Large Veblen ordinal: Difference between revisions
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The '''large Veblen ordinal''', also called the '''great Veblen number''',<ref>Rathjen, https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, p.10</ref> is a large extension of the [[small Veblen ordinal]]. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the [[Veblen hierarchy]] used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,\ldots,0,0) \), with \( \omega \) many zeroes. The new limit of this system is the large Veblen ordinal, or the fixed point of \( \alpha \mapsto \varphi\begin{pmatrix} 1 \\ \alpha \end{pmatrix} \), where \( \begin{pmatrix} 1 \\ \alpha \end{pmatrix} \) denotes a one followed by \( \alpha \) many zeroes. This is another milestone of the recursive ordinals, and may be represented in ordinal collapsing functions as the collapse of \( \Omega^{\Omega^\Omega} \). This system has further been extended to a system known a dimensional Veblen, where one can diagonalize over the amount of zeroes with expressions such as \( \varphi\begin{pmatrix} 1 \\ (1,0) \end{pmatrix} \) (should there be parentheses in the second row?), which represents the large Veblen ordinal.<ref> |
The '''large Veblen ordinal''', also called the '''great Veblen number''',<ref>Rathjen, https://www1.maths.leeds.ac.uk/~rathjen/ICMend.pdf, p.10</ref> is a large extension of the [[small Veblen ordinal]]. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the [[Veblen hierarchy]] used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,\ldots,0,0) \), with \( \omega \) many zeroes. The new limit of this system is the large Veblen ordinal, or the fixed point of \( \alpha \mapsto \varphi\begin{pmatrix} 1 \\ \alpha \end{pmatrix} \), where \( \begin{pmatrix} 1 \\ \alpha \end{pmatrix} \) denotes a one followed by \( \alpha \) many zeroes. This is another milestone of the recursive ordinals, and may be represented in ordinal collapsing functions as the collapse of \( \Omega^{\Omega^\Omega} \). This system has further been extended to a system known a dimensional Veblen, where one can diagonalize over the amount of zeroes with expressions such as \( \varphi\begin{pmatrix} 1 \\ (1,0) \end{pmatrix} \) (should there be parentheses in the second row?), which represents the large Veblen ordinal.<ref>https://arxiv.org/abs/2310.12832v1</ref> This system's limit is the [[Bachmann-Howard ordinal]]. |
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Latest revision as of 03:38, 21 October 2023
The large Veblen ordinal, also called the great Veblen number,[1] is a large extension of the small Veblen ordinal. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the Veblen hierarchy used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,\ldots,0,0) \), with \( \omega \) many zeroes. The new limit of this system is the large Veblen ordinal, or the fixed point of \( \alpha \mapsto \varphi\begin{pmatrix} 1 \\ \alpha \end{pmatrix} \), where \( \begin{pmatrix} 1 \\ \alpha \end{pmatrix} \) denotes a one followed by \( \alpha \) many zeroes. This is another milestone of the recursive ordinals, and may be represented in ordinal collapsing functions as the collapse of \( \Omega^{\Omega^\Omega} \). This system has further been extended to a system known a dimensional Veblen, where one can diagonalize over the amount of zeroes with expressions such as \( \varphi\begin{pmatrix} 1 \\ (1,0) \end{pmatrix} \) (should there be parentheses in the second row?), which represents the large Veblen ordinal.[2] This system's limit is the Bachmann-Howard ordinal.