Mahlo cardinal: Difference between revisions
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Ord is Mahlo has interesting consistency strength, as we've mentioned. Recall from [[Reflection principle|here]] that a cardinal \( \kappa \) is sound if \( V_\kappa \) is a full elementary substructure of \( V \). Such cardinals are massive, but their existence is provable in \( \mathrm{ZFC} \), due to the reflection principle. In particular, for all \(n\), we have a club of cardinals which are \(\Sigma_n\)-sound, and thus their intersection is also club. Meanwhile, say a cardinal \( \kappa \) is totally reflecting if it is sound and strongly inaccessible. Such cardinals are hyper-inaccessible and larger than virtually any other large cardinal axiom size-wise other than possibly stationary superhuges or Reinhardt cardinals. However, their consistency strength is not particularly high: it turns out that Ord is Mahlo has the same consistency strength as the existence of a totally reflecting cardinal, which shows that slight modifications of reflection principles can give large consistency strength. |
Ord is Mahlo has interesting consistency strength, as we've mentioned. Recall from [[Reflection principle|here]] that a cardinal \( \kappa \) is sound if \( V_\kappa \) is a full elementary substructure of \( V \). Such cardinals are massive, but their existence is provable in \( \mathrm{ZFC} \), due to the reflection principle. In particular, for all \(n\), we have a club of cardinals which are \(\Sigma_n\)-sound, and thus their intersection is also club. Meanwhile, say a cardinal \( \kappa \) is totally reflecting if it is sound and strongly inaccessible. Such cardinals are hyper-inaccessible and larger than virtually any other large cardinal axiom size-wise other than possibly stationary superhuges or Reinhardt cardinals. However, their consistency strength is not particularly high: it turns out that Ord is Mahlo has the same consistency strength as the existence of a totally reflecting cardinal, which shows that slight modifications of reflection principles can give large consistency strength. |
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Furthermore, let \( \mathrm{MP}(\mathbb{R}) \), the maximality principle for the real numbers be the following statement: "assume \( r \) is a real number and \( \varphi \) is a formula. Then if there is a forcing extension \( V[G] \) so that \( \varphi(r) \) and \( \varphi(r) \) persists, i.e. remains true in all subsequent extensions \( V[G][H] \), then \( \varphi(r) \) is already true in the universe". Essentially, the theory of the real numbers is already maximal, and it's not possible to |
Furthermore, let \( \mathrm{MP}(\mathbb{R}) \), the maximality principle for the real numbers be the following statement: "assume \( r \) is a real number and \( \varphi \) is a formula. Then if there is a forcing extension \( V[G] \) so that \( \varphi(r) \) and \( \varphi(r) \) persists, i.e. remains true in all subsequent extensions \( V[G][H] \), then \( \varphi(r) \) is already true in the universe". Essentially, the theory of the real numbers is already maximal, and it's not possible to for a statement with real numbers to be false but, in extensions of the universe, possibly necessary. The statement \( \mathrm{MP}(\mathbb{R}) \) has less consistency strength than \( \mathrm{MP}(V) \), where \( r \) is an arbitrary set, and is actually equiconsistent with Ord is Mahlo. |
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