Rigorous and revolutionary. Set theory, the proof that the reals are uncountable, and the hierarchy of infinities are airtight and foundational — modern mathematics is built on the ground Cantor cleared.
Rejected by his peers. Kronecker called it nonsense and blocked him; others called set theory a "disease." Cantor, who suffered recurring breakdowns, died in a sanatorium, undervalued — vindication came only after.
One question he couldn't close. Is there a size between ℵ₀ and the reals? His Continuum Hypothesis turned out undecidable — Gödel (1940) and Cohen (1963) proved it can be neither proved nor disproved from the standard axioms.
Picture an infinite list of infinite binary sequences — your claim that you've listed "all of them." Cantor builds a single new sequence by walking the diagonal: take row 1's first bit, row 2's second, row 3's third… and flip each one. The result differs from row 1 at position 1, from row 2 at position 2 — from every row at the one place that row was consulted. So it cannot be anywhere in your list. Your list was incomplete; and no list can be complete. Shuffle the rows and run it.
"The essence of mathematics lies precisely in its freedom." — Georg Cantor
The naturals, the integers, even the fractions — all the same size, because each can be put in a single endless queue, one-to-one with 1, 2, 3, …. The smallest infinity.
The reals, the points of a line, the infinite binary sequences — provably larger; no queue reaches them all (the diagonal escapes every one). A bigger infinity.
And it doesn't stop: the set of all subsets of any set is always strictly larger than the set itself (Cantor's theorem), so there is an endless tower of ever-greater infinities, ℵ₀ < ℵ₁ < ℵ₂ < …. The one question Cantor could not answer — is the continuum the very next infinity after ℵ₀, with nothing in between? — is the Continuum Hypothesis, and the twentieth century's answer was the strangest possible: it is independent of the standard axioms, true in some mathematical universes and false in others. The first great undecidable statement, and a direct ancestor of [[kurt-godel]]'s limits.
Gate kept on. The mathematics is secure — uncountability and the hierarchy of infinities are not opinions but proofs, and they reshaped every field that touches the infinite. What needs honesty is the reception and the open edge. In his lifetime Cantor was attacked with unusual cruelty: his former teacher Leopold Kronecker, who held that only the whole numbers were real, called the work humbug and worked to block his publications and career, and Cantor — a devout man who half-believed God had shown him these infinities — suffered repeated depressive breakdowns, dying in a sanatorium in 1918, malnourished during wartime, his worth still disputed. Hilbert's verdict came after: "No one shall expel us from the paradise that Cantor has created." And the Continuum Hypothesis remains the honest open door — not unsolved for want of effort, but provably undecidable from the usual axioms (Gödel showed it can't be disproved, Cohen that it can't be proved), the first place where mathematics found a true statement it had genuinely free choice about. The infinite has structure; some of that structure we get to decide.