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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 ›

7.4 Arithmetic of Cardinals

Cardinal arithmetic studies how the sizes of sets behave under natural set theoretic operations. Addition corresponds to disjoint union, multiplication corresponds to Cartesian product, and exponentiation corresponds to sets of functions. For finite sets these operations agree with ordinary arithmetic, but for infinite sets they have behavior that is at first surprising and later becomes one of the central tools of set theory. Cardinal Numbers A cardinal number is a...
logic set theory cardinals cardinal arithmetic infinity 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