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I know that there are statements that are neither provable nor disprovable within some set of axioms, and I also know that such statements are called undecidable. Please allow me to call these statements to be undecidable to the first order, or belong to $U_1$.

I was wondering if there is some kind of generalization of this concept. Are there any conjectures/statements of which we can prove that we cannot prove whether it is decidable or not? Such a statement would be undecidable to the second order, or belong to $U_2$. Generalizing even further:

What about statements that are in $U_\infty$ ?

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I have added [higher-order-logic], I am not too sure about [meta-math] tag. @Max, why do you think this has anything to do with metamathematics? –  Asaf Karagila Aug 1 '11 at 15:08
    
@Asaf: err I guess I thought this because I don't really know what metamathematics is, yet. I just suspected it. Feel free to edit the tags. –  Max Muller Aug 1 '11 at 15:09
    
Possible duplicate: Is there a statement whose undecidability is undecidable? –  Zhen Lin Aug 1 '11 at 15:13
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By contradiction, $U_{n+1}\subseteq U_n$. Presumably at each $U_n$ one must step back to an even higher-order formal system. Ugh. @Zhen: OP's question is only a (half-)duplicate at $n=2$, and it is not specific to ZFC. –  anon Aug 1 '11 at 15:28
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I don't think this question is about higher-order logic in the usual sense. The OP unfortunately uses the word "order" in the context "undecidable to second order" to mean something quite different from its meaning in "second-order logic". –  Andreas Blass Nov 10 '12 at 15:02

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