Small Veblen ordinal: Difference between revisions
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The Small Veblen ordinal is the limit of a finitary, variadic extension of the [[Veblen hierarchy]]. In particular, after the basic stage \( \varphi(\alpha, \beta) \), one lets \( \varphi(1,0,\alpha) \) enumerate fixed points of \( \beta \mapsto \varphi(\beta,0) \) - i.e. [[Veblen hierarchy|strongly critical ordinals]] - followed by \( \varphi(1,1,\alpha) \) enumerating its fixed points, and so on. The Small Veblen ordinal, very commonly abbreviated to SVO, is the least ordinal not reachable from below via this function, namely the limit of \( \omega \), \( \varepsilon_0 \), \( \Gamma_0 \), \( \varphi(1,0,0,0) \) (the Ackermann ordinal), ... In ordinal collapsing functions, in particular Buchholz's psi function, it is considered the countable collapse of \( \Omega^{\Omega^\omega} \), and may be denoted by \( \psi_0(\Omega^{\Omega^\omega}) \). |
The Small Veblen ordinal is the limit of a finitary, variadic extension of the [[Veblen hierarchy]]. In particular, after the basic stage \( \varphi(\alpha, \beta) \), one lets \( \varphi(1,0,\alpha) \) enumerate fixed points of \( \beta \mapsto \varphi(\beta,0) \) - i.e. [[Veblen hierarchy|strongly critical ordinals]] - followed by \( \varphi(1,1,\alpha) \) enumerating its fixed points, and so on. The Small Veblen ordinal, very commonly abbreviated to SVO, is the least ordinal not reachable from below via this function, namely the limit of \( \omega \), \( \varepsilon_0 \), \( \Gamma_0 \), \( \varphi(1,0,0,0) \) (the Ackermann ordinal), ... In ordinal collapsing functions, in particular Buchholz's psi function, it is considered the countable collapse of \( \Omega^{\Omega^\omega} \), and may be denoted by \( \psi_0(\Omega^{\Omega^\omega}) \). |
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Latest revision as of 16:46, 25 March 2024
The Small Veblen ordinal is the limit of a finitary, variadic extension of the Veblen hierarchy. In particular, after the basic stage \( \varphi(\alpha, \beta) \), one lets \( \varphi(1,0,\alpha) \) enumerate fixed points of \( \beta \mapsto \varphi(\beta,0) \) - i.e. strongly critical ordinals - followed by \( \varphi(1,1,\alpha) \) enumerating its fixed points, and so on. The Small Veblen ordinal, very commonly abbreviated to SVO, is the least ordinal not reachable from below via this function, namely the limit of \( \omega \), \( \varepsilon_0 \), \( \Gamma_0 \), \( \varphi(1,0,0,0) \) (the Ackermann ordinal), ... In ordinal collapsing functions, in particular Buchholz's psi function, it is considered the countable collapse of \( \Omega^{\Omega^\omega} \), and may be denoted by \( \psi_0(\Omega^{\Omega^\omega}) \).