Not to be confused with ε numbers, or Hyper-E notation.

ε function (ε関数 in Japanese) is a googological system introduced by a Japanese Googologist Jason, 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 It was submitted to a Japanese googological events, but was withdrawn because of several errors and the insufficiency of time to analyse it.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.

## 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:

$$\varepsilon$$ 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)

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