Ordinal numbers, transfinite induction, and the Continuum Hypothesis
Counting does not stop at infinity. Past every natural number comes ω, then ω+1, and a ladder of larger infinities, topped by a question the standard axioms cannot answer.
Step forward and back along the ordinal line from 0 through ω, ω+1, ω·2, and ω². Compare 1 + ω with ω + 1 and notice they differ.
Step through ordinals from 0 to omega squared
Beyond finite numbers lie the ordinal numbers, which extend the concept of "position in a sequence" into the transfinite. While cardinals measure how many, ordinals measure what order -- and the two diverge dramatically in the infinite realm.
Key insight: Ordinal arithmetic is not commutative: 1 + ω = ω but ω + 1 ≠ ω. The first adds one at the start (absorbed by the infinite tail), while the second adds one at the end (creating a new position).
Press Start Induction or Step to prove the property one ordinal at a time. Watch what happens at ω and the other limits, which have no predecessor.
Proving a property P holds for all ordinals up to omega^2
Key insight: Transfinite induction requires three cases: the base case (zero), the successor case (α → α+1), and the limit case (handling ordinals like ω that are not successors). The limit case is what makes it genuinely new.
Click through the timeline from Cantor to Cohen, then drag the slider to imagine how many cardinals could sit between the countable and the continuum if CH fails.
Is there a cardinality between the naturals and the reals?
The remarkable result:
CH is independent of ZFC. Both "CH is true" and "CH is false" are consistent with the standard axioms of set theory. No proof or disproof is possible from ZFC alone. This is not a limitation of our techniques -- it is a fundamental feature of the axiom system.
Cantor's theorem: |P(S)| > |S| for any set S. So 2^Aleph-0 > Aleph-0. The question is whether 2^Aleph-0 = Aleph-1 (the next cardinal) or something larger. ZFC cannot decide.
The capstone question of set theory is the Continuum Hypothesis: is there an infinity between the countable and the continuum? Remarkably, this question is independent of ZFC -- it can be neither proved nor disproved from the standard axioms.
Key insight: The Continuum Hypothesis (CH) states that there is no set whose cardinality is strictly between that of the integers and the reals. Godel (1940) showed CH is consistent with ZFC; Cohen (1963) showed its negation is also consistent. CH is genuinely undecidable in ZFC.
Spot a mistake or have an idea?