Buchholz's ordinal explained
In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal, is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem
-CA
0 of
second-order arithmetic;
[1] [2] this is one of the "big five" subsystems studied in
reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of
, the theory of finitely
iterated inductive definitions, and of
,
[3] a fragment of
Kripke-Platek set theory extended by an axiom stating every set is contained in an
admissible set. Buchholz's ordinal is also the order type of the segment bounded by
in Buchholz's ordinal notation
. Lastly, it can be expressed as the limit of the sequence:
\varepsilon0=\psi0(\Omega)
,
,
, ...
Definition
See main article: article.
, and
for
n > 0.
is the closure of
under addition and the
function itself (the latter of which only for
and
).
is the smallest ordinal not in
.
- Thus, ψ0(Ωω) is the smallest ordinal not in the closure of
under addition and the
function itself (the latter of which only for
and
).
References
- G. Takeuti, Proof theory, 2nd edition 1987
- K. Schütte, Proof theory, Springer 1977
Notes and References
- 1986-01-01. A new system of proof-theoretic ordinal functions. Annals of Pure and Applied Logic. en. 32. 195–207. 10.1016/0168-0072(86)90052-7. 0168-0072. Buchholz . W. . free.
- Book: Simpson, Stephen G.. Subsystems of Second Order Arithmetic. 2009. Cambridge University Press. 978-0-521-88439-6. 2. Perspectives in Logic. Cambridge.
- T. Carlson, "Elementary Patterns of Resemblance" (1999). Accessed 12 August 2022.