I'm sorry if this is a duplicate in any way. I doubt it's an original question. Due to my ignorance, it's difficult for me to search for appropriate things.
Motivation.
This question is inspired by Exercise 1.2.16 of these logic notes by S. G. Simpson. Here is a shortened version of that exercise for convenience.
Brown, Jones, and Smith are suspected of a crime. They testify as follows:
Brown: Jones is guilty and Smith is innocent.
Jones: If Brown is guilty then so is Smith.
Smith: I’m innocent, but at least one of the others is guilty.
a) Are the three testimonies consistent?
b) The testimony of one of the suspects follows from that of another. Which from which?
c) Assuming everybody is innocent, who committed perjury?
d) Assuming all testimony is true, who is innocent and who is guilty?
e) Assuming that the innocent told the truth and the guilty told lies, who is innocent and who is guilty?
I like to challenge friends and family with similar problems. It's fun to make up scenarios and the solutions are fairly easy to those familiar with basic mathematical logic. I can vary the testimonies, the number of suspects, the questions about the testimonies, etc.; it's good stuff.
Lately, though, I've wondered what it would mean to have (at least countably) infinitely many suspects. To make the problem tractable the testimonies would need some sort of defining rule and the questions ought to address appropriate groups of suspects.
The Question.
With this in mind, here's my scenario.
On the morning of the first night at Hilbert's hotel, when all the rooms were taken, the receptionist was found dead at his desk; it looked extremely suspicious. Was he murdered? The police interviewed all the tenants and staff, and concluded that the staff couldn't possibly have been involved in the death. However, the tenants had some interesting testimonies which amounted to the following.
$[\dots ]$
Okay, so I've given this some thought and I suspect that the original set up is at least similar to letting $$\begin{align} \text{Brown}&\mapsto [1]_3:=\{n\in\mathbb{N}\mid n\cong 1\pmod{3}\}, \\ \text{Jones}&\mapsto [2]_3, \\ \text{Smith}&\mapsto [3]_3, \end{align}$$ where each number $n$ represents the tenant in Room $n$, then changing the testimonies accordingly. (I'll leave that as an exercise for the reader (ha!): this is too long already.)
Immediately, I'm reminded of the notion of presentations and freeness. The above smells like a presentation (or perhaps some kind of homomorphism). I suppose my main bunch of questions here are:
What is this thing? What's a better, more formal way of describing the mathematics behind this scenario? What similar things have been done before?
The reason why I included the (number-theory) tag is that I'm curious now as to what number theoretic problems, if any, can be phrased this way. (Does that make any sense?)
Thoughts and Clarification.
This is based on the comments.
It's more about the maps between the infinite and finite cases.
A thorough answer would include a mathematical description of what the infinite case is, a mathematical description of how the infinite case relates to finite cases, details on what similar things have been done before, and perhaps a number-theoretic problem phrased using the above.
One has to take into account negations in the infinite case in such a way that the structure of the given finite case is preserved.
I suspect that they're just different models of the same theory, where the infinite case is in some sense "free"; that maps like the one given above are somehow related to the notion of a presentation; and that at least some trivial Number Theory problems can be stated this way.
:)