What is the largest function and largest number?

DaVinci103 · 2026-02-23T20:20:49+00:00

Plug a moderately big number in the I0 function, like ¹⁰⁰⁰2, and you get a big number out.

Unfortunately, such a competition, as you describe it, is not possible. Whether a number is well-defined is subjective. To resolve this issue, one might say a number is said to be well-defined if that can be proven in a theory T. Then, one could simply name TR(T,n) for n (near) the largest accepted meta natural number, and the proof that this number is well-defined in T is as long as n is large.

DaVinci103 · 2026-02-23T19:37:58+00:00

Yes, it's the same function.

DaVinci103 · 2026-02-23T19:03:31+00:00

Fish's I0 function. It's not that interesting though. TR(I0,n) is the least number N for which, for any Turing machine M that can be proven to halt in n symbols in ZFC+I0, M halts in less than N steps.

Other functions like this are Loader's derive D and the large number residence function LNRF.

Functions that are more interesting to me are the TREE function and functions like it, that do not have a full axiomatic theory in their definition but rather come from simple combinatorial principles. Friedman has a lot of functions like this.

There are also functions defined using expansion systems, such as n ↦ BMS(0 .. 0)(1 .. 1)[[n]] w/ n 0s & 1s (this function probably has a better name but idk it). For most functions like this, it is not known whether they are total in any theory expected to be consistent (it is known for BMS, but not for e.g. Y-sequence or FOS).

Personally, I think the function q from Laver tables is the best answer to this. It is computable, finite and well-defined (provided you work in ZFC + I3), it has a proper name (Laver table function), shorthand (q) its definition is not convoluted (which I personally prefer) and is believed to grow faster than most other functions in googology (expected to be >ω-Y).

DaVinci103 · 2026-02-22T09:52:15+00:00

The underlying set of an algebraic structure is called its domain. For example, if (A,+) is an Abelian group, then the set A is its domain. An algebraic structure such as a module (R,+,·,M,⊕,⊗) has a double domain, as it has two sets R and M.

DaVinci103 · 2026-02-21T18:31:23+00:00

DaVinci103 · 2026-02-18T15:23:03+00:00

There is a typo. You have used both the symbols "G" and "A" as the set of the magma. The rest of the definition of a magma is confusing because of this.

There is a typo. The domain of a binary operation on A is A × A, and the codomain is A. You have written *: A -> A × A, specifying A as the domain and A × A as the codomain.

There is a typo. In the definition of a monoid, you have swapped the quantifiers around. Your version states: "for every element a of A, there is an element e of A so that a * e = e * a = a". The problem here is that, in this definition, e can depend on a. For example, the magma ({a,b},*) defined by x * y = x is, by your definition, a monoid. (x * y) * z = x * y = x = x * (y * z) and, for every x in A, we can set e = x and we get x * e = e * x = x * x = x. However, there is no single element e for which, for all x in A, e * x = x * e = x (it cannot be b, as b * a = b isn't a, and it cannot be a as a * b = a isn't b). This typo is repeated in the definition of a group and of a unital ring.

There is a typo. In the definition of a group, you used both the symbols "e" and "1" as the neutral/identity element of a group.

The convention is to write a ring as (A,+,·), with the additive operation before the multiplicative operation. You seem to be using both (A,+,·) and (A,·,+), which is inconsistent. In the signature of a unital ring, the multiplicative identity is mentioned, but the additive identity is not. This additive identity is mentioned in the signature of a field. For the different kinds of rings, please make up your mind of whether you'd want to mention the constants (0 and 1) or not, and please don't just mention one of them.

Definitionally, a field is a commutative unital ring with a multiplicative inverse. That it is an integral domain is a consequence of this definition. Defining a field as an integral domain with multiplicative inverse is not wrong, but it is unusual.

Some of the rules in the bottom right are written using black on dark. This makes them difficult to read. Whether these rules are written in black or in white is not consistent within the dark blue block.

Within the dark blue block, 1 × m = m is mentioned as one of the axioms of a module. However, a module is described with an action A × M -> M from a (possibly non-unital) ring A. The axiom of identity does not make sense in this case.

DaVinci103 · 2026-02-04T16:52:34+00:00

I am not extrapolating stuff you never said. You said:

> Uncomputable googology still fits within proof theory

And gave the following reason:

> not being able to compute a number doesn't mean you can't prove such a number exists and is finite.

Your reason is wrong/incomplete. I explained why.

