A Second order logic and completeness

  • Thread starter jordi
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Goldrei's Propositional and Predicate Calculus states (in my words; any mistake is mine) that first-order logic is complete, i.e. any logic deduction from a set of axioms (written in first-order logic) is equivalent to proving the theorem for all models satisfying the axioms.

Completeness is good to have, and for this reason, both the Peano and the Real numbers axioms, which by themselves are second-order logic, are usually stated within set theory, where they become first-order logic.

But then my question is: when axioms are in second-order logic, it is not true that "any logic deduction from a set of axioms is equivalent to proving the theorem for all models satisfying the axioms"?

Is there an example of a logical deduction from a set of axioms (written in second-order logic), such that the resulting theorem is not valid for at least one model satisfying the axioms?
 

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