GCH counts Easton head conditions and subset names

Preliminary computation: if $\lambda >{\mathrm{\aleph }}_0$ is a cardinal and $0<\mu <\mathrm{cf}(\lambda )$ is a cardinal, then ${\lambda }^{\mu }=\lambda$; consequently ${\lambda }^{<\rho }=\lambda$ whenever $1<\rho <\mathrm{cf}(\lambda )$. Fix a cofinal sequence $⟨{\lambda }_{\alpha }:\alpha <\mathrm{cf}(\lambda )⟩$ of ordinals below $\lambda$. A function $f:\mu \to \lambda$ has bounded range because $\mu <\mathrm{cf}(\lambda )$, so its range lies in some ${\lambda }_{\alpha }$. Let ${\rho }_{\alpha }=\mathrm{\mid }{\lambda }_{\alpha }\mathrm{\mid }$ and ${\tau }_{\alpha }=\max \{{\rho }_{\alpha },\mu ,{\mathrm{\aleph }}_0\}<\lambda$. Under GCH, ${\rho }_{\alpha }^{\mu }\le {\tau }_{\alpha }^{{\tau }_{\alpha }}=2^{{\tau }_{\alpha }}={\tau }_{\alpha }^{+}\le \lambda$. Thus ${\lambda }^{\mu }\le {\sum }_{\alpha <\mathrm{cf}(\lambda )}{\rho }_{\alpha }^{\mu }\le \lambda$ by Choice and cardinal absorption; the reverse inequality follows from constant functions. The case $\mu =0$ has value $1$ and does not affect the displayed supremum, which includes $\mu =1$.

Write $B_{\gamma }=\{(\gamma ,\alpha ,\beta ):\alpha <\gamma ,\beta <F(\gamma )\}$ for each $\gamma \in \mathrm{dom}(F)$ with $\gamma \le \kappa$; all block indices below are restricted to these $\gamma$. A condition $p\in P^{\le \kappa }$ is a partial bit function on ${\bigcup }_{\gamma \le \kappa }{B}_{\gamma }$ satisfying the Easton support bounds. In particular, the graph of its $\gamma$-block is a subset of $B_{\gamma }\times 2$ of size below $\gamma$. There are at most $\kappa$ indices $\gamma$, $\mathrm{\mid }{B}_{\gamma }\times 2\mathrm{\mid }=F(\gamma )\le F(\kappa )$, and by [F4] every antichain of the head has cardinality at most $\kappa$.

$\mid P^{\le \kappa }\mid \ge F(\kappa )$: for each $\beta <F(\kappa )$ the singleton bit condition $\{(\kappa ,0,\beta )\mapsto 1\}$ satisfies every Easton support bound. These $F(\kappa )$ conditions are distinct, giving the lower bound.

$\mid P^{\le \kappa }\mid \le F(\kappa )$: by step 1.2 the graph of each $\gamma$-block has size below $\gamma$, so the number of possible blocks is at most ${\sum }_{\eta <\gamma }(2\cdot F(\gamma ){)}^{\mathrm{\mid }\eta \mathrm{\mid }}\le \gamma \cdot F(\gamma )=F(\gamma )$ by step 1.1 and absorption; the term for $\eta =0$ is $1$. Here the cofinality of $F(\gamma )$ exceeds $\gamma$. Coding a condition by its at most $\kappa$ blocks therefore gives $\mathrm{\mid }{P}^{\le \kappa }\mathrm{\mid }\le {\prod }_{\gamma \le \kappa }F(\gamma )\le F(\kappa {)}^{\kappa }=F(\kappa )$.

Steps 2.1 and 1.3 give injections in both directions between $P^{\le \kappa }$ and $F(\kappa )$, so $\mid P^{\le \kappa }\mid =F(\kappa )$ by antisymmetry of cardinal comparison.

There are at most $F(\kappa )$ nice $P^{\le \kappa }$-names for subsets of $\kappa$: by [F5] such a name is coded by the function $\kappa \to \{A\subseteq P^{\le \kappa }:A$ is an antichain$\}$ sending $a\mapsto A_a$, and two such functions that differ at an $a$ with $A_a\mathrm{\ne }{A}_a^{\mathrm{\prime }}$ give different names, since then either $A_a\setminus A_a^{\mathrm{\prime }}$ or $A_a^{\mathrm{\prime }}\setminus A_a$ contains some $p$ and $⟨\check{a},p⟩$ lies in exactly one of the two names; by step 1.2 every antichain has cardinality at most $\kappa$, so there are at most $\mid P^{\le \kappa }{\mid }^{\kappa }=F(\kappa {)}^{\kappa }=F(\kappa )$ antichains by steps 3.1 and 1.1, and at most $F(\kappa {)}^{\kappa }=F(\kappa )$ such coding functions by step 1.1. The exponent laws and products used are those of [F6], which hold under the Axiom of Choice [F8].

Steps 3.1 and 4.1 give $\mid P^{\le \kappa }\mid =F(\kappa )$ and at most $F(\kappa )$ nice $P^{\le \kappa }$-names for subsets of $\kappa$, which is the statement. ∎