Sentence examples for theoretic strength from inspiring English sources

The limited proof theoretic strength of CZF compared with IZF has often been considered one of the main advantages of constructive over intuitionistic set theory.

In fact, CZF and CZFExp have the same proof theoretic strength (Griffor and Rathjen 1994); therefore to investigate the relation between subset collection and exponentiation in constructive set theory one needed to develop tools other then proof theoretic methods.

In many cases the proof theoretic strength of a system has been determined by a chain of interpretations between constructive and classical systems, and by using a variety of tools, from relisability to more "traditional" proof theoretic techniques, as ordinal analysis (see, for example, Beeson 1985; Griffor and Rathjen 1994; Rathjen 2012b).

Syntactically, this means that given a set B, we can form a new set {X ∈ B : φ(X)} by separation only if any quantifier in φ is bounded, that is, it occurs only in contexts ∀X (X∈Y →...) and ∃X(X∈Y ∧...), for some set Y. We can see that this constraint on separation is efficacious since the proof theoretic strength of CZF, which has only restricted separation, is within the range of predicativity.

It is not surprising, then, that investigations on the proof-theoretic strength of constructive and intutionistic set theories have been a crucial meta-theoretical tool in the attempt to understand these theories and the relations between them.

If either direction of the rule is dropped but the other kept, FS retains its proof-theoretic strength (Sheard 2001).

As it turns out, this choice may considerably affect the proof-theoretic strength of the resulting theory.

The proof-theoretic strength of second-order Arithmetic is way above all ramified extensions of Arithmetic considered by Schütte.

For a definition of the notion of proof-theoretic strength and for surveys on proof theory see for example (Rathjen 1999, 2006b).

The proof-theoretic strength of theories of inductive definitions goes well beyond Feferman and Schütte's bound (Buchholz, Feferman, Pohlers and Sieg 1981).

In fact, we can show that a theory cannot prove a specific principle because adding it to the theory would increase the theory's proof-theoretic strength.