Kreisel ordinal

Kreisel ordinal
Name	Kreisel ordinal
Field	Mathematical logic
Introduced	1960s
Introduced by	Georg Kreisel

Kreisel ordinal is a countable ordinal associated with canonical proof-theoretic or computability-theoretic measures introduced in the mid-20th century. It serves as a benchmark in classifying consistency strength and recursion-theoretic complexity for formal systems and is studied in connection with ordinal analysis, proof theory, and recursion theory. The concept interacts with work by major figures and institutions in mathematical logic, set theory, and computability theory.

Definition and basic properties

A Kreisel ordinal is defined in terms of well-foundedness and effective presentations arising from analyses of formal systems such as those studied by Georg Kreisel, Gerald Sacks, Seymour Ginsburg, Stephen Kleene, Hilary Putnam, and Solomon Feferman. It is characterized via effective notations related to ordinal notations developed by W. W. Tait, Gerhard Gentzen, Kurt Gödel, Harvey Friedman, and Michael Rathjen. Typical properties include countability, computable (or arithmetical) presentations under assumptions from Kurt Gödel-style constructibility or from models like those studied at Institute for Advanced Study and Princeton University; closure properties mirror those of ordinals analyzed in work by John Burgess, Dana Scott, Hugh Woodin, and Akihiro Kanamori. The ordinal is frequently compared with ordinals arising in analyses by Gerhard Gentzen, Wilfried Buchholz, Jean-Yves Girard, Howard Friedman, and Jech.

Historical background and origin

The origin of the concept traces to correspondence and publications by Georg Kreisel and contemporaries including Kurt Gödel, Gerhard Gentzen, Stephen Kleene, Alan Turing, Alonzo Church, and André Weil during the 1950s and 1960s. Developments at Princeton University, University of Cambridge, Harvard University, Oxford University, University of Chicago, and University of California, Berkeley influenced formal definitions; discussions in seminars led by Paul Cohen, Dana Scott, Saul Kripke, and W. V. O. Quine contributed to clarifying connections with admissible ordinals and recursion theory. Subsequent refinements involved work by Michael Rathjen, Wilfried Sieg, Simpson, Daniel Friedman, and John Steel in the contexts of Reverse mathematics and proof-theoretic ordinal assignments for theories like those studied at Carnegie Mellon University and University of Oxford.

Examples and notable Kreisel ordinals

Concrete instances often arise as the supremum of ordinals with particular effective presentations linked to systems studied by Gerald Sacks, Harvey Friedman, Stephen Simpson, Michael Rathjen, Wilfried Sieg, Terry Tao, and W. Hugh Woodin. Notable related ordinals include those compared with the Howard ordinal examined by William Ackermann scholars, the Bachmann–Howard ordinal analyzed by Heinz Bachmann and Wilhelm Howard, and ordinals appearing in studies by Jean-Yves Girard, Wilfried Buchholz, Georg Kreisel, Simpson, and Feferman. Examples are often set alongside ordinals such as the Church–Kleene ordinal omega_1^CK associated with Stephen Kleene, admissible ordinals studied by Dana Scott, and projective ordinals investigated by Alexander Kechris and Yiannis Moschovakis.

Computability and definability aspects

Questions of computability and definability involve work by Alonzo Church, Alan Turing, Emil Post, Gerald Sacks, Harvey Friedman, Anthony Dorroh, Harvey Friedman, and Stephen Kleene. Kreisel ordinals are often characterized as suprema of recursive ordinals subject to constraints examined in recursion-theoretic contexts at University of Illinois Urbana-Champaign and University of California, Berkeley. Definability is explored via hierarchies such as arithmetical and analytical hierarchies studied by Kurt Gödel, Solomon Feferman, Gerald Sacks, Alexander S. Kechris, and Yiannis Moschovakis. Related definability frameworks include admissible sets linked to Dana Scott, constructible hierarchy connections studied by Kurt Gödel in his constructible universe work, and descriptive set theoretic perspectives advanced by Donald A. Martin and John R. Steel.

Relationships to other ordinals and hierarchies

Kreisel ordinals are compared with classical proof-theoretic ordinals studied by Gerhard Gentzen, Wilhelm Ackermann, Heinz Bachmann, William Howard, Jean-Yves Girard, and Wilfried Buchholz. They interface with hierarchies such as the arithmetical hierarchy of Stephen Kleene, the analytical hierarchy of Kurt Gödel, and the large countable ordinals appearing in work by Michael Rathjen, Harvey Friedman, Marek Balcerzak, and Simpson. Interactions with the Church–Kleene ordinal and admissible ordinals link to research by Dana Scott, Sacks, Kreisel, and Solomon Feferman. Comparative studies often cite programs at Institute for Advanced Study, Princeton University, and University of Cambridge where ordinal analyses relate to structural results by John Burgess and Hugh Woodin.

Applications in proof theory and reverse mathematics

Applications occur in proof-theoretic ordinal analyses of theories examined by Gerhard Gentzen, Georg Kreisel, Solomon Feferman, Wilfried Sieg, Stephen Simpson, Simpson, Michael Rathjen, Harvey Friedman, and Wilfried Buchholz. Kreisel ordinals provide benchmarks in reverse mathematics programs involving subsystems studied at Harvard University, Princeton University, and University of Oxford and in consistency proofs related to systems discussed by Kurt Gödel and Paul Cohen. They inform calibration of strength for theories appearing in work by Jean-Yves Girard, John R. Steel, W. Hugh Woodin, and Gerald Sacks and contribute to analyses carried out at institutions such as Institute for Advanced Study and Carnegie Mellon University.

Category:Ordinals