But when applied to arithmetic, it is argued, the imposition of a conservativeness requirement leads to a serious objection to deflationism, for the Godel sentence for Peano Arithmetic
(PA) is not a theorem of PA, but becomes one when PA is extended by adding plausible principles governing truth.
The authors begin by covering the development of Peano arithmetic
, mathematical induction and recursion theory.
In par+ticular, I argue that Hale and Wright have not given enough conditions to separate appropriate implicit definitions such as Hume's Principle from rival implicit definitions like Second-Order Peano Arithmetic
Consider (I do not assume that Peano arithmetic
The book Frege's Theorem focuses, obviously, on Frege's Theorem: that second-order logic supplemented with a cardinality principle also called "Hume's Principle" (HP) allows to derive the axioms of Peano Arithmetic
The way Godel established this is by formalising the entire preceding argument for the first incompleteness theorem in Peano Arithmetic
Those lecture series have been boiled down to four refereed essays on countable models and the theory of Boral equivalence relations, model theory of difference fields, some computability-theoretic aspects of reals and randomness, and weak fragments of Peano arithmetic
For example if we add to first-order Peano arithmetic
the statement that Peano arithmetic
is inconsistent, the resulting provability logic fails to count as modal by Koslow's criterion.
We add Priest's dialetheic semantics to ordinary Peano arithmetic
PA, to produce a recursively axiomatized formal system P[A.
Godel's second theorem is included because it involves an iteration of deviant coextensive provability operations that satisfy different principles in Peano arithmetic
, relevant to the mechanistic arguments in the philosophy of mind.