Measuring large numbers - Revision history

Loader3229: /* Theories */

2026-09-25T23:43:04Z

Theories

← Older revision		Revision as of 23:43, 25 September 2026
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	The proof-theoretic ordinal of a theory is defined as supremum of the order types of all ordinal notations that can be shown to be well-founded within the theory.		The proof-theoretic ordinal of a theory is defined as supremum of the order types of all ordinal notations that can be shown to be well-founded within the theory.

	Some examples of proof theoretical ordinals include <math>\operatorname{PTO}(Q) = \omega</math>, <math>\operatorname{PTO}(PA) = \~~epsilon_0~~</math>, and <math>\operatorname{PTO}(\operatorname{KP}) = \psi_0(\Omega_\omega)</math>.		Some examples of proof theoretical ordinals include <math>\operatorname{PTO}(Q) = \omega</math>, <math>\operatorname{PTO}(PA) = \varepsilon_0</math>, and <math>\operatorname{PTO}(\operatorname{KP}) = \psi_0(\Omega_\omega)</math>.

	The proof theoretic ordinal of second order arithmetic is unknown (relative to any ordinal notation), but is known to vastly exceed that of <math>\operatorname{KP}</math>.		The proof theoretic ordinal of second order arithmetic is unknown (relative to any ordinal notation), but is known to vastly exceed that of <math>\operatorname{KP}</math>.

Loader3229: /* Fast-growing hierarchy */

2026-09-25T23:42:35Z

Fast-growing hierarchy

← Older revision		Revision as of 23:42, 25 September 2026
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	It is suspected that in this wiki, and in most other spaces that deal with googology, the fundamental sequences described by Maksudov (2017) are used to define the fast-growing hierarchy.		It is suspected that in this wiki, and in most other spaces that deal with googology, the fundamental sequences described by Maksudov (2017) are used to define the fast-growing hierarchy.

	In this manner, the value of <math>S(3,5)</math> can be more concisely (though more weakly) written as <math>S(3,4) > f_\omega(2 \uparrow^{15} 3)</math>.		In this manner, the value of <math>S(3,5)</math> can be more concisely (though more weakly) written as <math>S(3,5) > f_\omega(2 \uparrow^{15} 3)</math>.

	Other busy beaver values that necessitate the use of the fast-growing hierarchy include:		Other busy beaver values that necessitate the use of the fast-growing hierarchy include:
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	<math>\operatorname{BB\lambda}(49) > f_{\omega + 1}\left(\frac{2 \uparrow \uparrow 6}{2}\right)</math>		<math>\operatorname{BB\lambda}(49) > f_{\omega + 1}\left(\frac{2 \uparrow \uparrow 6}{2}\right)</math>

	<math>\operatorname{S}(51) > f_{\~~epsilon_0~~ + 1}(8)</math>		<math>\operatorname{S}(51) > f_{\varepsilon_0 + 1}(8)</math>

	<math>\operatorname{BB\lambda}(100) > f_{\psi_0(\Omega_\omega) + 1}(4)</math>		<math>\operatorname{BB\lambda}(100) > f_{\psi_0(\Omega_\omega) + 1}(4)</math>

Δ⁵: /* Theories */ Z_ω is not nth order arithmetic

2026-09-16T01:07:38Z

Theories: Z_ω is not nth order arithmetic

← Older revision		Revision as of 01:07, 16 September 2026
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	A notable example of a theory which is directly used to create fast growing functions is the Calculus of Constructions (CoC), which has a proof-theoretic ordinal greater than that of any nth order arithmetic (<~~math~~>~~Z_\omega~~</~~math~~>); well-formed statements in CoC can also be easily detected, and are strongly normalizing (TODO: explain what that means in this context).		A notable example of a theory which is directly used to create fast growing functions is the Calculus of Constructions (CoC), which has a proof-theoretic ordinal greater than that of any nth order arithmetic (Z<sub>n</sub>); well-formed statements in CoC can also be easily detected, and are strongly normalizing (TODO: explain what that means in this context).

