The Universe of Discourse

Thu, 28 Oct 2021

Kuratowski pairs and Wiener pairs

I mentioned a couple of years back that Principia Mathematica was bloated with repetitive material because they hadn't been able to unify the idea of a relation and a set, because the ordered pair hadn't been invented yet. There's a section that defines set union, !!\cup!!, and then proves that it is commutative and associative and so on, and later there is a separate section that defines relation union, !!\dot\cup!!, and proves the exact same theorems in the same way. In 2021 (or even in 1921) we would say that a relation is a set of ordered pairs, and that relation union is just a special case of set union.

To do this we have to interpret ordered pairs set-theoretically. The method we usually use for this was invented by Kazimierz Kuratowski:

$$\langle a, b\rangle = \{\{a\}, \{a,b\}\}$$

But there were earlier developments that also sufficed. Hausdorff suggested the more intuitive, but technically more complex:

$$\langle a, b\rangle = \{\{a, 1\}, \{b, 2\}\}$$

where !!1!! and !!2!! are any two objects that are not among the things we want to include in our ordered pairs. And even earlier, the first interpretation of pairs as sets was in 1911 by Norbert Wiener. In modern notation, Wiener's definition is:

$$\langle a, b\rangle = \{\{\{a\}, \emptyset\}, \{\{b\}\}\}.$$

Wiener actually used the notation of Principia Mathematica, which I reproduce for your amusement:

$$\def\i{\iota`} \i(\i\i a\cup \i\Lambda)\cup\i\i\i b $$

The !!\i x!! notation means essentially the same as !!\{x\}!!. I thought Principia Mathematica had a better way to write !!\{x, y\}!! than as !!\i x\cup\i y!!, but if so I can't remember what it is.

Someone once told me that Wiener's definition is more complicated than Kuratowski's because it had to function in the context of Whitehead and Russell's type theory. Kuratowski was working later, in set theory, so could use a simpler definition that wouldn't function in type theory because the types didn't match up. I had never thought carefully about this until now but it seems to be wrong. The Kuratowski pair requires !!a!! and !!b!! to be the same type, or else you can't put them both into the class !!\{a, b\}!!. But the Wiener pair requires this also. Say !!a!! and !!b!! have type !!n!!. Then !!\{a\}!! and !!\{b\}!! have type !!n+1!!, and !!\{\{a\}, \emptyset\}!! and !!\{\{b\}\}!! have type !!n+2!!. And because they have the same type we can put them both into the class !!\{\{\{a\}, \emptyset\}, \{\{b\}\}\}!!. But for this to work, !!a!! and !!b!! have to have the same type to begin with.

I wanted to find out what Wiener said about this, and Wikipedia referred me to his paper A Simplification of the logic of relations, and helpfully pointed out that it was reprinted in van Heijenoort's Source Book in Mathematical Logic, which I have on the shelf. (I love when this happens. It makes me feel like a scholar.) Wiener agrees: !!a!! and !!b!! must have the same type. But, he points out, if !!a!! and !!b!! have different types you can still make it work by adjusting the nesting level of !!a!! or !!b!! accordingly. For example, if !!b!! had a type one higher than !!a!!, you could use !!\{\{\{a\}, \emptyset\}, \{b\}\}!! instead.

In any case the Kuratowski thing is still simpler. I wonder why Wiener didn't think of it first. But he does say “the particular method selected of doing this is largely a matter of choice”, so perhaps he didn't consider the details important. As in fact they aren't. The important point, and the real point of Wiener's paper, is that you can now construe a two-place relation as a class of ordered pairs.

The paper ends by observing that this fixes the !!\cup!!-versus-!!\dot\cup!! extravagance of Principia Mathematica, since now !!R\cup S = R\,\dot\cup\, S!!. Similarly the class of relations is now a subclass of the class of classes, and so forth.

[ Addendum 20211030: This construction makes the cartesian product nonassociative, but nobody cares. ]

[ Addendum 20211101: One way in which Wiener pairs are simpler than Kuratowski pairs. ]

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