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== Example == |
== Example == |
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| − | Since definition shifting is considered as a generalisation of side nesting, [[Side nesting#Examples|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. |
+ | Since definition shifting is considered as a generalisation of side nesting, [[Side nesting#Examples|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. Three other examples are given by Jason: |
| + | * [[δOCF]] |
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| + | * [[δφ]] |
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| + | * [[ε function]] |
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| − | Although neither of them has been |
+ | Although neither of them has been fully defined yet, they are expected to be significantly strong if they will be appropriately formalised. |
=== δOCF === |
=== δOCF === |
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| + | {{main|δOCF}} |
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| − | 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. |
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| ⚫ | 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.<ref>Jason, [https://drive.google.com/file/d/1nT3MBXtKyRyVXepSSHfoqoEUbCfaErH9/view δ関数の展開ルール], Google Drive.</ref><ref>Jason, [https://docs.google.com/presentation/d/1lyVkKA6i0EE_vWVlO26w15RdXou5ejuJwsDHpGddjdM/edit#slide=id.p1 δ解説 もっと詳しく], Google Document.</ref><ref>Jason, [https://twitter.com/D13jason2/status/1127226941205954561 グラハム数〜legend of δ〜], twiiter.</ref> |
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=== δφ === |
=== δφ === |
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| + | {{main|δφ}} |
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| − | δφ 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. |
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| ⚫ | δφ is the second system defined by shifting definition, and is intended to perform as an ordinal function associated to [[Veblen function]] through shifting definition.<ref>Jason, [https://docs.google.com/spreadsheets/d/1hGbOqFrMu1SergIDwLlO9yFx_8yGYwuB0Mi2qlBm3cA/edit#gid=0 δφ対応表], Google Document.</ref><ref>Jason, [https://googology.wikia.org/ja/wiki/%E3%83%A6%E3%83%BC%E3%82%B6%E3%83%BC%E3%83%96%E3%83%AD%E3%82%B0:%E3%81%98%E3%81%87%E3%81%84%E3%81%9D%E3%82%93/%CE%B4%CF%86%E3%81%AE%E3%81%84%E3%82%8D%E3%81%84%E3%82%8D δφのいろいろ], Japanese Googology WIki user blog.</ref> |
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| − | 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. |
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| − | 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. |
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| + | === ε function === |
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| − | 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. |
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| + | {{main|ε function}} |
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| − | 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 \(A +_{\delta \varphi} A\). |
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| + | ε function is the third system defined by shifting definition, and it intend to perform as a notation inclusing shifting definition and higher system called ''meta-shifting definition'', ''meta-meta-shifting definition'', and so on.<ref>Jason, [https://docs.google.com/document/d/1VoDgcsa-4pv1_R20aNPNCm-fTkeOzHz1wxYqtss7LEw/edit ε関数定義試作], Google Document. (Trial to formalise the notation.)</ref><ref>Jason, [https://docs.google.com/spreadsheets/d/1dvsZtGQomMTQolqerGmtrnAWjIlSJFa3zxl28-eLg6Y/edit#gid=272325030 ε関数成長記録], Google Document. (Comparison to other notations.)</ref><ref>Jason, [https://docs.google.com/document/d/15ETSEe2xMIz7wxE97qAhHcEhQT16rssIHcz_rffK6qo/edit ε関数 ver ε.0.1.0], Google Document. (The current version.)</ref><ref>Jason, [https://googology.wikia.org/ja/wiki/%E3%83%A6%E3%83%BC%E3%82%B6%E3%83%BC%E3%83%96%E3%83%AD%E3%82%B0:%E3%81%98%E3%81%87%E3%81%84%E3%81%9D%E3%82%93/%CE%B5%E9%96%A2%E6%95%B0%E3%81%AE%E8%A7%A3%E8%AA%AC%E3%82%92%E8%A9%A6%E3%81%BF%E3%82%8B ε関数の解説を試みる], Japanese Googology Wiki user blog. (Trial to explain ε function)</ref> |
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| − | 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 \(A +_{\varphi \delta} A\). |
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| − | 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: |
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| − | * \(\delta \varphi(A +_{\delta \varphi} \delta \varphi(0))\), 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 \(\varphi(\omega,x_1)\). |
