Here are links to fuller discussions of semiotics.
The approach described here is based on what I regard as the core definition of sign relations, one explicit enough to support a consequential theory of signs.
C.S. Peirce • On the Definition of Logic (as depending on the definition of a sign)
C.S. Peirce • Logic as Semiotic
The sign domain is the representamen domain. Peirce was fussy about terminology. The sign is the unity of the 3 domains (object, representamen, interpretant).
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Peirce sometimes fussed over that term but more often not. The shorter word will serve us well enough as it did him and it has the benefit of being far less off-putting.
The more important thing is to understand triadic sign relations as a defined species of triadic relations, which are in turn a species of mathematical relations denoted by relative terms.
For a taste of how Peirce treated the logic of relative terms see my discussion of his 1870 Logic Of Relatives, which is where I first cut my teeth on the subject. Historically speaking this is also where we got our
first real breakthrough in reasoning about the mathematics of relations.
Peirce’s 1870 Logic Of Relatives
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JA> It's a common mistake to confound infinite with unbounded. A process can continue without end and still be "bounded in a nutshell". So a sign process can pass from sign to interpretant sign to next interpretant sign ad infinitum without ever leaving a finite set of signs.
That's not a mistake. It's the definition of countably infinite: I'm sure you know that, but it's important to state the definition in a way that distinguishes countably infinite sets (e.g., the integers) from uncountable sets (e.g., the real numbers).
The set of integers is countably infinite. Therefore, any set of integers that you can reach by counting is always finite. But the last sentence above is confusing because it's impossible to count "ad infinitum". However, you could specify an ordering of the integers that could produce an infinite set if you could run the machine "ad infinitum".
The real numbers are uncountably infinite. If you tried to count them, as you would with the integers, you could only get a finite set. And even if you had a machine that could run "ad infinitum", you could only get a countable subset. You could never get all the real numbers.