Omega^omega: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is: * The least \( \alpha \) so that \( \alpha \) is the \( \alpha \)th limit ordinal. * The least limit of additive principal ordinals. * The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator. * The proof-theoretic ordin...") |
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{{DISPLAYTITLE:\( \omega^\omega \)}} |
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The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is: |
The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is: |
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* The least limit of additive principal ordinals. |
* The least limit of additive principal ordinals. |
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* The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator. |
* The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator. |
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* The proof-theoretic ordinal of |
* The proof-theoretic ordinal of [[Second-order arithmetic|\(\mathrm{RCA}_0\)]] |
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* The proof-theoretic ordinal of |
* The proof-theoretic ordinal of [[Second-order arithmetic|\(\mathrm{WKL}_0\)]]. |
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* The proof-theoretic ordinal of Peano arithmetic, with induction restricted to \( \Sigma^0_1 \)-formulae. |
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* The proof-theoretic ordinal of primitive recursive arithmetic. |
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Latest revision as of 23:02, 22 March 2024
The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is:
- The least \( \alpha \) so that \( \alpha \) is the \( \alpha \)th limit ordinal.
- The least limit of additive principal ordinals.
- The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator.
- The proof-theoretic ordinal of \(\mathrm{RCA}_0\)
- The proof-theoretic ordinal of \(\mathrm{WKL}_0\).
- The proof-theoretic ordinal of Peano arithmetic, with induction restricted to \( \Sigma^0_1 \)-formulae.
- The proof-theoretic ordinal of primitive recursive arithmetic.