Why not $\{\{a\},\{b, \emptyset \}\}$ as the ordered pairs?

Question

The ordered pair $\{\{a\},\{b,\emptyset \}\}$ seems to be very simple, neat, and highly intuitive ordered pair. So why Kuratowski's pairs were preferred?

Answer 1

This is a nice question, with a perhaps surprisingly subtle answer. Many different set-theoretic implementations of ordered pair have been proposed, especially in the early days of set theory; a good condensed history is given in Kanamori’s paper The empty set, the singleton, and the ordered pair (2003, Bulletin of Symbolic Logic; paywalled there but findable un-paywalled if you search a bit). The first proposal (well, probably equal-first with Hausdorff’s) was Norbert Wiener’s in 1914, defining $(x,y)$ as $\{\{\{x\},\emptyset\},\{\{y\}\}\}$ — similar to your proposal, but with an extra layer of brackets around $x$ and $y$, which I’ll come back to later.

Why did Kuratowski’s emerge as the standard? As discussed in this related question, the choice is to some extent an accident of history — we don’t care about the exact implementation used, so long as it satisfies some essential properties. However, proving those properties will of course depend on what theory/formalism we’re working in — so it’s not just a matter of “anything satisying those properties (in ZF) is equally good”. We have a spectrum of desiderata for any proposed implementation of pairs, from precise mathematical properties through to more context-dependent and subjective criteria:

1. (essential) for all $x,y$, the specification of $(x,y)$ is a valid definition, i.e. specifies some uniquely existing object;
2. (essential) for all $x,y,x',y'$, $(x,y) = (x',y')$ if and only if $x = x'$ and $y = y'$;
3. (near-essential) for any sets $X, Y$, the product set $X \times Y = \{ (x,y) \mid x \in X,\, y \in Y\}$ exists;
4. (important) the proofs of properties 1–3 should need only a small core fragment of the axioms of set theory, so that it works in other set theories, not just ZFC
5. (varying with context) the definition should also work in other formalisms, not just ZFC-like first-order theories
6. (very subjective) general aesthetics: simplicity, symmetry, etc.

Your definition is great on criteria 1–4 — you can prove 1–3 for them with just extensionality, existence of unordered pairs, and the empty set. (And it doesn’t need excluded middle; the proof can be made fully constructive.) And it’s arguably good on criterion 6 too — simpler than Wiener’s, and without the duplication of Kuratowski’s.

However, your proposal falls down on criterion 5. In the modern ZF-style set-theoretic universe, where everything is a set and any of objects can be collected together, it’s fine — but in the early days, that viewpoint wasn’t established immediately, and certainly wasn’t taken for granted until rather later. Much early set theory assumed (first informally, then formalised in systems like Russell’s theories of types) a worldview where there are objects of different types — in the simplest version, levels, with just atoms at the lowest level, then sets of atoms, then sets of sets of atoms, and so on — and a set can only collect together objects from a given type. So forming $\{b,\emptyset\}$ as in your definition is only possible when $b$ is a set — it wouldn’t be admissible if $b$ is an atom, since no sets exist the lowest level, and in particular, no empty set.

This is why Wiener’s version uses $\{\{x\},\emptyset\}$ not just $\{x,\emptyset\}$: he takes singletons to raise all the objects to the same level. That was essential in his original version (A simplification of the logic of relations, 1914, Proc. Cam. Phil. Soc.), since he was working in the typed system of Russell and Whitehead’s Principia. Later, in ZF-style theories, the typing discipline isn’t required, but all other early proposed implementations I’ve seen still follow it.

There have been later definitions which break this typing discipline, such as the definition in Scott and McCarty, Reconsidering ordered pairs (2014, Bulletin of Symbolic Logic); but such definitions are usually proposed when the authors have specific technical requirements that Kuratowski pairing is unsatisfactory (and so are other established implementations) — e.g. in the Scott–McCarty case, to provide pairs of proper classes, not just sets.

So, to summarise:

• I think early researchers in set theory still preferred to work with notions satisfying the typing discipline of Russell-style systems, even when it was no longer formally required since they’d moved to modern ZF-style systems;
• your definition doesn’t fit that discipline, which would explain why it wasn’t proposed in the early days;
• in most later work, the Kuratowski implementation is completely satisfactory and there’s usually no reason to prefer anything else;
• when people do consider other implementations, it’s for specific technical purposes.