Ultrafinitism argues infinity is unobservable and therefore suspect, and that calculus, combinatorics, and computation already work fine without it.
Key Takeaways
Doron Zeilberger (Rutgers, combinatorics) contends infinity can be eliminated from math entirely with no real loss; computers already handle math with finite digit allowances.
Skewes’ number (e^e^e^79) has never been written in decimal form; ultrafinitists question whether unwritable values qualify as numbers at all.
Alexander Esenin-Volpin formalized the position in the 1960s-70s: numbers exist only if constructible within real resource limits like time, memory, or proof length.
Zermelo-Fraenkel set theory embeds actual infinity as a foundational default; ultrafinitism challenges that root axiom, not just edge cases.
Edward Nelson’s 1976 attempt to rebuild arithmetic without infinity axioms revealed the resulting system is remarkably weak, exposing how load-bearing the infinite assumption is.
