Shifting definition

Shifting definition' (定義ずらし in Japanese) is a method to generate a googological system introduced by a Japanese Googology Wiki user Jason. It is regarded as a generalisation of side nesting by its creator mrna.

Feature
Similar to side nesting, a \(2\)-ary function symbol \(+\) in a notation defined by shifting definition does not necessarily work as the addition. Namely, even if a valid expression \(a\) corresponds to a countable ordinal \(\alpha\) and \(a + a\) is also a valid expression, \(a + a\) does not necessarily corresponds to \(\alpha + \alpha\).

Another specific feature of shifting definition is that it is intended to be a correspondence which assigns a new defining formula to a defining formula of ordinal functions in a certain class. In order to formalise this strategy, we need to encode defining formulae in set theory in some way. Although the method has not been formalised yet, Jason intends to encode definining formulae into ordinals. There are two examples of unformalised googological systems which are intended to be justified by shifting definition through the encoding of defining formulae into ordinals. We will explain them in the next section.

Example
Since definition shifting is considered as a generalisation of side nesting, the notations in the articles of side nesting, which has not been formalised yet, are examples of notations associated to ordinal functions defined by shifting definition. Two other examples are given by Jason: Although neither of them has been formalised yet, they are expected to be significantly strong if they will be appropriately formalised.
 * δOCF
 * δφ

δOCF
δOCF is the first system defined by shifting definition, and is intended to perform as an ordinal function associated to an OCF through the method. Although it was rejected by a referee of a Japanese googological event to which Jason submitted δOCF, it is expected to be a powerful function which goes beyond many other OCFs if it will be appropriately formalised. Since it is quite complicated for others to understand from the explanation of its intended behaviour, Jason created δφ as a simpler ordinal function without using an OCF.

δφ
δφ is intended to perform as an ordinal function associated to Veblen function through shifting definition. The 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. 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 \(\delta \varphi([d]0,x_2)\). 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: 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 \(\delta \varphi([\varphi([\cdots \varphi([A]A +_{\delta \varphi} \delta \varphi(0)) \cdots]A +_{\delta \varphi} \delta \varphi(0))]A +_{\delta \varphi} \delta \varphi(0))\).
 * \(\delta \varphi(A +_{\delta \varphi} \delta \varphi(0))\), 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.

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: 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 \(+\), 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 \(o(\alpha) = \beta\).
 * 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 \(\varepsilon_0 \leq \alpha < A\).