Like Gottlob Frege, Russell's basic idea for defending logicism was that numbers may be identified with classes of classes and that number-theoretic statements may be explained in terms of quantifiers and identity.
According to Gödel's incompleteness theorems, the statement that Peano Arithmetic (PA) is consistent, in its guise as a number-theoretic statement (given the technique of Gödel numbering), cannot be derived in PA itself.
The question is how this problem formulation can be reconciled with a decision-theoretic problem statement that involves utilities.
Anti-realists have one in the Model-Theoretic Argument.
For the model-theoretic actualist, the statement that it is possible that Socrates exists, ◇E!s, is evaluated straightaway as true at M just in case there is a Tarski model M in M at which E s is true — the "actual" model M@ indexed by @, for instance.
The independence techniques were so powerful that set theorists soon found themselves preoccupied with the meta-theoretic enterprise of proving that certain fundamental statements could not be proved or refuted within ZFC.
The result was triggered by Church himself in 1934, when he used the Richard paradox to prove a kind of incompleteness theorem (with respect to statements asserting the totality of number theoretic functions).
In order to address this difficulty, we propose and examine rigorous complexity-theoretic models of the statement 'the decision problem D has a phase-transition'.
For example, in the time of Copernicus, when the Ptolemaic and Copernican theory were observationally indistinguishable, theoretical reasons could still be given for the Copernican theory over the Ptolemaic theory.[15] Similarly in the set theoretic case the non-pluralist sees theoretical reason at play at higher level, beyond the theorems and the intrinsically plausible statements.
Thus much discussion in this area has concerned the comparative merits of the transworld identity and counterpart-theoretic accounts as interpretations, within a possible-worlds framework, of statements of what is possible and necessary for particular individuals.
