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. I also suggest that this task can only be performed adequately if one of the proposed conditions is that every implicit definition be univocal.

Consider (I do not assume that

Peano arithmetic is first-order)

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. Frege achieved this in GgA, but the system of GgA is inconsistent.

Footnotes have been added by the author because Feferman does assume prior understanding of some formal-language concepts, for example: "

Peano Arithmetic" (PA) and "recursive".

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. Each is indexed separately.

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.sup.[starf]] that contains its own truth predicate.

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.