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jacquesb
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The binary Veblen function [itex] \varphi_\alpha(\beta) [/itex] or [itex] \varphi(\alpha,\beta) [/itex] (see https://www.physicsforums.com/threa...-for-the-veblen-hierarchy-of-ordinals.933538/) can be generalized to finitely many variables, for example [itex] \varphi(\alpha,\beta,\gamma) [/itex] which can also be written [itex] \varphi(\alpha_2,\beta_1,\gamma_0) [/itex] (often written [itex] \varphi(\gamma_0,\beta_1,\alpha_2) [/itex] or [itex] \varphi(0 \rightarrow \gamma, 1 \rightarrow \beta, 2 \rightarrow \alpha) [/itex], but here I prefer to use reverse order to stay consistent with the convention I used for binary Veblen function). Indices represent the position of the variable, 0 may be ommited, for example [itex] \varphi(\alpha,0,\beta) [/itex] may be written [itex] \varphi(\alpha_2,\beta_0) [/itex]. With this notation we can generalize Veblen function to transfinitely many variables with a finite number different from 0, for example [itex] \varphi(1_\omega) [/itex] which is the limit of [itex] \varphi(1)=\varphi(0,1), \varphi(1,0), \varphi(1,0,0), \varphi(1,0,0,0), ... [/itex]. The set of all ordinals which can be reached with this Veblen function with transfinitely many variables (or the least ordinal which cannot be reached with it) is called the large Veblen ordinal.
For more explanations see for example :
- Wikipedia : https://en.wikipedia.org/wiki/Veblen_function
- Googology : http://googology.wikia.com/wiki/Ordinal_notation
- Veblen's article : http://www.ams.org/journals/tran/1908-009-03/S0002-9947-1908-1500814-9/S0002-9947-1908-1500814-9.pdf
My question is : is it possible to go further with this formalism ?
It seems that generally, when people want to go further, they use different formalism like Schütte's Klammersymbols or bracket, or ordinal collapsing functions, generally using [itex] \Omega [/itex], the first uncountable ordinal, to define countable ordinals. Is it because it is impossible to go furthen within the Veblen formalism, for example starting by enumerating the fixed points of [itex] \alpha \rightarrow \varphi(1_\alpha) [/itex] (the first one seems to me to be the large Veblen ordinal), or is it easier to use other formalisms beyond the large Veblen ordinal ?
For more explanations see for example :
- Wikipedia : https://en.wikipedia.org/wiki/Veblen_function
- Googology : http://googology.wikia.com/wiki/Ordinal_notation
- Veblen's article : http://www.ams.org/journals/tran/1908-009-03/S0002-9947-1908-1500814-9/S0002-9947-1908-1500814-9.pdf
My question is : is it possible to go further with this formalism ?
It seems that generally, when people want to go further, they use different formalism like Schütte's Klammersymbols or bracket, or ordinal collapsing functions, generally using [itex] \Omega [/itex], the first uncountable ordinal, to define countable ordinals. Is it because it is impossible to go furthen within the Veblen formalism, for example starting by enumerating the fixed points of [itex] \alpha \rightarrow \varphi(1_\alpha) [/itex] (the first one seems to me to be the large Veblen ordinal), or is it easier to use other formalisms beyond the large Veblen ordinal ?
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