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The book Roads to Infinity: The Mathematics of Truth and Proof stated that, "All the standard mathematical theories have axioms that can be expressed in predicate logic." Predicate logic generally refers to any formal system like first-order, second-order logic, many-sorted logic, or infinitary logic, but it sometimes refers to just first-order logic (Wikipedia). The book never specified which it was referring to. I understand that not all axioms can be expressed in first-order logic, but what about in second-order logic, many-sorted logic or infinitary logic?

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    $\begingroup$ The axioms of ZFC can be expressed in first order logic (with equality) where the language is extended by a single binary relation symbol $\in$. $\endgroup$ – Hagen von Eitzen Jul 10 '15 at 17:10
  • $\begingroup$ Axioms are sentences of some particular language, so I'm not sure what kind of notion of expressability you have in mind. $\endgroup$ – Malice Vidrine Jul 10 '15 at 17:33
  • $\begingroup$ About the "expressive power" , see Second-order and Higher-order Logic and Stewart Shapiro, Foundations without Foundationalism : A Case for Second-Order Logic (1991). $\endgroup$ – Mauro ALLEGRANZA Jul 10 '15 at 19:25
  • $\begingroup$ @MaliceVidrine I'm asking if there are sentences expressed in mathematics that have no equivalent expression in predicate logic (in the general sense of the word). $\endgroup$ – Kelmikra Jul 10 '15 at 22:01
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    $\begingroup$ @Kyth'Py1k: My point was, what does "expressed in mathematics" mean; and if they were not expressed in a predicate logic to start with, what kind of "equivalence" with a sentence of predicate logic do you have in mind? I think the question is ill posed. $\endgroup$ – Malice Vidrine Jul 11 '15 at 2:05

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