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Not to be confused with ε numbers, Hyper-E notation, or [[:{{{3}}}]].

ε function (ε関数 in Japanese) is a googological system introduced by a Japanese Googologist Jason,[1][2][3][4][5] and is based on shifting definition. It was submitted to a Japanese googological events, but was withdrawn because of several errors and the insufficiency of time to analyse it. Later, the formalisation up to the limit of the realm of $$\varepsilon(0)$$ has been officially completed since 30/10/2020.[4]

Notation

The system consists of several function symbols: $$E(x)$$, $$\varepsilon(x)$$, $$EE_{\alpha}(x)$$, and $$+$$. As the $$1$$-ary $$\delta \varphi(x)$$ in δφ admits two distinct ways to receive arguments, i.e. the usual substitution $$x \mapsto \delta \varphi(x)$$ and the bracketed substitution $$x \mapsto \delta \varphi([x])$$, $$\varepsilon$$ admits two distinct ways to receive arguments, i.e. the usual substitution $$x \mapsto \varepsilon(x)$$ and the bracketed substitution $$x \mapsto \varepsilon([x])$$. Unlike the addition symbol $$+$$ in side nesting, the addition symbol $$+$$ works as the addition with respect to the correspondence to ordinals.

Analysis

Although the termination has not been verified yet, it is expected to be very powerful if it actually terminates. For example, Jason has the following expectations of the correspondence to ordinals:[3]

$$\varepsilon$$ function Rathjen's $$\psi$$ function
$$EE_A(A)$$ $$\varphi_{1}(0)$$
$$EE_A(A+EE_{O_2}(A+EE_{O_2}(A+O_1)+O_1))$$ $$\psi_{\chi_0(0)}(\chi_0(0))$$
$$EE_A(A+EE_{A2}(0))$$ $$\psi_{\chi_0(0)}(\Phi_1(0))$$
$$EE_A(A2)$$ $$\psi_{\chi_0(0)}(\Phi_2(0))$$
$$EE_A(A2+EE_B(A2+E(\varepsilon([A2]))))$$ $$\psi_{\chi_0(0)}(\Phi_3(0))$$
$$EE_A(A2+EE_B(A2+EE_B(0)))$$ $$\psi_{\chi_0(0)}(\Phi_{\omega}(0))$$
$$EE_A(A2+EE_B(A2+EE_B(A2+E(\varepsilon([A2])))))$$ $$\psi_{\chi_0(0)}(\psi_{\chi_1(0)}(0))$$
$$EE_A(A2+EE_B(A2+EE_B(A2+E(\varepsilon([A2]))))+E(\varepsilon([A2])))$$ $$\psi_{\chi_0(0)}(\chi_1(0))$$
$$EE_A(A+A+A)$$ $$\psi_{\chi_0(0)}(\chi_2(0))$$
$$EE_A(A+A+A+A)$$ $$\psi_{\chi_0(0)}(\chi_3(0))$$
$$EE_A(EE_{E(\varepsilon(0)×(1)+\varepsilon([A])×(1))}(0))$$ $$\psi_{\chi_0(0)}(\psi_{\chi_{\omega}(0)}(0))$$
$$EE_A(EE_{E(\varepsilon(0)×(1)+\varepsilon([A])×(1))}(A))$$ $$\psi_{\chi_0(0)}(\psi_{\chi_M(0)}(0))$$
$$EE_A(EE_{E(\varepsilon(0)×(1)+\varepsilon([A])×(1))}(A2))$$ $$\psi_{\chi_0(0)}(\chi_M(0))$$
Limit $$\psi_{\chi_0(0)}(\psi_{\chi_{M+1}(0)}(0))$$

Here, $$A$$ is the shorthand of $$E(\varepsilon(0) \times (1))$$, $$O_1$$ is the shorthand of $$E(\varepsilon([A]) \times (1))$$, $$O_2$$ is the shorthand of $$E(\varepsilon([A]) \times (1+1))$$, $$A2$$ is the shorthand of $$A+A$$, $$B$$ is the shorthand of $$Ε(\varepsilon([Α2])+\varepsilon([Α]))$$, $$\varphi$$ is Veblen function, $$\chi$$ is Rathjen's $$\chi$$ function, $$M$$ is the least weakly Mahlo cardinal, and $$\psi$$ is Rathjen's ordinal collapsing function based on $$M$$. According to Jason's impression, $$\varepsilon(0)$$ yields shifting definition, $$\varepsilon(1)$$ yields meta-shifting definition, and $$\varepsilon(2)$$ yields meta-meta-shifting definition.

