Unbounded Functions

Abstract

In this section κ is a regular cardinal and C_α (α < κ⁺) is a fixed C-sequence with the property that tp(C_α) ≤ κ for all α < κ⁺. When the C-sequence is necessarily coherent, then it is natural to define the corresponding mapping

$$ \rho :[\kappa ^ + ]^2 \to \kappa $$

(9.1.1)

as follows:

$$ \rho (\alpha ,\beta ) = \sup \{ tp(C_\beta \cap \alpha ),\rho (\alpha ,\min (C_\beta \backslash \alpha )),\rho (\xi ,\alpha ): \xi \in C_\beta \cap \alpha \} , $$

(9.1.2)

with the boundary value ρ(α, α) = 0 for all α < κ⁺, a definition that is slightly different from the one given above in (7.3.2) above. Clearly,

$$ \rho (\alpha ,\beta ) \geqslant \rho _1 (\alpha ,\beta ) for all \alpha < \beta < \kappa ^ + , $$

(9.1.3)

and so, using Lemma 6.2.1, we have the following fact.