Sentence examples for consistency statements from inspiring English sources

For this, we present the following results: (i) A proof of the necessary Lloyd-Max conditions for optimality of a graph quantizer, (ii) consistency statements for optimal graph quantizer design, and (iii) an accelerated version of competitive learning graph quantization.

The consistency statements for exponentially large Frege and extended Frege proofs can be used to characterize the provably total functions of second-order theories of bounded arithmetic for polynomial space and exponential time.

"In a progression of theories based on an initial theory of numbers, one attempts to overcome incompleteness by successively adding consistency statements or stronger principles reflecting on the systems in question," Feferman explained in an e-mail interview.

But neither of these consistency statements can be said to be intensionally correct.

We have already seen one way of climbing the hierarchy of the degrees of interpretability, namely, by adding consistency statements.

It is known that formalized consistency statements are unprovable whenever the provability predicate obeys certain general derivability conditions.

Second, the consistency statement does not take one very far up the hierarchy.

But PA can be strengthened by adding this consistency statement or by stronger axioms.

As a response to the Bezboruah and Sheperdson result, Pudlák gives (1996) an intensionally correct consistency statement for Robinson's theory Q, which Q fails to prove.

That is, we should grant the meta-theoretical claim that a theory T cannot prove its own consistency only when there is a sentence both which T "recognizes" as a consistency statement, and which T cannot prove.

First, if one starts with a theory that "arises in nature" and adds the consistency statement one lands in a degree that has no known representative that "arises in nature".