	CoC gives way to the fast growing function <math>D(n)</math> (defined by Ralph Loader in 2003 to create the eponymous Loader's number = <math>D^5(99)</math>), which computes something similar to (but larger than) "The largest number that is a minimal x where the statement <math>\phi(x)</math> is true, and where <math>\exists x \phi(x)</math> can be written in less than n symbols and shown to be true in CoC." <math>D(n)</math> outgrows all functions definable in <math>Z_\omega</math>; the growth rate of <math>D(n)</math> is sometimes informally written as <math>f_{\operatorname{PTO}(Z_\omega)}(n)</math>, though no fundamental sequence has yet been defined for all ordinals under <math>\operatorname{PTO}(Z_\omega)</math>.		CoC gives way to the fast growing function <math>D(n)</math> (defined by Ralph Loader in 2003 to create the eponymous Loader's number = <math>D^5(99)</math>), which computes something similar to (but larger than) "The largest number that is a minimal x where the statement <math>\phi(x)</math> is true, and where <math>\exists x \phi(x)</math> can be written in less than n symbols and shown to be true in CoC." <math>D(n)</math> outgrows all functions definable in <math>Z_\omega</math>; the growth rate of <math>D(n)</math> is sometimes informally written as <math>f_{\operatorname{PTO}(Z_\omega)}(n)</math>, though no fundamental sequence has yet been defined for all ordinals under <math>\operatorname{PTO}(Z_\omega)</math>.

HelpMe: changed some words.

2026-09-15T13:45:30Z

changed some words.

← Older revision		Revision as of 13:45, 15 September 2026
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	The busy beaver functions are known to grow faster than any computable function, as do many of their variants. Because of this, values of the function, or lower bounds on the function can be very, very large, so it is tempting to find ways of representing these numbers; this article will cover these methods.		The busy beaver functions are known to grow faster than any computable function, as do many of their variants. Because of this, values of the function, or lower bounds on the function can be very, very large, so it is tempting to find ways of representing these numbers; this article will cover these methods.

	Paradoxically, although the Kolgomorov complexity of stating "The runtime of program X" to define the runtime of program X is typically very low, it is often not useful as it does not give us good information on the size of the number. Informally, more "natural" or "combinatorial" ways of measuring large numbers are desired. Tools like ~~the~~ [[Fast-Growing Hierarchy\|fast-growing hierarchy]] ~~can help us achieve this, as do Laver tables,~~ or the diagonalization of an axiomatic foundation of mathematics.		Paradoxically, although the Kolgomorov complexity of stating "The runtime of program X" to define the runtime of program X is typically very low, it is often not useful as it does not give us good information on the size of the number. Informally, more "natural" or "combinatorial" ways of measuring large numbers are desired. Tools like Knuth's up-arrow notation can help us achieve this, as well as more powerful methods such as [[Fast-Growing Hierarchy\|fast-growing hierarchy]] or the diagonalization of an axiomatic foundation of mathematics.

	== Scientific notation ==		== Scientific notation ==
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	<math>\operatorname{BB\lambda}(100) > f_{\psi_0(\Omega_\omega) + 1}(4)</math>		<math>\operatorname{BB\lambda}(100) > f_{\psi_0(\Omega_\omega) + 1}(4)</math>

	== ~~Ordinal~~ notations and ordinal collapsing functions ==		=== Defining fundamental sequences: ordinal notations and ordinal collapsing functions ===
	TODO<!-- somewhere in this section should mention ~~that these often supplement the fast growing hierarchy~~		Individually defining fundamental sequences for a transfinite number of ordinals may be less than ideal; thus, to define the fast-growing hierarchy effectively, ordinal notations, which define a computable mapping between sequences of symbols and ordinals can be used to define fundamental sequences instead.TODO<!-- somewhere in this section should mention
	primitive ~~(+ hyper primitive? pair?)~~ sequence system or kirby paris hydra		cantor form
	buccholz ocf + extension to p0(Lambda)		primitive sequence system or kirby paris hydra
	~~bms~~		buccholz ocf and/or extension to p0(Lambda)
	after that, go, be free. live. -->		after that, go, be free. live. -->

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	An even stronger but simpler function, <math>q(n)</math>, has been derived from the ~~theory~~ <math>~~\operatorname{ZFC} +~~ \operatorname{I3}</math>, though it is unknown if this function ~~is well-defined~~, ~~or if~~ it grows faster than the aforementioned <math>D(n)</math>. It is used in the following (conjectured) bounds:		An even stronger but simpler function, <math>q(n)</math>, has been derived from the structure of <math>\operatorname{I3}</math> cardinals, though it is unknown if this function can be shown to be total in <math>\operatorname{ZFC}</math> alone, and whether it grows faster than the aforementioned <math>D(n)</math>. It is used in the following (conjectured) bounds:

	<math>\operatorname{BB\lambda}(201) > q(5)</math>		<math>\operatorname{BB\lambda}(201) > q(5)</math>

