While the elements of a set can be totally arbitrary, the sets that mathematicians generally use are not (there are rare exceptions, mostly if you are a set theorist trying to build something like a non-measurable set). That is why they use notation like {x | …} where the … is some property. So the cited passage actually expresses the common reality. Actually, for the standard development of set theory in which the elements of a set must themselves be sets, there is no set whose elements are “the sky”, “the number 7” and “Sherlock Holmes” because two of those three entities are not sets.
If we’re going to persist in the fantasy that the number 7 is a set, we might as well make up encodings under which the sky and Sherlock Holmes are sets as well…
We all know the ordinary language “set” concept exists more generally than strictly in reference to sets of sets of sets of sets. And this more general understanding of “sets” is, I would say, a good place to start, pedagogically. There’s more to life than the cumulative hierarchy, despite the restrictive account of the term “set theory” which the caprice of history has standardized upon.
FTR, I was taught set theory by a world reknowned leading expert in the philosophy of logic and mathematics, and she was very explicit in saying that sets can contain anything as a member. The thing with all sets containing only sets as members was introduced as a special model which is standardly used to model numbers.
Is this not the standard line?
Penelope Maddy, I’m guessing?
If “the standard line” is the account of “sets” as given by ZFC and its ilk, then it says that sets contain only other sets as members. I think most mathematicians would acknowledge that the word “set” is also reasonably used to refer to collections in a more general sense, but still, the use of the term “set” as meaning specifically “A set of sets of sets of sets of…” is quite dominant (often, it makes no difference, but in those particular contexts where it matters, that’s typically what the jargon “set” means).
ZFC implies the non-existence of sets which have as members, non-sets?
I find this a bit odd. Of the ones that wouldn’t acknowledge such, what do they say about cases where people go ahead and define “sets” that have non-sets as elements? That is really a “zet” not a set, or that set doesn’t actually exist, or something else? I guess the same question can be asked about competing “set theories”; if you are a ZFC guy do you think that, say, New Foundations is just wrong about what sets there are, or isn’t talking about sets, or something else.
Yes. We could take a look at the definition of “railroad” and hypothesize a railroad (i.e. a road made of iron or steel rails) that starts from one deserted section of Death Valley and goes to another deserted section of Death Valley two miles away, with stations only at the termini. Such a railroad would have little practical value today and a transportation planner, railroad engineer, conductor, or traveler is extremely unlikely to encounter such a railroad in real life unless major changes come to California. But if it DID exist somehow, would we say therefore that it is “not a railroad”? Of course not. The same with sets. The fact that you can define a useless set doesn’t mean that the definition of “set” should be changed to mean only sets that are useful according to some human standard which could change with time.
If you read ZFC to be talking about "set"s, sure. The particular way ZFC is set up as a one-sorted theory lets us use its axiom of extensionality to conclude that every set x is equal to {{z | z in y} | y in x}. Accordingly, every member of x is itself a set. If you want to make models of a ZFC-esque theory with urelements, you have to tweak the axiom of extensionality.
(As an example of the difference this makes, compare the ease with which one demonstrates the relative consistency of the negation of the axiom of choice with ZF relaxed to allow urelements (Fraenkel-Mostowski permutation models), vs. the greater difficulty of doing so without urelements (the same idea, but one now has to additionally find a way to encode it into “pure sets” via forcing…))
Don’t ask me. I suppose different people would say different things. Although I also suppose one person could say all those things (“Those aren’t really 'set’s, as I understand and use the term, and so they don’t exist as sets, though you can call them 'zet’s if you like, and similarly, New Foundations is wrong about what sets there are, though I suppose if it has models, you could take it to be talking about something other than sets…”).
Like I said, I think most people would acknowledge the use of the word “set” more broadly, but consider the sort of thing which happened just in this thread (“there is no set whose elements are ‘the sky’, ‘the number 7’ and ‘Sherlock Holmes’ because two of those three entities are not sets.”; that may not be how Hari personally feels about the word “set”, but it’s an attitude prevalent enough to note).
Because extensionality just says any two blah are identical iff _____; as opposed to any two *sets *are identical iff _____. I guess one could still say ZFC was talking about some but not all sets.
Sure; that’s why I say “If you read ZFC to be talking about ‘sets’”. But it does seem to be a thing people often do.
I think the lack of disagreement is fairly high here. I was just spelling out things in another way, for no particular reason.
I always get confused when people say they are talking about sets, as opposed to “whatever it is that fits these conditions”. That includes when I go ahead and do it myself.
Ah, gotcha.
Heh, me too…
This is horrible, absolutely godawful textbook writing. It fails literally, it fails metaphorically, and it fails as an analogy. If you gave me a week I might be able to think of a more confusing way to introduce the concept of sets, but it would be difficult.
Yeah, but mine refers to pets and family, it doesn’t talk about science at all so therefore it’s not educational and will rot our children’s brains!
At least, that’s the impression I’ve gotten of sub-college level textbooks over the years. An incorrect, confusing mess of a reference to SCIENCE(!) is leagues better than a correct, illustrative example that refers to things like cute puppies or video games.
ETA: I like the books I’ve had in college more if for no reason other than they seem to actually be allowed to have examples like dropping eggs on your professor’s head as a rudimentary kinematics problem.
Heh, at least one person agrees with me.
Are you kidding me? This entire thread is sitting on the far right seat of the couch which it has placed in a state of eternal dibs.
mmm