Epsilon numbers: Difference between revisions
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'''Epsilon numbers''' are [[fixed point|fixed points]] of the function \(\alpha\rightarrow\omega^\alpha\). They can also be informally visualized as \(\omega^{\omega^{\omega^{\cdots}}}\), though this represents all the epsilon numbers identically and doesn't have a formal definition. Since the function \(\alpha\rightarrow\omega^\alpha\) 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
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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