Yep, and are talking at cross purposes.

Supose our domain of discourse - what we are talking about - contains only the letters "a" and "b". How many things are in that domain?

If we listen to Frank, then we have a, and we have b, of course; two things. But we also have the set {a,b}. So there are three things: a, b and {a,b}. But then we also have {a,b,{a,b}} - so there are four things in our domain - a, b, {a,b}, and {a,b,{ab}} - and off we go. I hope folk see the problem inherent in counting a set as a different thing to it's elements.

No, a set is no more than the things it contains.

There's also the problem that if a set consists in a criteria rather than it's members, then we can construct the criteria "the set of sets that do not contain themselves", and upset Bertrand Russell. No, set consists in its elements, not in a description of those elements.

That's part of the axiom of extensionality. We can have innumerably many descriptions of some set - the set of odd number, the set of all numbers one less than an even number, the set of numbers that are two more than the last number in the set, starting at one. These are not three different sets, but three different descriptions of the very same set. It's the elements that count (pun intended), not the description.

What this does is to define what we mean when we say that a set is an abstract object - the set {a ,b} is not something else in addition to it's elements, but a different way of talking about a and b. A bit of extra language, not a bit of extra ontology. We talk as if the set were a new thing, but it isn't one of the things in the domain.

Similarly, when we talk of the red of the sports car and the red of the sunset, we haven't thereby added a new thing to the word - the property of redness. Redness is just a new way of talking about the car and the sunset.

Properties dissolved by analysis. Tim will love it. Not.

Supose our domain of discourse - what we are talking about - contains only the letters "a" and "b". How many things are in that domain? — Banno

I'm thinking "none" at that level of abstraction.

Or perhaps the opposite in reflection.

If our domain of discourse consists of only two letters then, on the first iteration, there is nothing to be said.

However, just that I understood "contains only the letters "a" and "b" " indicates some meta position wherein I can say things like "contains" etc etc.

Properties dissolved by analysis. Tim will love it. Not. — Banno

I suspect no one will love what I have to say, but I say it cuz I think it's true.

I'd hoped that it was pretty clear there were two things - a and b. Kinda what I stipulated. You can say "the domain consists of a and b" to avoid boxes if you like.

OK, yes. "a" and "b" are two things, as stipulated. (and, yes, I like avoiding boxes)

:smile:

Pardon the intrusion. I haven't read all the posts. You have a set {1,3,5} , and then you have another set that has as elements the properties shared by the elements of the first set. If this derived set is a singleton, then one could identify the elements of the first set.

Not really set theory, otherwise we would need TonesinDeepFreeze.

I don't think that a property is a collection. Redness is not the collection of all red things but something that is had by all red things. — litewave

Glad to see you've since taken the vows of nominalism!

Everybody in this thread has their own private set theory. :lol:

Except me.

Tones — jgill

Types?

@litewave

... Nominalism in the sense of, as @Banno says, dissolving properties by analysis.

But not necessarily in any strict sense.

E.g. not in the sense of, as Goodman says, "hyper-extensionalism", i.e. allowing no more than the power set to be defined upon the elements in the domain.

Rather, just in the sense of, as you (and @frank, I think? I need to study the thread) correctly say, extensionalism, i.e. allowing the definition of sets according to their extension to iterate indefinitely. E.g. on the domain {a, b} the sets { }, {a}, {b}, {a, b}, {{a}, a}, {{a}, b}, {{a}, a, b}, {{b}, a}, {{b}, b}, {{b}, a, b}, {{a, b}, a}, {{a, b}, b}, {{a, b}, a, b}, {{{a}}, a}, etc.

Not sure what @Banno is thinking there :chin:

It does indeed continue ad nauseum, but that is indeed set theory.

So much so, that structures in set theory are typically constructed entirely from the empty set.

we would need Tones. — jgill

Yes, where is he!

AH!

I missed it.

Yep. Tones can tell me, rightly, how all the stuff I've said here is a gross oversimplification.

@TonesInDeepFreeze?

Hasn't been seen for eight months.

What this does is to define what we mean when we say that a set is an abstract object - the set {a ,b} is not something else in addition to it's elements, but a different way of talking about a and b. A bit of extra language, not a bit of extra ontology. We talk as if the set were a new thing, but it isn't one of the things in the domain. — Banno

Rereading I want to highlight this bit as a better explanation of what I've been saying.

Naturally I'd accept @TonesInDeepFreeze, though at this point I wonder if that's too much pressure on them.

Cool.

Tone's last few posts expressed frustration with a particularly recalcitrant contributor. :worry:

The set of all of our theories of set theory is public, but here we are attempting to figure out what the members of that set are.

No, a set is no more than the things it contains. — Banno

Enter the prompt "Is a set identical to its elements?" in ChatGPT, Claude or Gemini. They will all give you the answer No.

Pardon the intrusion. I haven't read all the posts. You have a set {1,3,5} , and then you have another set that has as elements the properties shared by the elements of the first set. If this derived set is a singleton, then one could identify the elements of the first set. — jgill

"the properties shared by the elements of the first set" might be where @litewave is coming from.

Your expertise is not an intrusion at all.

I think @Banno is there. That's why he posited a difference of hierarchies between elements, sets, sets of sets, etc.

To say "a set is no more than its members" is to say that there is no condition of inclusion aside from being part of the set.

Yep. As explained earlier, "identical" has a very specific definition here. The set is not extensionally identical to it's elements.

Try "Is a set completely determined by its elements?"

Glad to see you've since taken the vows of nominalism! — bongo fury

Alas, I have broken the vows in the course of this thread. Although it was not really nominalism about properties; I still regarded them as real separate objects, I just wanted to identify them with sets.

The set of all of our theories of set theory is public, but here we are attempting to figure out what the members of that set are. — Moliere

:up:

As explained earlier, "identical" has a very specific definition here. The set is not extensionally identical to it's elements. — Banno

So the set is another object, in addition to its elements.

Redacted. I can't help you here.

I can't help you here. — Banno

You can, easily. Yet choose not to. Like always.

I'll go so far as to say "another logical object", yet the addition would still amount to 2 elements.

Heh. If you can help our new friend easily then by all means -- jump in!

I don't think set theory is easy, intuitive, etc.

:yawn: .

So how many individuals are there in the set {a,b}?

I say two. — Banno

Depends on what you mean by individual. There are obviously two elements in this set: a, b. By the way, in pure set theory these two elements are always sets as well.

By the way, in pure set theory these two elements are always sets as well. — litewave

I'm interested.

Where can I learn more on this?

I'm a naive set theory boi who reads logic texts and that's it.

I don't know what pure set theory is.

Yeah, I deleted that response becasue it doesn't make the teaching point I would like. It does depend on what we mean by individual, just as it depends on what we mean by identical or by completely determined.