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| − | * \(\delta \varphi(\delta \varphi(A +_{\delta \varphi} \delta \varphi(0)))\), which is often abbreviated to \(A^{\omega}\) although it does not coincides with \(\varepsilon_0^{\omega}\), is intended to coincide with [[Small Veblen ordinal|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. |
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| − | * \(\delta \varphi(\delta \varphi(A +_{\delta \varphi} A))\), which is often abbreviated to \(A^A\) although it does not coincides with \(\varepsilon_0^{\varepsilon_0}\), is intended to coincide with [[Large Veblen ordinal|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. |
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| − | * \(\delta \varphi([A]A +_{\delta \varphi} \delta \varphi(0))\), which might be abbreviated to \(\varepsilon_{A+1}\) although it does not coincides with \(\varepsilon_{\varepsilon_0+1}\), is intended to coincide with [[Bachmann-Howard ordinal|the Bachmann-Howard ordinal]]. |
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| − | 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 \(\delta \varphi([\varphi([\cdots \varphi([A]A +_{\delta \varphi} \delta \varphi(0)) \cdots]A +_{\delta \varphi} \delta \varphi(0))]A +_{\delta \varphi} \delta \varphi(0))\). |
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| − | 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: |
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| − | 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: |
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| − | # \(\delta \varphi(x_1)\) coincides with \(\omega^{x_1}\) for any \(x_1\). |
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| − | # \(\delta \varphi(1,0)\) is a sufficiently large ordinal such as \(\Omega\). |
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| − | # The normality of an expression of ordinals below \(\varepsilon_0\) coincides with the normality in Cantor normal form. |
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| − | # There is no ordinal \(\alpha \in C\) satisfying \(\varepsilon_0 \leq \alpha < A\). |
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| − | 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 \(o(\alpha) = \beta\). |
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| + | * [[side nesting]] |
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| − | == See Also == |
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{{Googology in Asia}} |
{{Googology in Asia}} |
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[[Category:Notations]] |
[[Category:Notations]] |
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[[Category:Functions]] |
[[Category:Functions]] |
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[[Category:Googology in Asia]] |
[[Category:Googology in Asia]] |
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| − | [[Category: |
+ | [[Category:Shifting definition]] |
| + | [[Category:Higher computable level]] |
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Revision as of 12:18, 14 November 2020
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.[1] 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. Three other examples are given by Jason:
Although neither of them has been fully defined yet, they are expected to be significantly strong if they will be appropriately formalised.
δOCF
- Main article: 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.[2][3][4]
δφ
- Main article: δφ
δφ is the second system defined by shifting definition, and is intended to perform as an ordinal function associated to Veblen function through shifting definition.[5][6]
ε function
- Main article: ε function
ε function is the third system defined by shifting definition, and it intend to perform as a notation inclusing shifting definition and higher system called meta-shifting definition, meta-meta-shifting definition, and so on.[7][8][9][10]
References
- ↑ The user page of Jason in Japanese Googology Wiki.
- ↑ Jason, δ関数の展開ルール, Google Drive.
- ↑ Jason, δ解説 もっと詳しく, Google Document.
- ↑ Jason, グラハム数〜legend of δ〜, twiiter.
- ↑ Jason, δφ対応表, Google Document.
- ↑ Jason, δφのいろいろ, Japanese Googology WIki user blog.
- ↑ Jason, ε関数定義試作, Google Document. (Trial to formalise the notation.)
- ↑ Jason, ε関数成長記録, Google Document. (Comparison to other notations.)
- ↑ Jason, ε関数 ver ε.0.1.0, Google Document. (The current version.)
- ↑ Jason, ε関数の解説を試みる, Japanese Googology Wiki user blog. (Trial to explain ε function)
See Also
Indian counting system: Lakh · Crore · Tallakshana · Uppala · Dvajagravati · Paduma · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra · Nirabhilapya nirabhilapya parivarta · Jaghanya Parīta Asaṃkhyāta
Chinese, Japanese and Korean counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
See also: Template:Googology in Japan