HelpMe: where the f word did i get 4 from

2026-09-13T15:04:31Z

where the f word did i get 4 from

← Older revision		Revision as of 15:04, 13 September 2026
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	<math>g_n = \begin{cases}		<math>g_n = \begin{cases}
	3 \uparrow \uparrow \uparrow \uparrow 4, & \text{if }n = 1		3 \uparrow \uparrow \uparrow \uparrow 3, & \text{if }n = 1
	\\ 3 \uparrow^{g_{n - 1}} 3, & \text{otherwise.}		\\ 3 \uparrow^{g_{n - 1}} 3, & \text{otherwise.}
	\end{cases}</math>		\end{cases}</math>

HelpMe: added a comment also corrected avogadro's number

2026-09-13T14:21:32Z

added a comment also corrected avogadro's number

← Older revision		Revision as of 14:21, 13 September 2026
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	Scientific notation is one of the most common ways of representing exceedingly large numbers; numbers written in standard scientific notation symbolically consist of a multiplicand (taking a value between 1 and the base, though is sometimes omitted), a base (commonly 10) and an index (usually an integer, though sometimes takes non-integer values if the multiplicand is omitted).		Scientific notation is one of the most common ways of representing exceedingly large numbers; numbers written in standard scientific notation symbolically consist of a multiplicand (taking a value between 1 and the base, though is sometimes omitted), a base (commonly 10) and an index (usually an integer, though sometimes takes non-integer values if the multiplicand is omitted).

	For example, <math>8.68 \cdot 10^6</math>represents the number 8,680,000, whereas <math>6.022 \cdot 10^{23}</math>represents the number 602,~~000~~,000,000,000,000,000,000.		For example, <math>8.68 \cdot 10^6</math> represents the number 8,680,000, whereas <math>6.022 \cdot 10^{23}</math> represents the number 602,200,000,000,000,000,000,000.

	In cases where the index is not an integer, the representation usually represents an irrational number, though sometimes it can represent a rational number, or even an integer.		In cases where the index is not an integer, the representation usually represents an irrational number, though sometimes it can represent a rational number, or even an integer.
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	<math>\operatorname{BB\lambda}(100) > f_{\psi_0(\Omega_\omega) + 1}(4)</math>		<math>\operatorname{BB\lambda}(100) > f_{\psi_0(\Omega_\omega) + 1}(4)</math>

	== Ordinal notations ==		== Ordinal notations and ordinal collapsing functions ==
	TODO		TODO<!-- somewhere in this section should mention that these often supplement the fast growing hierarchy
			primitive (+ hyper primitive? pair?) sequence system or kirby paris hydra
			buccholz ocf + extension to p0(Lambda)
			bms
			after that, go, be free. live. -->

	== Theories ==		== Theories ==

HelpMe: fixed stuff

2026-09-13T14:14:20Z

fixed stuff

← Older revision		Revision as of 14:14, 13 September 2026
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	A notable example of a theory which is directly used to create fast growing functions is the Calculus of Constructions (CoC), which has a proof-theoretic ordinal greater than that of any nth order arithmetic (<math>Z_\omega</math>); well-formed statements in CoC can also be easily detected, and are strongly normalizing (TODO: explain what that means in this context).		A notable example of a theory which is directly used to create fast growing functions is the Calculus of Constructions (CoC), which has a proof-theoretic ordinal greater than that of any nth order arithmetic (<math>Z_\omega</math>); well-formed statements in CoC can also be easily detected, and are strongly normalizing (TODO: explain what that means in this context).

	CoC gives way to the fast growing function <math>D(n)</math> (defined by Ralph Loader in 2003 to create the eponymous Loader's number = <math>D^5(99)</math>), which computes something similar to (but larger than) "The largest number that is a minimal x where the statement <math>\phi(x)</math>is true, and where <math>\exists x \phi(x)</math>can be written in less than n symbols and shown to be true in CoC." <math>D(n)</math> outgrows all functions definable in <math>Z_\omega</math>; the growth rate of <math>D(n)</math> is sometimes informally written as <math>f_{\operatorname{PTO}(Z_\omega)}(n)</math>, though no fundamental sequence has yet been defined for all ordinals under <math>\operatorname{PTO}(Z_\omega)</math>.		CoC gives way to the fast growing function <math>D(n)</math> (defined by Ralph Loader in 2003 to create the eponymous Loader's number = <math>D^5(99)</math>), which computes something similar to (but larger than) "The largest number that is a minimal x where the statement <math>\phi(x)</math> is true, and where <math>\exists x \phi(x)</math> can be written in less than n symbols and shown to be true in CoC." <math>D(n)</math> outgrows all functions definable in <math>Z_\omega</math>; the growth rate of <math>D(n)</math> is sometimes informally written as <math>f_{\operatorname{PTO}(Z_\omega)}(n)</math>, though no fundamental sequence has yet been defined for all ordinals under <math>\operatorname{PTO}(Z_\omega)</math>.

