Comparing Cardinals

Set Theory · Axiom Academy

Understanding the ordering and comparability of infinite cardinalities 1. Defining Cardinal Inequality We extend the notion of "less than or equal to" from natural numbers to arbitrary cardinals using injections. Why this definition? An injection from A to B means we can find a distinct element in B for each element of A. This captures the intuition that B is "at least as large" as A. |ℕ| ≤ |ℝ| because the identity map is an injection ℕ → ℝ | 1,2,3 | ≤ | a,b,c,d,e | via any injection from the first to second |(0,1)| ≤ |ℝ| via the identity inclusion map We can also define when one cardinal is strictly less than another, meaning the sets have genuinely different sizes. |A| ≤ |B| (there exists an injection A → B), AND |A| ≠ |B| (there is no bijection A → B) Cantor's Theorem: For any set A, we have |A| < |P(A)|. The power set is always strictly larger! |ℕ| < |ℝ| because ℕ ⊂ ℝ (injection exists) but ℝ is uncountable (no bijection) This gives us an infinite hierarchy: |ℕ| < |P(ℕ)| < |P(P(ℕ))| < ... A remarkable property of cardinals is that they can always be compared - but this requires the Axiom of Choice. This means the cardinals form a total order - any two can be compared. This is not obvious! For general partial orders, elements need not be comparable. Proof Sketch: The Axiom of Choice allows us to well-order any set. Once sets are well-ordered, we can compare their order types (ordinals), which induces a comparison on their cardinals. 4. The Well-Ordered Class of Cardinals

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