Reflection principle: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that, for all \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of...") |
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This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example.
== Alternate meaning ==
An alternate type of reflection principle instead asserts that, instead of properties of an inner model reflecting down to a level of a cumulative hierarchy, properties of a single level of a cumulative hierarchy reflect down to a lower level of the same cumulative hierarchy. In particular, this gives rise to [[stability]], reflecting ordinals and indescribable or shrewd cardinals.
Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\).
By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and [[Kripke-Platek set theory|admissible]] are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory.
Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals.
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