Strong increase and bounding projections in countable ordinal products

Work in ZFC, with AC as in The Axiom of Choice. Let $A$ be infinite, $I$ a proper ideal on $A$, $\lambda$ an infinite regular cardinal, and $(f_{\alpha }{)}_{\alpha <\lambda }$ a strictly ${<}_I$ increasing sequence of ordinal-valued functions. Comparisons, and the finite-ideal notation ${<}^{\ast },{\le }^{\ast },{=}^{\ast }$, are as in Reduced products, true cofinality and scales; their laws were proved in Progressive products and true cofinality transfers. Regularity uses $\mathrm{cf}(\alpha )\le \alpha$; $\mathrm{cf}(0)=0$ and $\mathrm{cf}(\alpha +1)=1$; for a limit ordinal $\lambda$ the value $\mathrm{cf}(\lambda )$ is an infinite cardinal with $\mathrm{cf}(\mathrm{cf}(\lambda ))=\mathrm{cf}(\lambda )$, so it is regular; and every cofinal subset of $\lambda$ has cardinality at least $\mathrm{cf}(\lambda )$, a value that is attained.

A subsequence with indices $L\subseteq \lambda$ is strongly increasing if there are sets $Z_{\alpha }\in I$, for $\alpha \in L$, such that for every $\alpha <\beta$ in $L$ and every $a\mathrm{\notin }{Z}_{\alpha }\cup Z_{\beta }$ one has $f_{\alpha }(a)<f_{\beta }(a)$. For a regular cardinal $\kappa \le \lambda$, property $(\ast {)}_{\kappa }$ says: every unbounded $U\subseteq \lambda$ contains a set $L$ of order type $\kappa$ on which the sequence is strongly increasing. The witnesses are individual small sets, not a single common exceptional set.

For nonempty sets $S(a)$ of ordinals define $s(a)=\sup S(a)$. The ceiling projection of $f$ has value

$p_f(a)=\min \{t\in S(a):f(a)\le t\}$

when the displayed set is nonempty, and value $\min S(a)$ otherwise. This latter value is a specified fallback. In particular if $f(a)<s(a)$ the ceiling exists: otherwise every $t\in S(a)$ would be less than $f(a)$, implying $s(a)\le f(a)$. Thus $f{<}_{I}s$ makes the possible fallback set $I$-small. Assertions modulo $I$ are independent of the values chosen on that exceptional set. Empty $S(a)$ are not allowed; if they occur on a discarded small support, replace them by a nonempty singleton there before using this convention.

The sequence has the $\kappa$ bounding-projection property if, for every such family of nonempty ordinal sets with $\mathrm{\mid }S(a)\mathrm{\mid }<\kappa$ for all $a$ and $f_{\alpha }{<}_{I}s$ for every $\alpha <\lambda$, there is $\alpha <\lambda$ whose projection $p_{f_{\alpha }}$ strictly ${<}_I$ bounds every term of the sequence. This is a quantified property, not an assertion that a bounding projection always exists.

An exact upper bound is an ordinal function $h$ with $f_{\alpha }{\le }_{I}h$ for every $\alpha$, and such that for every ordinal function $g{<}_{I}h$ there is $\alpha <\lambda$ with $g{<}_I{f}_{\alpha }$. Its later existence theorem will supply a positive limit-valued representative and prove leastness and uniqueness. No leastness, existence, or cofinality conclusion is assumed just by giving the terminology.

The countable specialization takes $\mathrm{\mid }A\mathrm{\mid }={\mathrm{\aleph }}_0$ and $I$ the ideal of finite subsets of $A$. Strong increase is then witnessed by finite $Z_{\alpha }$, and all comparisons in the preceding paragraphs become eventual comparisons. The general definitions allow ideals that are not countably complete and do not contain all singletons.