The explicit definition of an ordered pair ((a,b)={{a},{a,b}}) is frequently relegated to pathological set theory...
It is easy to locate the source of the mistrust and suspicion that many mathematicians feel toward the explicit definition of ordered pair given above. The trouble is not that there is anything wrong or anything missing; the relevant properties of the concept we have defined are all correct (that is, in accord with the demands of intuition) and all the correct properties are present. The trouble is that the concept has some irrelevant properties that are accidental and distracting. The theorem that (a,b)=(x,y) if and only if a=x and b=y is the sort of thing we expect to learn about ordered pairs. The fact that {a,b}∈(a,b), on the other hand, seems accidental; it is a freak property of the definition rather than an intrinsic property of the concept.
The charge of artificiality is true, but it is not too high a price to pay for conceptual economy. The concept of an ordered pair could have been introduced as an additional primitive, axiomatically endowed with just the right properties, no more and no less. In some theories this is done. The mathematician’s choice is between having to remember a few more axioms and having to forget a few accidental facts; the choice is pretty clearly a matter of taste. Similar choices occur frequently in mathematics...
--Paul R. Halmos, Naïve Set Theory (1960, p. 24-5)
Modern type theory mostly solves this blemish of set theory and is highly economic conceptually to boot. Most of the adherence of set theory is historical inertia—though some aspects of coding & presentations is important. Future foundations will improve our understanding on this latter topic.
Modern type theory mostly solves this blemish of set theory and is highly economic conceptually to boot. Most of the adherence of set theory is historical inertia—though some aspects of coding & presentations is important. Future foundations will improve our understanding on this latter topic.