Post #2061455

2026-04-09 14:21 UTC

@ncf It's often striking to me how, for so many people, the entire conceptual universe is circumscribed by ZF(C). That theory has become so dominant over last last 100 years that many people can't escape it. Their motto, and they really believe it, is “everything is a set”. For them there's no collection that isn't a ZF set, and it's incoherent to suggest the idea of a mathematical object that isn't somehow isomorphic to a ZF set. J.H. Conway laments this in an epigraph to _On Numbers and Games_. For example, he says that one should be able to postulate the existence of ordered pairs without being expected to identify a particular ZF construction with the required properties. Of course he's right, ZF is irrelevant here. But I've had many conversations with people on Math SE who said that no, an ordered pair _is by definition_ a set of the form {{x}, {x,y}}, nothing more or less.

Replies (2)

@ncf@types.pl 2026-04-09 14:45

@mjd What annoys me most about this is that it seems like the only available way for beginners to learn about foundations, which leads to endless confusion about the thoroughly chaotic and unprincipled way that basic notions are encoded in material sets. Latest example to date but this happens every other week on MSE. Most people still view type theory as a fringe topic or an advanced area of research, not something for beginners to learn, and this makes me very sad.

Open ##2061456
@skewray@mathstodon.xyz 2026-04-09 15:02

@mjd @ncf As someone who studies probability, it seems obvious that an experiment has a specific result, and so individual objects exist, and that one wants to compute chances for sets of those. Looks like a set theory. But as a probablist, I know ZFC is a non-starter, since it doesn't play well with measure theory, which is no more than assigning sane values of chance to sets. So, I am left with ¬(ZFC).

Open ##2061459