Epsilon numbers: Difference between revisions
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'''Epsilon numbers''' are [[fixed point|fixed points]] of the function \(\alpha\rightarrow\omega^\alpha\). Since the function is continuous in the order topology, they are the same as the closure points. Using the [[Veblen hierarchy]], they are enumerated as \(\varphi(1,\alpha)\). The least epsilon number is the limit of "predicative" [[Cantor normal form]], since, as we mentioned, it can't be reached from below via base-\(\omega\) exponentiation. And, in general, \(\varphi(1,\alpha+1)\) is the least ordinal that can't be reached from \(\varphi(1,\alpha)\). |
'''Epsilon numbers''' are [[fixed point|fixed points]] of the function \(\alpha\rightarrow\omega^\alpha\). Since the function is continuous in the order topology, they are the same as the closure points. Using the [[Veblen hierarchy]], they are enumerated as \(\varphi(1,\alpha)\). The least epsilon number is the limit of "predicative" [[Cantor normal form]], since, as we mentioned, it can't be reached from below via base-\(\omega\) exponentiation. And, in general, \(\varphi(1,\alpha+1)\) is the least ordinal that can't be reached from \(\varphi(1,\alpha)\). |
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By Veblen's fixed point lemma, the enumerating function of the epsilon numbers is normal and thus also has fixed points - these are denoted \(\varphi(2,\alpha)\) or \(\zeta_\alpha\). By iterating Cantor normal form and the process of taking (common) fixed points, the [[Veblen hierarchy]] is formed. This induces a natural normal form, called Veblen normal form. Its limit is not \(\zeta_0\), but a much larger ordinal, denoted \(\Gamma_0\). And in general, the ordinals that can't be obtained from below via Veblen normal form are called strongly critical. They are important in ordinal analysis. |
By Veblen's fixed point lemma, the enumerating function of the epsilon numbers is normal and thus also has fixed points - these are denoted \(\varphi(2,\alpha)\) or \(\zeta_\alpha\). (Use of the letter \(\zeta\) seems a bit difficult to find, for example sometimes it is called \(\kappa_\alpha\): https://mathoverflow.net/questions/243502) By iterating Cantor normal form and the process of taking (common) fixed points, the [[Veblen hierarchy]] is formed. This induces a natural normal form, called Veblen normal form. Its limit is not \(\zeta_0\), but a much larger ordinal, denoted \(\Gamma_0\). And in general, the ordinals that can't be obtained from below via Veblen normal form are called strongly critical. They are important in ordinal analysis. |
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Revision as of 23:04, 31 August 2023
Epsilon numbers are fixed points of the function \(\alpha\rightarrow\omega^\alpha\). Since the function is continuous in the order topology, they are the same as the closure points. Using the Veblen hierarchy, they are enumerated as \(\varphi(1,\alpha)\). The least epsilon number is the limit of "predicative" Cantor normal form, since, as we mentioned, it can't be reached from below via base-\(\omega\) exponentiation. And, in general, \(\varphi(1,\alpha+1)\) is the least ordinal that can't be reached from \(\varphi(1,\alpha)\).
By Veblen's fixed point lemma, the enumerating function of the epsilon numbers is normal and thus also has fixed points - these are denoted \(\varphi(2,\alpha)\) or \(\zeta_\alpha\). (Use of the letter \(\zeta\) seems a bit difficult to find, for example sometimes it is called \(\kappa_\alpha\): https://mathoverflow.net/questions/243502) By iterating Cantor normal form and the process of taking (common) fixed points, the Veblen hierarchy is formed. This induces a natural normal form, called Veblen normal form. Its limit is not \(\zeta_0\), but a much larger ordinal, denoted \(\Gamma_0\). And in general, the ordinals that can't be obtained from below via Veblen normal form are called strongly critical. They are important in ordinal analysis.