BusyBeaver(6) Is Quite Large

Original Article ↗ Hacker News Discussion ↗

Scaling, tetration, and “rounding away” the universe

  • Commenters unpack why dividing a gargantuan tetration like 10↑↑10,000,000 by “grains per universe” barely changes it in that notation: adjacent tetration values differ by numbers like “(previous value) to the 10th power,” so any fixed factor (even ~10²⁰⁰) is negligible.
  • Others push back that even if ratios are tiny in that notation, they can matter conceptually (e.g., universes almost empty vs packed with sand); “differences never matter” is framed as context-dependent, not absolute.

Capacity of the observable universe

  • Several responses argue the observable universe cannot store BB(6) in full: information bounds (via Bekenstein) give ~10¹²⁰ bits, vastly below what’s needed even for much smaller towers.
  • Some nuance appears about whether “writing down” requires simultaneous existence of all digits and how relativity and infinite time might complicate that, but the consensus is still “no” in any reasonable physical sense.

Independence from ZFC and nonstandard models

  • A long subthread clarifies: the integer BB(748) exists (in classical math), but ZFC cannot prove which integer it is, because certain 748‑state machines are constructed to halt iff ZFC is inconsistent.
  • Independence is explained via models: there can be models of ZFC where a machine “halts after Q steps,” with Q a nonstandard “finite” number larger than any standard integer; in the standard model that machine never halts.
  • This leads to discussion of how first‑order Peano arithmetic and ZFC cannot uniquely pin down the standard natural numbers, and why “finite” is model-dependent in first-order logic.

Numbers vs computability and ultrafinitism

  • Some argue BB(748) is “just a concept,” not really a number in any usable sense, because no algorithm can decide inequalities involving it; constructive and ultrafinitist perspectives are invoked.
  • Others insist it is exactly as much a number as 12: there is a specific finite integer; we just can’t feasibly know or prove which one within ZFC.
  • Distinctions are drawn between:
    • BB as a total function ℕ→ℕ versus its noncomputability in general.
    • Individual finite integers (always “computable” in the trivial sense “print n”) versus uncomputable reals.
  • Several people stress the mismatch between “definable in classical logic” and “what we usually mean by number” in more practical or constructive contexts.

Gödel, halting, and the role of axioms

  • Multiple comments connect BB to Gödel’s incompleteness and the halting problem:
    • If BB were computable, the halting problem would be decidable.
    • Specific machines are built that halt iff “ZFC is inconsistent,” so ZFC cannot prove whether they halt if it is consistent.
  • There is debate over whether mathematics should have pursued richer axiom systems more aggressively post‑Gödel, versus the practical fact that most everyday mathematics fits into very weak systems (e.g., elementary arithmetic).
  • Second‑order PA and reverse mathematics are mentioned to illustrate that:
    • ZFC is “overkill” for most of math, yet
    • Still inadequate to settle many natural independence phenomena (e.g., large BB values, continuum hypothesis).

Busy Beaver growth vs other huge numbers

  • Commenters compare BB(6) and hypothetical BB(7) to famous large numbers like Graham’s number, speculating that BB(7) might exceed it.
  • There is back‑and‑forth over what “grows faster than any computable sequence” really means:
    • BB eventually dominates any fixed computable function of n, but this doesn’t straightforwardly bound specific small values like BB(7) versus particular fast‑growing constructions.
  • Related work on “functional” Busy Beaver (lambda terms) is cited to show surprisingly strong behavior in small encodings.

Meaning and communication of enormous numbers

  • Several people note that physical analogies (grains of sand, universes) fail to genuinely “visualize” these magnitudes; past a point these numbers encode consistency strength and logical power more than intuitive quantities.
  • Others comment on the article’s density and target audience: the BB(6) post is seen as aimed at readers with at least undergraduate‑level theory, blending expository writing with cutting‑edge results.
  • There is minor meta‑discussion about who is “on the forefront” of Busy Beaver research versus who is popularizing it, and links to bbchallenge.org and specialized blogs for deeper technical details.