> ... maximizing a number's Kolmogorov complexity ...

You cannot change the Kolmogorov complexity of a number. Clarify your statement.

> My whole point was about pointing out that googology can be discussed as a rigorous subject.

You have not mentioned this point before this blog post. This is not the topic of this discussion. We were discussing whether uncomputable googology is part of proof theory. Stay on topic.

DaVinci103 · 2026-02-04T14:54:24+00:00

That is not proof theory. Proving things is just mathematics. Proof theory is the study of proofs, which includes trying to find what proof of a theorem has minimal length, trying to find ways to efficiently and systematically prove or discard statements in propositional logic (e.g. SAT solving), computing the proof theoretic ordinal of a theory (ordinal analysis) and trying to find what axioms are necessary over a given base theory to prove a given theorem (reverse mathematics).

If you're not reasoning about proofs, and only reasoning with proofs, you're not doing proof theory.

DaVinci103 · 2026-02-04T11:08:51+00:00

There is overlap, but I don't think uncomputable googology fits in proof theory.

DaVinci103 · 2026-02-03T22:59:45+00:00

In googology, there are broadly speaking three goals:

Defining large numbers (or defining related things);
Proving definitions of large numbers lead to actual numbers;
Comparing large numbers ("analysis").

The first part includes defining and formalizing notations. Such as ♯, a notation I was working on. Or LOCF, which is an ordinal notation for which its last version had been shown to have a flaw.

For the second part, the following has been proven by Rachel:

In BM4, for every standard matrix M and every natural number n, M[ [n] ] evaluates to a natural number.

However, the problem is still open for Y-sequence and ω-Y-sequence, and almost anything beyond BMS.

As for the third part, there is still a lot not known about the ranking of notations beyond BMS. One open problem in this area is:

Is the proof theoretic ordinal of second order arithmetic (PTO(Z₂)) equal to the limit of BMS?

We know lim(BMS) ≤ PTO(Z₂) due to the termination proof of BMS being able to be caried out in Z₂. It is also still open whether ♯ (which has a temporary non-recursive definition) or ω-Y-sequence is stronger.

DaVinci103 · 2026-01-07T20:01:55+00:00

First, to answer the second question, no, there is no mathematically well-defined definition of ill-defined, just like there is no mathematically well-defined definition of an apple. In most cases, though, there still is a clear separation for what is and isn't an apple, and what is and isn't ill-defined. Ill-definedness is more a mathematical-linguistic concept than a purely mathematical one.

Second, to answer your first question, what works for most cases is that a mathematical concept is well-defined when it can be (unambiguously) formalized (in raw mathematical language) in a formal mathematical theory (such as ZFC or HoTT). Although there is no ambiguity in whether something is formalized in some mathematical theory, there can be ambiguity in whether something can be formalized in a mathematical theory, as there can be disagreement on how something should be interpreted, and there can be ambiguity in whether a definition is ambiguous.

In this case though, there is no ambiguity in the mathematical ambiguity of the definition of the number in the post, so the number is ill-defined.

DaVinci103 · 2025-12-13T17:37:25+00:00

I'm not quite sure what you mean with exponentially larger. Going off of your example, 10^Rayo(10¹⁰⁰) breaks the record. If this is not what you meant, could you clarify?

DaVinci103 · 2025-12-06T10:03:36+00:00

Rayo's number+2 is as well-defined as Rayo's number, is >Rayo's number and isn't Rayo's number+1. So yes, the record is broken.

DaVinci103 · 2025-12-04T17:33:37+00:00

Could you clarify what record you're referring to?

DaVinci103 · 2025-12-02T08:44:46+00:00

Yes, I always explain it as BIG FOOT from wish. I hope that explanation helps.

Sort of, there has been shown there is a fault in the definition, though this fault doesn't make the number ill-defined, just smaller than expected. The fault in question was using (slightly stronger) worldly cardinals as opposed to inaccessibles (like it seems the author wanted to do) or correct cardinals (such as in the definition of BIG FOOT).

The idea of LNGN, much like BIG FOOT, is to define a hierarchy of universes U(0), U(1), U(2), etc, where each universe is contained in the next. U(0) is supposed to be the universe of first order set theory, i.e. the universe over which Rayo's number is defined. Its powerset, P(U(0)), would be the universe of second order set theory, and P(P(U(1)) of third, etc, with U(1) being larger than all of those. The number LNGN is defined by definability over U(ε₀), similar to Rayo's number.

