I'm trying to wrap my head around some of Harvey Friedman's recent, unpublished work on his tangible incompleteness project, and I'm trying to see the link between his "tangible statements" (propositions about subsets of $\mathbb{Q}[0,n]^k:=\{x\in\mathbb{Q}\;|\;0\leq x\leq n\}^k$ for $n,k\in\mathbb{N}$) and the large cardinals in the subtle/$k$-SRP range.

An example of the sort of statement he considers is the following (from this document):

Definition 1. $S\subseteq\mathbb{Q}[0,n]^k$ is an emulator of $E\subseteq\mathbb{Q}[0,n]^k$ if each element of $S^2$ is order equivalent to some member of $E^2$ (considered as subsets of $\mathbb{Q}[0,n]^{2k}$), where $x,y\in\mathbb{Q}[0,n]^k$ are said to be order equivalent if for any $1\leq i,j\leq k$, $x_i<x_j\leftrightarrow y_i<y_j$.

Definition 2. $A\subseteq\mathbb{Q}[0,n]^k$ is stable if and only if for all $p<1$, $\langle p,1,\ldots,k-1\rangle\in A\leftrightarrow\langle p,2,\ldots,k\rangle\in A$.

Example statement (called "MES" by Friedman). For all $k$, and all finite $E\subseteq\mathbb{Q}[0,k]^k$, there is a stable, $\subseteq$-maximal emulator of $E$.

Apparently MES implies $\mathrm{Con}(\mathsf{SRP})$ (where $\mathsf{SRP}$ is $\mathsf{ZFC}$+"there is a $k$-SRP cardinal" as an axiom scheme in $k$), so somehow there is a lot of strength happening in stable maximal emulators. And the relation to $k$-SRP cardinals, specifically, makes me think that emulators must be hiding a Ramsey condition under the hood, but I haven't been able to spot it yet.

How can I see what is happening with stable maximal emulators of finite subsets of $\mathbb{Q}[0,n]^k$ ($n,k>2$) that their guaranteed existence implies the consistency of the existence of large cardinals? [Edited to reflect JDH's correction in the comments.]

(For a reference on $k$-SRP cardinals, see Friedman, Harvey. Subtle cardinals and linear orderings. (English summary) Ann. Pure Appl. Logic 107 (2001), no. 1-3, 1–34.)

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    $\begingroup$ This might be more appropriate at mathoverflow - this is some seriously technical material (and I think people capable of answering it might be more likely to see it there). $\endgroup$ Jan 15, 2020 at 4:07
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    $\begingroup$ Not sure, honestly - crossposting immediately is generally frowned on, but in this case I would imagine deleting here and asking there should be fine (since it hasn't been up here for long enough for a move to be construed as wasting someone's time). But that's just my opinion; certainly there's no harm asking it here and moving to MO after a bit if no answers show up. $\endgroup$ Jan 15, 2020 at 4:09
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    $\begingroup$ "...their guaranteed existence implies the existence of large cardinals" is not right, because no arithmetic statement can imply the existence of large cardinals. You mean to imply the consistency of the existence of the large cardinals. $\endgroup$ Jan 16, 2020 at 8:20
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    $\begingroup$ @TimothyChow - I suppose I didn't totally answer your question: I'm most interested in the line of thought behind formulating statements of this kind, less with k-SRP cardinals specifically. There's clearly something that leads Friedman to choose stable maximal emulators (or various other similar objects) as good objects for bumping up consistency strength, but I have no clue what the guiding idea is. $\endgroup$ Jan 23, 2020 at 9:08
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    $\begingroup$ @MaliceVidrine I'm currently working through Finite Functions and the Necessary Use of Large Cardinals, which Friedman published around the same time. Broadly speaking the argument there involves using a strong finitary Ramsey's theorem and compactness to build a structure that has the ability to code some basic set theory and has a special cofinal indiscernible sequence of elements. This indiscernible sequence then gives you enough reflection to show that the interpreted model of basic set theory satisfies ZFC. I would guess that the story in this paper is broadly similar. $\endgroup$ Oct 13, 2023 at 20:26


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