9
$\begingroup$

Given regular cardinal $\kappa>\omega,$ we can define the canonical functions $f_\alpha: \kappa\to \kappa,$ for $\alpha<\kappa^+.$

Some of their properties are presented in Chapter 22 of the book "Problems and Theorems in Classical Set Theory" by Péter Komjáth and Vilmos Totik.

(A) I am looking for more references and additional properties of these functions.

(B) I also want to know what are the main applications of these functions, and how they can be used in proving theorems (in particular $ZFC$ results).

$\endgroup$
2
  • $\begingroup$ Canonical functions can also be found in Jech Ch. 24 Lem 24.5. $\endgroup$ Oct 20, 2014 at 11:51
  • $\begingroup$ Also, the handbook of set theory page 99 refers to canonical functions as well along with other places. $\endgroup$ Oct 20, 2014 at 12:03

1 Answer 1

5
$\begingroup$

In belated partial response to (A), canonical functions are certainly already defined in the Jech-Shelah paper: A note on canonical functions, where reference is made to the work F. Galvin and A. Hajnal, Inequalities for cardinal powers, Annals of Math. 101 (1975), 491-498. The name "canonical" appears to derive from the property that if they exist, they are unique up to equivalence modulo a club. Below $\aleph_2$, they always exist; at or above, they may or may not.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.