I don't know what pure set theory is. — Moliere

A set is pure if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the von Neumann universe of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets.

https://en.m.wikipedia.org/wiki/Set_theory#Ontology

Alas, I have broken the vows in the course of this thread. — litewave

:rofl:

Although it was not really nominalism about properties; I still regarded them as real separate objects, I just wanted to identify them with sets. — litewave

Right, but, identify them with sets in the way that model theory maps predicates to sets? (Sets defined on elements in a domain.)

Then you can be at least extensionalist about properties just by replacing them with predicates? Or with the corresponding sets if you prefer, yes!

Hyper-extensionalism is a further economy:

The nominalist cancels out the property and treats the predicate as bearing a one-many relation directly to the several things it applies to or denotes. — Goodman, p49

And in this case cancelling out the property is indeed a matter of cancelling out the set. And satisfying @Banno's thirst for a restriction to individuals. See Goodman's "calculus of individuals". (Mereology as @Moliere alluded.)

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. — Banno

Why would there be a problem in counting a set as a different thing to its elements?

https://st.openlogicproject.org/settheory-screen.pdf

Heh.

Got it.

That's the next logic textbook cuz of you :D

There’s no formal problem in set theory with counting sets as different from their elements. The “problem” arises only if one has an intuition that collections shouldn’t add to ontology—that a set should “just be” its members. In that case, the proliferation looks like an unnecessary or suspicious multiplication of entities.

That's the next logic textbook cuz of you :D — Moliere

Sorry.

While I understand the apology I'm a sucker for pain.

At least, when it comes to Truth and Logic.

One of the problems here was a classic for Tones - the move from formal exposition to philosophical exposition. The OP makes that move, by equating the formality of set theory with the informality of properties. So we are a bit stuck with it.

Right, but, identify them with sets in the way that model theory maps predicates to sets? — bongo fury

I wanted to say that the set is the common property of its elements.

Yes, and so does a model theorist? (And earlier theorists of the semantics of first order logic too.) I thought this was what @Banno was pointing out to you 4 years ago? I may have misunderstood.

There’s no formal problem in set theory with counting sets as different from their elements. The “problem” arises only if one has an intuition that collections shouldn’t add to ontology—that a set should “just be” its members. In that case, the proliferation looks like an unnecessary or suspicious multiplication of entities. — Banno

But in set theory, sets do add to ontology. And in pure set theory all elements of a set are sets too.

But in set theory, sets do add to ontology. — litewave

That's why nominalists (e.g. Quine) didn't like taking it for granted in logic.

Yes, so a model theorist? I — bongo fury

I have not studied model theory.

I wanted to say that the set is the common property of its elements. — litewave

Thanks for the clarity.

To my eye, this reifies the property, making it a thing alongside the elements of the set.

That is, you now have the set and the property, separately, and are apparently defining the set in terms of the property.

But of course we could then stipulate a set with no common properties.

I had taken you as proposing to eliminate properties in favour of sets. I would agree with that. But it seems you have something else in mind.

And I'm not at all sure what.

I thought this was what Banno was pointing out to you 4 years ago? — bongo fury

Me, too.

But in set theory, sets do add to ontology. — litewave

What does this mean?

Here's one way to look at it. We have the domain <a,b>. The only items in that domain are a and b. Constructing the set {a,b} does not add to the domain. It does not add {a,b} to the domain.

Are you eliminating properties in favour of sets (which I would support), or making sets into reified metaphysical entities that ground properties?

or making sets into reified metaphysical entities — Banno

Well, they are, a bit. :wink:

That's why nominalists (e.g. Quine) didn't like taking it for granted in logic. — bongo fury

Mary Tiles (a philosopher of math) says she can imagine mathematicians ditching set theory someday.

That is, you now have the set and the property, separately, and are apparently defining the set in terms of the property. — Banno

Yes, because my attempt to treat the set and the property as one and the same object seems to have failed.

But in set theory, sets do add to ontology.
— litewave
What does this mean? — Banno

That sets are objects in the ontology of set theory.

Well, they are, a bit. :wink: — bongo fury

:lol:

Which is why these threads are neverending.

Mary Tiles (a philosopher of math) says she can imagine mathematicians ditching set theory someday. — frank

And I hope has the sense not to ditch it yet?

:cheer:

That you're willing to say as much is a credit to you.

And I hope has the sense not to ditch it yet? — Banno

Why do you say that?

Yes, because my attempt to treat the set and the property as one and the same object seems to have failed. — litewave

But I hope you see that your intuition - that having the property of being red and being a member of the set of red things say much the same thing - remains valid?

That sets are objects in the ontology of set theory. — litewave

And so long as you do not expect to bump in to them as you walk down the street, that's fine, isn't it? What is needed is to keep track of which domain is which.

I think the intuition in the OP is quite right, and in a rough line with Quine and indeed pretty much all of more recent logic.

Cheers. Respect.

Are you eliminating properties in favour of sets (which I would support), or making sets into reified metaphysical entities that ground properties? — Banno

I have always treated sets as real metaphysical entities. So if properties were sets, then properties would be real too. If properties are not sets, I am not sure if properties are real, but I tend to think they are.

Odd.

You think she should throw it out on the basis of of her imaginings?

But I hope you see that your intuition - that having the property of being red and being a member of the set of red things say much the same thing - remains valid? — Banno

Yes.

That sets are objects in the ontology of set theory.
— litewave
And so long as you do not expect to bump in to them as you walk down the street, that's fine, isn't it? — Banno

I interact with collections of objects all the time.

I have always treated sets as real metaphysical entities. So if properties were sets, then properties would be real too. If properties are not sets, I am not sure if properties are real, but I tend to think they are. — litewave

Do you want me to go on?

What is a real metaphysical entity as contrasted with a real entity? What does the word "metaphysical" do here?

What is a real metaphysical entity as contrasted with a metaphysical entity? What does the word "real" do here?

So what more do we have then "I have always treated sets as entities", which seems quite agreeable.

Just leave aside the baggage of "real" and "Metaphysical".

So if properties were sets, then properties would be real too. — litewave

So instead think about these issues in terms of sets, with all the clarity of the formal apparatus that invokes, and just drop the use of "property", or use it as an anachronistic approximation.

If properties are not sets, I am not sure if properties are real, but I tend to think they are. — litewave

What does it mean to say they are real? What more can we do with real properties that we can't do just with properties? Or much better, with talk of sets or predicates?

This stuff:

What's curious here is how "the property of..." serves to confuse things. The very grammar of "the property of..." encourages us to think we're talking about entities when we're really just manipulating linguistic constructions. — Banno

This is the legacy of syllogistic logic. Since it can only deal in terms of "All S are P", "Some S are P", and so on, it obliges the user to think in terms of substances having properties. It squeezes the world in to an ontology of things and properties. Scholastic metaphysics elaborated on this logical limitation by inventing essences, accidents, substance and so on.

We now have better logical tools for dealing with all of this stuff. The answer on offer to ↪litewave is not to identify properties with sets but to drop talk of properties for talk of sets and predication and extension. Indeed, that is probably the intuition behind the OP.
a day ago — Banno

I interact with collections of objects all the time. — litewave

Sure. Just not in the way you interact with chairs. Different domains.