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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.
3. Jason, δφのいろいろ, Japanese Googology WIki user blog.

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