In first-order logic, all valid statements are provable. The statements about natural numbers not provable from given axioms are also independent of the axioms. Both PA+Con(PA) and PA+NOT(Con(PA)) are consistent.
Does Con(PA) mean “the consequences of the peano axioms? That’s the only meaning I can think of, but then I don’t see how you could avoid inconsistency if you accept some axioms and then deny their consequences (PA + NOT (Con(PA)).
Con(PA) means the consistency of PA. It can be formalized within PA as a statement about integers, something like “there’s no integer encoding a proof in PA that 1=0”.
For an explanation of Nesov’s counterintuitive statement that PA+NOT(Con(PA)) is a consistent theory (or rather, equiconsistent with PA itself), see paragraph 48.2 in this PDF. For an explanation of how to build a model for that preposterous theory, see paragraph 2.1 in this PDF.
In first-order logic, all valid statements are provable. The statements about natural numbers not provable from given axioms are also independent of the axioms. Both PA+Con(PA) and PA+NOT(Con(PA)) are consistent.
Does Con(PA) mean “the consequences of the peano axioms? That’s the only meaning I can think of, but then I don’t see how you could avoid inconsistency if you accept some axioms and then deny their consequences (PA + NOT (Con(PA)).
Con(PA) means the consistency of PA. It can be formalized within PA as a statement about integers, something like “there’s no integer encoding a proof in PA that 1=0”.
For an explanation of Nesov’s counterintuitive statement that PA+NOT(Con(PA)) is a consistent theory (or rather, equiconsistent with PA itself), see paragraph 48.2 in this PDF. For an explanation of how to build a model for that preposterous theory, see paragraph 2.1 in this PDF.