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It is a very basic question, but could not find it elsewhere and I think is the source of many of the misunderstandings that I have. My understanding is that in principle you can define a hierarchy of formulas of the form $\Pi^m_n$ and $\Sigma^m_n$, for any math theory, provided you use the appropriate language. But I am not sure if when I see those symbols in the literature they always refer to formulas of PA.

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Might be related:… – Yoni Rozenshein Apr 20 '13 at 9:32
This is definitely not a question for first-order-logic! Especially when allowing $m>0$. :-) – Asaf Karagila Apr 20 '13 at 22:36
up vote 4 down vote accepted

No, the symbols are also used elsewhere. For example indescribable cardinals do not refer to cardinals which cannot be described by formulas in the language of arithmetics.

Taken from Kanamori, The Higher Infinite:

$\kappa$ is $Q$-indescribable iff for any $R\subseteq V_\kappa$ and $Q$ sentence $\varphi$ such that $\langle V_\kappa,\in, R\rangle\models\varphi$ there is $\alpha<\kappa$ such that $\langle V_\alpha,\in,R\cap V_\alpha\rangle\models\varphi$. (p. 58)

In this context $Q$ is $\Sigma^m_n$ or $\Pi^m_n$. But the language itself is an arbitrary language containing two symbols, and certainly not over $\Bbb N$.

You can read more in the first section, (p. 5) where Kanamori discusses hierarchies of formulas.

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One more: is (or can be modified to be) the definition of Q-indescribable that you gave above of the format $Pi_n^m$ or $\Sigma_n^m$, is it in some metalanguage, or does ZFC allow formulas that cannot be expressed in that format? – Wolphram jonny Apr 21 '13 at 1:04
julian, $Q$ is a language which is in the universe, and since we talk about set model for that language we can define its truth, and encode everything within the universe. So no need to go meta here. – Asaf Karagila Apr 21 '13 at 1:07
I do not mean Q but the whole definition given in gray, is that a formula in the language of set theory? – Wolphram jonny Apr 21 '13 at 1:09
Yes, because we can encode high-order formula of set-structures in the universe; and we can say when they are true or false in particular set-structures. – Asaf Karagila Apr 21 '13 at 1:14
Thanks for the reference, I 've got the book and I am gonna read it. in the mean time, could you tell me in advance if the definition in gray is of the format $\Sigma_n^m$ or $\Pi_n^m$ for some m, n? or it goes beyond predicate calculus+$\in$+axioms? – Wolphram jonny Apr 21 '13 at 19:33

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