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δφ is a googological system introduced by a Japanese Googologist Jason,[1][2][3] and is the second system defined by shifting definition. It has not been formalised.

It is intended to perform as an ordinal function associated to Veblen function through shifting definition. A restricted system of δφ consists of the constant term \(0\), an associative \(2\)-ary function \(+\), and variadic functions \(\delta \varphi(x_1,\ldots,x_k)\) and \(\delta \varphi([d]x_1,\ldots,x_k)\). The term \(d\) in the latter expression plays a role to indicate a definition, which is coded into an ordinal, of an ordinal function.


Explanation

When \(x_1 = 0\), then \(x_1\) is often omitted. For example, \(\delta \varphi([d])\) is a shorthand of \(\delta \varphi([d]0)\), and \(\delta \varphi([d],x_2)\) is a shorthand of \(\delta \varphi([d]0,x_2)\). Moreover, the expressions \(\delta \varphi(x_1,x_2)\) and \(\delta \varphi([d]0,x_2)\) in the restricted system themselves are shorthands of \(\delta \varphi(x_1,[0]x_2)\) and \(\delta \varphi(0,[d]0,[0,0]x_2)\), and hence the full system of δφ is much more complicated.

We explain how it is intended to work. As we noted above, the \(2\)-ary function \(+\) does not play a role of the addition. Therefore we denote by \(+_{\delta \varphi}\) the \(2\)-ary function \(+\) in order to distinguish it from the addition.

The \(1\)-ary \(\delta \varphi(x_1)\) coincides with \(\omega^{x_1} = \varphi(0,x_1)\), and the \(2\)-ary function \(x +_{\delta \varphi} y\) coincides with \(x + y\) as long as \(y\) is smaller than \(\varepsilon_0\). On the other hand, the \(2\)-ary function \(\delta \varphi(x_1,x_2)\) behaves in a tricky way, The initial value \(\delta \varphi(1,0)\), which we will denote by \(A\), coincides with \(\varphi(1,0) = \varepsilon_0\). However, \(A +_{\delta \varphi} A\) is intended to be \(\varphi(2,0) = \zeta_0\), which is much larger than \(A + A = \varepsilon_0 \times 2\). The ordinal \(\varepsilon_0 \times 2\) is expressed as \(A +_{\delta \varphi} \delta \varphi([A]0)\). Similarly, we have \(A +_{\delta \varphi} \delta \varphi([A]0) +_{\delta \varphi} \delta \varphi([A]0) = \varepsilon_0 \times 3\), and \(+_{\delta \varphi} \delta \varphi([A]0)\) plays a role of \(+ \varepsilon_0\). Although the expression \(\delta \varphi([A]0)\) itself is not a normal expression in this system, it is harmless to regard it as \(\varepsilon_0\) as long as we interpret occurrences of \(\delta \varphi([A]0)\) in a normal expression.

The next significant expression is \(A +_{\delta \varphi} \delta \varphi(\delta \varphi([A]0) +_{\delta \varphi} \delta \varphi(0))\). By \(\delta \varphi(0) = \omega^0 = 1\), \(x +_{\delta \varphi} y = x + y\) as long as \(y\) is smaller than \(\varepsilon_0 = A\), and \(\delta \varphi(x_1) = \omega^{x_1}\), it is the sum of \(A\) and \(\omega^{A+1} = A \times \omega\) with respect to \(+_{\delta \varphi}\). The value is intended to coincide with \(A + \omega^{A+1} = A \times \omega\), and hence we do not have to care about the difference of \(+_{\delta \varphi}\) and \(+\) in this realm. Indeed, we have \(A +_{\delta \varphi} y = A + y\) for any normal expression \(A +_{\delta \varphi} y\) smaller than \(A +_{\delta \varphi} A\).

The \(1\)-ary function \(\delta \varphi([A]x_1)\) plays a role of \(\varphi(1,x_1)\) although the expression itself is not normal. For example, we have \(A +_{\delta \varphi} \delta \varphi([A] \delta \varphi([A]0)) = A + \varepsilon_{\varepsilon_0} = \varepsilon_{\varepsilon_0}\). The limit of ordinals expressed by \(0\), a single occurrence of \(A\), \(+_{\delta \varphi}\), \(\delta \varphi(x_1)\), and \(\delta \varphi([A],x_1)\) is \(\zeta_0\), which is expressed as \(A +_{\varphi \delta} A\).

Similarly, \(\delta \varphi([A +_{\delta \varphi} A]x_1)\) plays a role of \(\varphi(2,x_1)\), and \(A +_{\varphi \delta} A +_{\varphi \delta} A\) is intended to coincide with \(\varphi(3,0) = \eta_0\). Continuing a similar computation, we obtain the following analysis:

