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Line 18: As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref> : It may be helpful to give ~~sine~~some informal arguments illustrating the use of reflection principles. : The simplest is perhaps: the universe of sets is inaccessible (i.,e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)). Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" />