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I don't know much about singular cardinal combinatorics, so I apologize in advance if I write something that is wrong or looks funny. First let me recall some basic definitions.

Let $\lambda$ be a singular cardinal and let $(\kappa_i : i < \text{cf}(\lambda))$ be a sequence of regular cardinals cofinal in $\lambda$. A sequence $(f_\alpha : \alpha < \lambda^+)$ of functions $f_\alpha \in \prod_{i < \text{cf}(\lambda)} \kappa_i$ is called a scale if it is:

(1) increasing modulo $<^*$, where $f <^* g$ means that $f(i) < g(i)$ for all but a bounded set of $i < \text{cf}(\lambda)$, and

(2) cofinal in $\prod_{i < \text{cf}(\lambda)} \kappa_i$ modulo $<^*$.

A scale is said to be good if it satisfies the additional property

(3) for club many $\alpha < \lambda^+$, if $\text{cf}(\alpha) > \text{cf}(\lambda)$ then $\alpha$ is a good point, meaning that there is an unbounded set $S \subset \alpha$ and an ordinal $i < \text{cf}(\lambda)$ such that the sequence $(f_\alpha \restriction [i,\text{cf}(\lambda)): \alpha \in S)$ is pointwise increasing.

The proof of the fact "if $\lambda$ is a singular cardinal and there is a strongly compact cardinal between $\text{cf}(\lambda)$ and $\lambda$, then there is no good scale of length $\lambda^+$" seems to use only properties (1) and (3) but not property (2). So my questions are:

(A) Do sequences of length $\lambda^+$ satisfying (1) and (3) have a name (e.g. "good pseudo-scale")?

(B) Is their existence equivalent to the existence of good scales?

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  • $\begingroup$ If it looks like a scale; and it proves like a scale; it's a duck. Big, red, scaly duck, which looks suspiciously like a dragon. $\endgroup$
    – Asaf Karagila
    Sep 1, 2014 at 17:27
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    $\begingroup$ (Also, after two votes up, your reputation is so last year! ;-)) $\endgroup$
    – Asaf Karagila
    Sep 1, 2014 at 17:27

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Your pseudo-good scale is indeed equivalent to a good scale, although the good scale possibly lives on a different product. This follows easily from the construction of a scale given in Cummings' Notes on Singular Cardinal Combinatorics.

A basic theorem of Shelah: If a $<^*$-increasing sequence of functions has stationary many good points of cofinality $\kappa$ for some $\kappa<\lambda$, then it has an exact upper bound $g$ such that $\mathrm{cf}(g(i))>\kappa$ for but boundedly many $i$.

Taking such an exact upper bound, you can thin to $X\subseteq \mathrm{cf}(\lambda)$ so that $i\mapsto \mathrm{cf}(g(i)): i\in X$ is increasing and unbounded in $\lambda$. Then you can fix cofinal subsets of $g(i)$ of order-type $\mathrm{cf}(g(i))$ for each $i\in X$ and modify the functions to live on $\prod_{i\in X} \mathrm{cf}(g(i))$. You can check that this preserves (3), and the resulting sequence of functions is cofinal in the new product.

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