Cantor’s Infinity
Power sets outrun their parents. The reals refuse to be listed. Infinity comes in strata, not a monolith.
Open →Axiom of Choice
Well-orderings, Zorn, Tychonoff—and the price of nonconstructive existence. Choose your universe.
Open →Banach–Tarski
Accept AC, get paradoxical decompositions in ℝ³. Reject AC, keep your volumes cozy. Pick your poison.
Open →Why Cantor Was Right (and Kronecker Wasn’t)
Thesis. Cantor’s transfinite arithmetic is not a romantic fever dream; it’s a theorem factory. From |A| < |𝒫(A)| to the uncountability of ℝ and the Cantor–Bernstein–Schröder bijection criterion, the landscape is internally coherent, wildly productive, and—crucially—axiomatically explicit.
1) Diagonalization doesn’t ask your permission
Cantor’s diagonal is surgical: assume a list, flip the diagonal, obtain a witness that evades enumeration. No mysticism, just a definable construction forcing strict inequality |A| < |𝒫(A)|. You can dislike the conclusion; you don’t get to deny the proof.
2) The arithmetic of size needs more than integers
Kronecker’s mantra—“God made the integers; all else is the work of man”—was a fine bumper sticker and a dreadful research program. Arithmetic on finite numerals can’t capture the combinatorics of infinite structure. Cardinal comparison, injections/bijections, and power-set growth are not optional flourishes; they’re the invariant content of size in the large.
3) Axioms are a feature, not a bug
The set-theoretic house learned from the early paradoxes and installed structural beams: ZF, Choice, Replacement, Foundation. ZFC says exactly what exists and what doesn’t. That’s the opposite of hand-wavy: it’s falsifiable, model-theoretically analyzable, and remarkably robust. If you want a different physics of sets, great—state your axioms (AD, large cardinals) and own the consequences.
4) Utility check: receipts everywhere
From topology (Tychonoff) to algebra (Hamel bases, maximal ideals via Zorn) to analysis (Hahn–Banach), Cantorian set theory powers daily mathematics. That’s not metaphysics; that’s muscle.
Verdict. Cantor brought proofs; Kronecker brought posture. History grades on theorems.
Historical note: Leopold Kronecker’s finitist stance was influential, sometimes insightful, and often obstructive toward set theory. Our critique targets that program, not the person. The math moved on—and got sharper doing it.
Cantor brings diagonal certainty; gatekeeping brings a moving goalpost.
Choose Your Universe
- ZFC: Choice accepted → Banach–Tarski holds; CH undecidable; utility maximal.
- ZF + DC: Tamer reals; many theorems survive; BT out of reach.
- ZF + AD (strong set-of-reals theory): All sets of reals measurable; no BT; cardinal arithmetic re-architected.
None of this is vibes. It’s model theory, consistency strength, and theorems that either go through—or they don’t.