Large Numbers and Infinity

Overview

This lecture explores the concept of very large numbers, the nature of infinity, cardinal and ordinal numbers, and how mathematics handles different sizes of infinity using set theory and axioms.

Large Numbers and Infinity

• Any finite number, no matter how large, can always be exceeded by adding one.
• Infinity is not a number but represents an unending quantity or process.
• Cardinal numbers count "how many" objects are in a set; ordinals describe the position or order of objects.

Cardinal Numbers and Set Theory

• The set of natural numbers (0, 1, 2, ...) is infinite, called "aleph-null" (ℵ₀), the smallest infinity.
• Two sets have the same cardinality if their elements can be paired one-to-one.
• The set of rational numbers (fractions) also has cardinality ℵ₀.
• The "power set" of any set is the set of all its subsets and always has a larger cardinality.
• The power set of the naturals, 2^ℵ₀, is strictly larger than ℵ₀.

Cantor's Diagonalization and Larger Infinities

• Cantor's diagonal argument shows some infinities (e.g., real numbers) are larger than ℵ₀.
• Repeated applications of the power set operation yield ever-larger infinities.

Ordinal Numbers and Order Types

• Ordinal numbers label positions in an ordered sequence; the first infinite ordinal is omega (ω).
• Omega + 1 is the next position after all naturals (omega), but not a larger cardinality.
• Arithmetic with infinite ordinals is not commutative: ω + 1 ≠ 1 + ω.

Axioms of Set Theory and Infinite Sets

• Axioms are assumptions in math; the "Axiom of Infinity" asserts the existence of an infinite set.
• The "Axiom of Replacement" lets us construct new sets by replacing each element in a set systematically.
• The process of creating larger ordinals and cardinals relies on these axioms.

Inaccessible Cardinals and Larger Infinities

• "Inaccessible cardinals" are infinities so large that they cannot be reached by operations on smaller sets.
• ℵ₀ is sometimes seen as inaccessible from finite numbers.
• Larger cardinals require additional axioms to be assumed into existence.

Open Problems and Philosophical Implications

• The Continuum Hypothesis asks whether there's a cardinality strictly between ℵ₀ and the power set of ℵ₀; its truth is still unresolved.
• Some infinities may be so large their existence in reality is questionable, but mathematically they can be assumed.

Key Terms & Definitions

• Cardinal number — A number expressing "how many" elements are in a set.
• Ordinal number — A number describing position in an ordered sequence.
• Aleph-null (ℵ₀) — The smallest infinity, cardinality of the natural numbers.
• Omega (ω) — The first transfinite (infinite) ordinal.
• Power set — The set of all subsets of a given set.
• Cantor's diagonal argument — A method demonstrating larger infinities exist.
• Axiom — A foundational assumption in mathematics.
• Inaccessible cardinal — A cardinal number unattainable from smaller sets by normal set operations.
• Continuum Hypothesis — The question of whether a set exists with cardinality strictly between ℵ₀ and the continuum.

Action Items / Next Steps

• Review Cantor's diagonal argument and its proof.
• Research the Continuum Hypothesis and its importance in set theory.
• Examine properties of ordinal arithmetic for further understanding.