### arnold beckmann's pages

## Applications of cut-free infinitary derivations
to generalized recursion theory

**File:** PDF-File

**Author:** Arnold Beckmann and Wolfram Pohlers

**Title:** Applications of cut-free infinitary derivations
to generalized recursion theory

**Journal:** Ann.Pure.Appl.Logic 1998, **94**: 7-19

**DOI:** 10.1016/S0168-0072(97)00063-8

**Abstract:**
In this paper we introduce an infinitary proof system
for a language of second order arithmetic which is complete for
Π
1
1
- sentences.
The main observation there is the Boundedness Theorem telling that
the order-type of a
Σ
1
1
- definable well-ordering
<
is less than or equal to the truth complexity
of
TI(<)
.
The commonly known Boundedness Principle of Generalized Recursion
Theory will be obtained as a consequence of our proof theoretical
Boundedness Theorem which in turn is a consequence of cut-freeness.
This yields a proof of the Boundedness Principle which does not use
the Analytical Hierarchy Theorem.