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My philosophical background going into set theory was heavily laden with Kantian and neo-Kantian elements, so one of my essential premises was read off the following passage from the first Critique [2nd ed., Meiklejohn translation]:

The same function which gives unity to the different representation in a judgement, gives also unity to the mere synthesis of different representations in an intuition; and this unity we call the pure conception of the understanding. Thus, the same understanding, and by the same operations, whereby in conceptions, by means of analytical unity, it produced the logical form of a judgement, introduces, by means of the synthetical unity of the manifold in intuition, a transcendental content into its representations, on which account they are called pure conceptions of the understanding, and they apply a priori to objects, a result not within the power of general logic.

So we could say that every general operator maps to a transcendental operator, its transcendental interpolant. This gives us a schematic for categories: take the general operators of the theory and interpolate their transcendental counterparts. Given that we have fairly different general operators at hand than the twelve functions of the understanding tabulated by Kant, we will end up with different categories accordingly, but anyway...

Going in to set theory, I was enchanted by the idea that the powerset function can be represented not only via exponents but also tetration. I tried to make a lot of my local arguments about CH turn on this fact and had occasion to develop a comprehensive theory and method of transfinite hyperoperations. Although I would eventually learn that almost all my work in this area was but more or less but a drop in the ocean of replacement's classical schematic power, refining my familiarity with these issues did provide me with a solid logical grounding for the axioms of union and powerset, which were interpreted as transcendental interpolants of the conjunction operation and an erotetic powerset-like relation (the relation between a set of questions and the set of all subquestions of a question).

You can imagine my chagrin when I found out, however, that the usual idea is to interpolate intersection with conjunction. I admit, I only half-understand/agree with the reasoning I've encountered as to why this is so. It seems to me as though union and addition are coupled pretty strongly even in mainstream representations, e.g. in the set-theoretic definition of the ordinal successor function as a U {a}. Moreover, there will be talk about theories T and T' and they will use the addition sign to represent their union. But so is addition not akin to conjunction? When A and B are both true, isn't all the information in both A and B true information, not just the intersection of the information common to both? In fact, what if their intersection is empty?

Maybe I'm just not seeing it, but this seems like the most peculiar dogma (forgive the harsh term) of standard mathematical logic I've ever seen. I mean people will openly doubt the LEM or the LNC or the axiom of choice or a myriad other things but not this?

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  • "conjunction interposed with intersection instead of union" Where? Apr 26, 2022 at 15:40
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    In set-theoretical terms, the intersection of two sets is the set of common elements: "common" means that they belong to both, i.e. they belong to the fits AND they belong to the second. Apr 26, 2022 at 15:41
  • That's trivial, though; the set of all elements of A and all elements of B is also characterized by "AND." If union implements succession/addition in the hyperoperator sequence, but if we represent whole theories T, T', etc. in statements like, "T + T'," aren't we representing their conjunction (as T, etc. are sentential)? Apr 26, 2022 at 16:32

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The reason in set theory is these identities:

A∩B={x∈U∣x∈A∧x∈B}
A∪B={x∈U∣x∈A∨x∈B}

In normal English, AND is used as a joining operator: "This activity will be enjoyable for boys and girls", but that's probably because of the fact that OR in normal English is viewed as exclusive. That is, "boys or girls" implies "not both", whereas AND implies both: "Able and Baker will be at the party".

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  • That makes perfect sense, actually, but I also suspect I know what I was doing, looking it in the opposite direction. I was thinking of the relation between A and B as sets, rather than looking at the identifying relation for x as an element, here. Apr 26, 2022 at 19:07

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