Large Number

For a non-zero expression $$Z$$, let $$A_Z$$ denote the expression $$E(\varepsilon(0) \times (Z))$$. In particular, we have $$A_1 = A$$. Let $$\mathbb{N}_{> 0}$$ denote the set of positive integers. Jason defined a large function \begin{eqnarray*} G \colon \mathbb{N}_{> 0} & \to & \mathbb{N}_{> 0} \\ n & \mapsto & G(n) \end{eqnarray*} as $$G(n) := 3 \uparrow^{EE_A \left( A_{A_{\cdot_{\cdot_{\cdot_A}}}} \right)} 3$$ ($$n$$ $$A$$s in the parentheses), and coined グラハム数ver ε.0.1.0 (Graham's number ver ε.0.1.0 in English) as $$G^{64}(4)$$. Although the termination of $$G(n)$$ has not been verified, the number is expected to be very large if it actually terminates.

Jason also coined First Shift Ordinal as the limit of ordinals corresponding to expressions of the form $$EE_A \left( A_{A_{\cdot_{\cdot_{\cdot_A}}}} \right)$$ (finitely many $$A$$s). If the system is actually well-founded and the expectation of the correspondence to ordinals is correct, First Shift Ordinal is a well-defined countable ordinal much bigger than $$\psi_{\chi_0(0)}(\chi_M(0))$$, and coincides with $$\psi_{\chi_0(0)}(\psi_{\chi_{M+1}(0)}(0))$$.

Sources

1. The user page of Jason in Japanese Googology Wiki.
2. Jason, ε関数定義試作, Google Document. (Trial to formalise the notation.)
3. Jason, ε関数成長記録, Google Document. (Comparison to other notations.)
4. Jason, ε関数 ver ε.0.1.0, Google Document. (The current version.)
5. Jason, ε関数の解説を試みる, Japanese Googology Wiki user blog. (Trial to explain ε function)

Original numbers, functions, notations, and notions

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's psi function
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4 original idea
By ふぃっしゅ (Fish): Fish numbers (Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7 · S map · SS map · s(n) map · m(n) map · m(m,n) map) · Bashicu matrix system 1/2/3/4 formalisation · TR function (I0 function)
By Gaoji: Weak Buchholz's function
By じぇいそん (Jason): Irrational arrow notation · δOCF · δφ · ε function
By 甘露東風 (Kanrokoti): KumaKuma ψ function
By koteitan: Bashicu matrix system 2.3
By mrna: 段階配列表記 · 降下段階配列表記 · 多変数段階配列表記 · SSAN · S-σ
By Naruyoko Naruyo: Y sequence formalisation · ω-Y sequence formalisation
By Nayuta Ito: N primitive · Flan numbers (Flan number 1 · Flan number 2 · Flan number 3 · Flan number 4 version 3 · Flan number 5 version 3) · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By Okkuu: Extended Weak Buchholz's function
By p進大好きbot: Ordinal notation associated to Extended Weak Buchholz's function · Ordinal notation associated to Extended Buchholz's function · Naruyoko is the great · Large Number Garden Number
By たろう (Taro): Taro's multivariable Ackermann function
By ゆきと (Yukito): Hyper primitive sequence system · Y sequence original idea · YY sequence · Y function · ω-Y sequence original idea

Methodology

By バシク (BashicuHyudora): Bashicu matrix system as a notation template
By じぇいそん (Jason): Shifting definition
By mrna: Side nesting
By Nayuta Ito and ゆきと (Yukito): Difference sequence system

Implementation of existing works into programs

Proofs, translation maps for analysis schema, and other mathematical contributions

By ふぃっしゅ (Fish): Computing last 10000 digits of mega · Translation map for primitive sequence system and Cantor normal form
By Kihara: Proof of an estimation of TREE sequence · Proof of the incomparability of Busy Beaver function and FGH associated to Kleene's $$\mathcal{O}$$
By koteitan: Translation map for primitive sequence system and Cantor normal form
By Naruyoko Naruyo: Translation map for Extended Weak Buchholz's function and Extended Buchholz's function
By Nayuta Ito: Comparison of Steinhaus-Moser Notation and Ampersand Notation
By p進大好きbot: Proof of the termination of Hyper primitive sequence system · Proof of the termination of Pair sequence number · Proof of the termination of segements of TR function in the base theory under the assumption of the $$\Sigma_1$$-soundness and the pointwise well-definedness of $$\textrm{TR}(T,n)$$ for the case where $$T$$ is the formalisation of the base theory

Entertainments

By 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
By koteitan: Dancing video of a Gijinka of Fukashigi · Dancing video of a Gijinka of 久界 · Storyteller's theotre video reading Large Number Garden Number aloud