Unbounded Functions

Part of the Progress in Mathematics book series (PM, volume 263)


In this section κ is a regular cardinal and Cα (α < κ+) is a fixed C-sequence with the property that tp(Cα) ≤ κ for all α < κ+. When the C-sequence is necessarily coherent, then it is natural to define the corresponding mapping
$$ \rho :[\kappa ^ + ]^2 \to \kappa $$
as follows:
$$ \rho (\alpha ,\beta ) = \sup \{ tp(C_\beta \cap \alpha ),\rho (\alpha ,\min (C_\beta \backslash \alpha )),\rho (\xi ,\alpha ): \xi \in C_\beta \cap \alpha \} , $$
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,
$$ \rho (\alpha ,\beta ) \geqslant \rho _1 (\alpha ,\beta ) for all \alpha < \beta < \kappa ^ + , $$
and so, using Lemma 6.2.1, we have the following fact.


Triple System Eventual Dominance Continuum Hypothesis Regular Cardinal Force Notion 
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© Birkhäuser Verlag AG 2007

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