8.4 Ordinal and Cardinal Arithmetic

Ordinal numbers describe order and position, while cardinal numbers measure set size. Both extend to infinite sets, introducing fascinating concepts like transfinite ordinals and aleph notation for infinite cardinals.

Arithmetic operations on ordinals and cardinals have unique properties, often differing from familiar finite arithmetic. These concepts are crucial for comparing infinite sets and solving cardinality problems in set theory and beyond.

Ordinal Numbers and Arithmetic

Ordinal numbers and operations

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  alternative proof - Definition of Ordinal (w/ Axiom of Regularity) (problem 37, page 208 ... View original

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• Ordinal numbers describe order or position represent well-ordered sets (1st, 2nd, 3rd)
• Comparison of ordinals uses order relations ($<$, $>$, $=$) exhibit transitive property (if a < b and b < c, then a < c)
• Arithmetic operations include addition, multiplication, and exponentiation
• Ordinal arithmetic properties non-commutative (a + b ≠ b + a), associative ((a + b) + c = a + (b + c)), and distributive on right side only (a * (b + c) = a * b + a * c)

Arithmetic of transfinite ordinals

• Transfinite ordinals extend beyond finite numbers ($\omega$, $\omega + 1$, $\omega \cdot 2$)
• Addition of transfinite ordinals: $\omega + n = \omega$ for finite n, $\omega + \omega = \omega \cdot 2$
• Multiplication: $\omega \cdot n = \omega$ for finite n > 0, $\omega \cdot \omega = \omega^2$
• Exponentiation: $\omega^n$ for finite n, $\omega^\omega$ represents larger transfinite ordinal

Cardinal Numbers and Arithmetic

Cardinal numbers and aleph notation

• Cardinal numbers measure set size relate to ordinals (|{a, b, c}| = 3)
• Aleph notation represents infinite cardinals: $\aleph_0$ (natural numbers), $\aleph_1$ (next larger infinite cardinal)
• Cardinal arithmetic operations include addition, multiplication, and exponentiation
• Cantor's theorem states power set has strictly larger cardinality (|P(A)| > |A|)

Properties of cardinal arithmetic

• Absorption property for infinite cardinals: $\kappa + \kappa = \kappa \cdot \kappa = \kappa$
• Non-commutativity of cardinal multiplication: $2 \cdot \aleph_0 \neq \aleph_0 \cdot 2$
• Other properties include associativity and distributivity

Applications of ordinal and cardinal arithmetic

• Compare cardinalities of infinite sets using bijective functions and Cantor-Schröder-Bernstein theorem
• Solve cardinality problems distinguishing countable vs. uncountable sets (ℚ vs. ℝ)
• Apply in set theory: continuum hypothesis, generalized continuum hypothesis
• Real-world applications include infinite data structures in computer science and analysis of infinite-dimensional spaces in mathematics