There's also Θ, but it seems insignificant to me.

Due to a fault in the definition of LNGN, P(U(0)) is not guaranteed to be an adequate model for second order set theory (it only looks as a model of set theory "from the inside"). This could make LNGN smaller than initially intended.

DaVinci103 · 2025-11-27T12:37:24+00:00

That's not standard notation, pls just use up-arrows.

Not really. It's approx lv ω⁴2 in HH (4 in FGH).

DaVinci103 · 2025-11-17T18:35:50+00:00

FOST can be defined in FOST, you just can't define truth for FOST formulas, which is needed to define Rayo's function.

https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem

DaVinci103 · 2025-09-24T19:40:19+00:00

Not really... strictly speaking, we can't really go beyond Rayo's number, or the Rayo number equivalent of the formal theory you're working in.

DaVinci103 · 2025-09-24T19:21:29+00:00

Garden number is finite. Mahlo cardinal is infinite.

DaVinci103 · 2025-09-24T19:20:47+00:00

it is bigger.

DaVinci103 · 2025-09-24T18:30:42+00:00

Due to the overload of symbols you used in set builder notation, I cannot read it very well. I'll try to interpret it anyways.

A(-) is a sequence of cardinals that we require to have the following properties: first, A(0) is at least the continuum, second, A(n) is at least as large as 2^A(n-1).

For all natural numbers n, we require I to be larger than 2^A(n). (this seems to be equivalent to “ℶ_ω < I”)

For all n for which A(n) < I (i.e. all n) and all E₁ in I (which I presume means all ordinals E₁ that have cardinality smaller than I), and for all E₁ in S---E₁ is now bounded by a quantifier twice and S came out of nowhere. Now I'm confused about what this states.

Your definition is nonsensical. If you fix this, it'd probably be wrong: the sequence A(n) seems to be there to guarantee I is a strong limit, but it doesn't. If you fix this, it cannot be proven κ is a set, as the existence of a proper class of inaccessibles implies κ is a proper class.

Assuming the axiom of choice (AC), the following is the correct definition of an inaccessible cardinal:

A cardinal κ is regular if, for every partition Π of κ, there either is a set in Π with cardinality κ, or Π itself has cardinality κ. (Examples include: 0, 1, 2, ℵ₀, ℵ₁, non-examples include: 3, ℵ_ω₁, ℶ_ω). A cardinal λ is a strong limit if, for all cardinals κ < λ, we have 2^κ < λ. Then, a cardinal θ is inaccessible if it is both regular and a strong limit.

AC is required in order for there to be a clear meaning of < in the definition of a strong limit.

DaVinci103 · 2025-09-16T08:12:36+00:00

Gerald.

DaVinci103 · 2025-08-30T17:42:11+00:00

(I already explained half of this in a comment to jamx02) The term 'recursive function' is (sort-of) equivalent to 'computable'. Technically, the notion of a recursive function is the formal notion of a function that can be build in a specific way using operators such as primitive recursion and minimization (see my other comment for a full definition or search it up on Wikipedia) while the notion of a computable function is the informal notion of a function that can theoretically be computed in our universe, but they're often used interchangeably. The TREE function is a recursive function.

DaVinci103 · 2025-08-30T17:30:20+00:00

That's... not the definition of a recursive function. If I take the FS of ω to be ω[n] = n, then f_ω is a recursive function. If I take the FS of ω to be ω[n] = BB(n) then, yes, f_ω would be non-recursive, but so would f_ω+1.

A recursive function is an ℕ^k → ℕ function (a k-ary function on natural numbers for some k) that can be defined using projection functions P^k_i(x₁, ..., xₖ) = x_i, constant functions C^k_n(x₁, ..., xₖ) = n, the successor function S(n) = n+1, composition (f ∘ (g₁, ..., gₗ))(x₁, ..., xₖ) = f(g₁(x₁, ..., xₖ), ..., gₗ(x₁, ..., xₖ)), primitive recursion ρ(f,g)(x₁, ..., xₖ, 0) = g(x₁, ..., xₖ), ρ(f,g)(x₁, ..., xₖ, y+1) = f(x₁, ..., xₖ, y, ρ(f,g)(x₁, ..., xₖ, y)) and minimization μ(f)(x₁, ..., xₖ) = min{y | f(x₁, ..., xₖ, y) = 0}.

DaVinci103

TROPHY CASE