	The simplicity of CoC and <math>D(n)</math>has been utilized to show the following bounds for certain busy beaver values:		The simplicity of CoC and <math>D(n)</math> has been utilized to show the following bounds for certain busy beaver values:

	<math>\operatorname{BB\lambda}(1850) > D^5(99)</math>		<math>\operatorname{BB\lambda}(1850) > D^5(99)</math>

HelpMe: fixed wording

2026-09-13T14:12:24Z

fixed wording

← Older revision		Revision as of 14:12, 13 September 2026
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	{{Stub}}		{{Stub}}
	The ~~Busy Beaver function is~~ known to grow faster than any computable function, as do many of ~~it's~~ variants. Because of this, values of the function or lower bounds on the function can be very, very large, so it is ~~important~~ to find ways of representing these numbers; this article will cover these methods.		The busy beaver functions are known to grow faster than any computable function, as do many of their variants. Because of this, values of the function, or lower bounds on the function can be very, very large, so it is tempting to find ways of representing these numbers; this article will cover these methods.

	Paradoxically, although the Kolgomorov complexity of stating "The runtime of program X" ~~for~~ the program X is typically very low, it is often not useful as it does not give us good information on the size of the number. Informally, more "natural" or "combinatorial" ways of measuring large numbers are desired. Tools like the [[Fast-Growing Hierarchy]] can help us achieve this, as do Laver tables, or the diagonalization of an axiomatic foundation of mathematics.		Paradoxically, although the Kolgomorov complexity of stating "The runtime of program X" to define the runtime of program X is typically very low, it is often not useful as it does not give us good information on the size of the number. Informally, more "natural" or "combinatorial" ways of measuring large numbers are desired. Tools like the [[Fast-Growing Hierarchy\|fast-growing hierarchy]] can help us achieve this, as do Laver tables, or the diagonalization of an axiomatic foundation of mathematics.

	== Scientific notation ==		== Scientific notation ==

HelpMe: added something. please do ordinal notations for me im too tired its 11pm

2026-09-13T14:03:30Z

added something. please do ordinal notations for me im too tired its 11pm

@@ Line 4: / Line 4: @@
 Paradoxically, although the Kolgomorov complexity of stating "The runtime of program X" for the program X is typically very low, it is often not useful as it does not give us good information on the size of the number. Informally, more "natural" or "combinatorial" ways of measuring large numbers are desired. Tools like the [[Fast-Growing Hierarchy]] can help us achieve this, as do Laver tables, or the diagonalization of an axiomatic foundation of mathematics.
-== Exponential ==
+== Scientific notation ==
 TODO
-== Tetrational ==
+== Theories ==
-TODO
+The proof-theoretic ordinal of a theory is defined as supremum of the order types of all ordinal notations that can be shown to be well-founded within the theory.
-== Ackermannian ==
+<math>\operatorname{S}(67) > q^{81}(q(5) - 2)</math>
 == Resources ==

Azerty: TODO sections

2026-09-13T11:48:19Z

TODO sections

← Older revision		Revision as of 11:48, 13 September 2026
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	Paradoxically, although the Kolgomorov complexity of stating "The runtime of program X" for the program X is typically very low, it is often not useful as it does not give us good information on the size of the number. Informally, more "natural" or "combinatorial" ways of measuring large numbers are desired. Tools like the [[Fast-Growing Hierarchy]] can help us achieve this, as do Laver tables, or the diagonalization of an axiomatic foundation of mathematics.		Paradoxically, although the Kolgomorov complexity of stating "The runtime of program X" for the program X is typically very low, it is often not useful as it does not give us good information on the size of the number. Informally, more "natural" or "combinatorial" ways of measuring large numbers are desired. Tools like the [[Fast-Growing Hierarchy]] can help us achieve this, as do Laver tables, or the diagonalization of an axiomatic foundation of mathematics.

			== Exponential ==
			TODO

			== Tetrational ==
			TODO

			== Ackermannian ==
			TODO

			== Resources ==
	The Googology wiki hosts good information on methods of describing large numbers or large upper bounds for numbers.		The Googology wiki hosts good information on methods of describing large numbers or large upper bounds for numbers.

			* https://googology.miraheze.org/wiki/Main_Page

	[[Category:Stub]]		[[Category:Stub]]