinality structures May 2, 2026