Re: Second order arithmetic

Aatu is exactly right.

What can cause confusion to the unwary here is the fact that the
second-order Peano Axioms do semantically entail all the truths of
arithmetic, even though we cannot derive all those truths from the
Peano axioms in some recursively axiomatized deductive system for
second-order logic, by Gödel's first theorem. (Second-order semantic
consequence is not deductively axiomatizable.)

There's a careful explanation in sections 22.1 to 22.6 of my Gödel
book if you want to follow that up in detail.

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www.godelbook.net


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Relevant Pages

  • Re: On Programs That Output Themselves
    ... >first-order framing of the Peano axioms. ... if he was referring to the second-order system then the ... first-order system then the claim he's saying is not justified ... >this and the FOL approximations to it. ...
    (sci.logic)
  • Re: On Programs That Output Themselves
    ... first-order framing of the Peano axioms. ... It is the second-order PA that does this, ... We don't restrict ourselves to FOL in practice. ... this and the FOL approximations to it. ...
    (sci.logic)
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