  • \(\delta \varphi(A +_{\delta \varphi} \delta \varphi(0))\), which is often abbreviated to \(A \times \omega\) although it does not coincides with \(\varepsilon_0 \times \omega\), is intended to coincide with \(\varphi(\omega,0)\), and \(\delta \varphi([\delta \varphi(A +_{\delta \varphi} \delta \varphi(0))]x_1)\) plays a role of \(\varphi(\omega,x_1)\).
  • \(\delta \varphi(\delta \varphi(A +_{\delta \varphi} \delta \varphi(0)))\), which is often abbreviated to \(A^{\omega}\) although it does not coincides with \(\varepsilon_0^{\omega}\), is intended to coincide with the small Veblen ordinal, and \(\delta \varphi([\delta \varphi(\delta \varphi(A +_{\delta \varphi} \delta \varphi(0)))]x_1)\) plays a role of the enumeration of fixed points of multivariable Veblen functions.
  • \(\delta \varphi(\delta \varphi(A +_{\delta \varphi} A))\), which is often abbreviated to \(A^A\) although it does not coincides with \(\varepsilon_0^{\varepsilon_0}\), is intended to coincide with the large Veblen ordinal, and \(\delta \varphi([\delta \varphi(\delta \varphi(A +_{\delta \varphi} A))]x_1)\) plays a role of the enumeration of fixed points of transfinte-variable Veblen functions.
  • \(\delta \varphi([A]A +_{\delta \varphi} \delta \varphi(0))\), which might be abbreviated to \(\varepsilon_{A+1}\) although it does not coincides with \(\varepsilon_{\varepsilon_0+1}\), is intended to coincide with the Bachmann-Howard ordinal.

We should recall that we have not used \(\delta \varphi(1,1)\) in the computation above. It is intended to coincide with \(\psi(\Omega_2^{\Omega_2})\) with respect to a certain OCF, and is expanded as \(\delta \varphi([\varphi([\cdots \varphi([A]A +_{\delta \varphi} \delta \varphi(0)) \cdots]A +_{\delta \varphi} \delta \varphi(0))]A +_{\delta \varphi} \delta \varphi(0))\).


Issue

In this section, we argue on an issue on δφ. In order to formalise δφ, we need to define values such as \(\delta \varphi([A]0)\), although it is not a normal expression. According to Jason, \(\delta \varphi([A],0)\) is expected to be smaller than \(A\), and \(A +_{\delta \varphi} \delta \varphi([A],0)\) is expected to coincide with \(\varepsilon_0 \times 2\), which is greater than \(A +_{\delta \varphi} \beta\) for any ordinal \(\beta\) below \(\varepsilon_0\). On the other hand, \(A\) is intended to "correspond" to \(\varepsilon_0\), and every ordinal below \(A\) can be expressed by \(0\), \(+_{\delta \varphi}\), and \(\delta \varphi(x_1)\). It implies that there is no ordinal \(\alpha\) such that \(\alpha\) is smaller than \(A\) but \(A +_{\delta \varphi} \alpha\) differs from \(A +_{\delta \varphi} \beta\) for any ordinal \(\beta\) below \(\varepsilon_0\). In order to avoid such an obvious contradiction, we need to justify the equalities above in terms of ordinal types in the following way:

Let \(C\) denote the set of ordinals which can be expressed by \(0\), the usual addition \(+\), and variadic functions \(\delta \varphi(x_1,\ldots,x_k)\) and \(\delta \varphi([d],x_1,\ldots,x_k)\) with respect to a certain restriction on the normality of expressions so that every ordinal in \(C\) admits a unique normal expression. For each \(\alpha \in C\), we denote by \(o(\alpha\)\) the ordinal type of the strict well-ordered set \((C \cap \alpha,\in) = (\{\beta \in C \mid \beta < \alpha\},\in)\). Forgetting the properties of \(\delta \varphi\) such as \(A = \varepsilon_0\) explained in the previous section, assume the following alternative conditions:

  1. \(\delta \varphi(x_1)\) coincides with \(\omega^{x_1}\) for any \(x_1\).
  2. \(\delta \varphi(1,0)\) is a sufficiently large ordinal such as \(\Omega\).
  3. The normality of an expression of ordinals below \(\varepsilon_0\) coincides with the normality in Cantor normal form.
  4. There is no ordinal \(\alpha \in C\) satisfying \(\varepsilon_0 \leq \alpha < A\).

Then we have \(o(A) = \varepsilon_0\). It does not contradict if we define \(\delta \varphi([A]0)\) as \(\varepsilon_0\) as long as it is not a normal expression. In particular, the properties \(o(A) = \varepsilon_0\), \(\delta \varphi([A]0) < A\), \(o(A + \delta \varphi([A]0)) = \varepsilon_0 \times 2\), and \(o(A + A) = \zeta_0\) are consistent. Since \(o\) does not necessarily commute with \(+\) as we expect \(o(A+A) = \zeta_0 \neq \varepsilon_0 \times 2 = o(A) + o(A)\), the usual addition plays the role of \(+_{\delta \varphi}\) without a modification. As a result, the equalities \(\alpha = \beta\) between expressions \(\alpha\) in δφ and actual ordinals \(\beta\) above should be regarded as short hands of the equalities \(o(\alpha) = \beta\).


Sources

  1. The user page of Jason in Japanese Googology Wiki.
  2. Jason, δφ対応表, Google Document.
  3. Jason, δφのいろいろ, Japanese Googology WIki user blog.


See also

By Aeton: Okojo numbers · N-growing hierarchy
By 新井 (Arai): Arai's \(\psi\)
By バシク (BashicuHyudora): Primitive sequence number · Pair sequence number · Bashicu matrix system 1/2/3/4
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 computation programmes · TR function (I0 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 小林銅蟲 (Kobayashi Doom): Sushi Kokuu Hen
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 computation programme · ω-Y sequence computation programme
By Nayuta Ito: N primitive · Flan numbers · Large Number Lying on the Boundary of the Rule of Touhou Large Number 4
By p進大好きbot: 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 · YY sequence · Y function · ω-Y sequence
See also: Template:Googology in Asia

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