She has never seen a universal though. But she has seen collections (sets), so she may know more about collections than about universals. — litewave

You think knowledge is limited to what you can see? If so, she's never seen a set. A set is an abstract object.

You think knowledge is limited to what you can see? If so, she's never seen a set. A set is an abstract object. — frank

A set is a collection of objects. An average person surely knows what a collection is. Not so surely a universal.

This is an intensional definition of a set, a definition by specifying a common property of the set's elements. An extensional definition of a set would be a definition by listing all the particular elements. — litewave

Also:
Extensional definition of Ship = {ferry, tankers, icebreakers}
Intensional definition of Ship = {large boat, travels on water}
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What about these two: the property of redness, and the property of being an instance of redness (or the property of having the property of redness). Both properties seem to be instantiated in all instances of redness, so the instances form one and the same set. — litewave

A common example of coextensive properties

The property of "having a heart" = {human, dog, cat}
The property of "having a kidney" = {human, dog, cat}

True, a heart is a distinct thing to a kidney.

However, the above example is invalid, as a human can exist using an external dialysis machines. With medical progress, hearts and kidneys are no longer necessary to what makes a human.

Is "being an instance of redness" referring to one thing

Taken at face value, an instance is one particular thing. This infers that "being an instance of redness" is also one particular thing, meaning that it cannot be a property.

Therefore, the expression "the property of being an instance of redness" is not a valid expression.

Is "being an instance of redness" referring to several things

However, if "being an instance of redness" is referring to several things, as in "being an exemplification of redness", then it means the same as "redness".

"Being an instance of redness" and "redness" are then not distinct as a heart and kidney are distinct, meaning that "the property of redness" and "the property of being an instance of redness" is not an example of coextensive properties.

There may not be examples of genuinely different coextensive properties, meaning that it doesn't prevent us from associating a property with a set.

A set is a collection of objects — litewave

Not in set theory. A set is criteria. It's an abstract object.

However, if "being an instance of redness" is referring to several things, as in "being an exemplification of redness", then it means the same as "redness". — RussellA

"Being an instance of redness" seems to be a property of all instances of redness, yet it seems to be a different property than redness itself. Both properties have exactly the same instances, which suggests that the properties could be one and the same, but "being an instance of redness" refers to an instance in relation to redness while redness refers only to redness.

Not in set theory. A set is criteria. It's an abstract object. — frank

Come on, objects that are included in a set satisfy certain criteria (have certain properties) but the set is a collection of those objects.

A set is an abstract object.

I don't think that's at odds, per se, with defining a set as a collection of objects, or individuals.

Though...
I'd put it to you that the collection of individuals is an abstract object. To use your cell phone example -- we can think about the cell phone as a collection of particular objects and then name this in accord with set-theory. I.e. we can make sets which refer to concrete individuals, but to treat something as a set is still an abstraction.

We can also treat the phone as an individual, from the logical point of view. Suppose the set of all of my possessions. Then, even though I can break my phone down into smaller parts in the case of the set of all of my possessions, the phone is merely an individual.

Whether something is within a set or not doesn't reflect upon its ontology -- I'd say that's more of a question for mereology (which the logic we choose to utilize may have implications for, but it's still different from the logic of sets)

I don't think that's at odds, per se, with defining a set as a collection of objects, or individuals. — Moliere

The point is a set isn't something you can see, anymore than you can see infinity.

A way to think about set theory --

It doesn't matter what's in the set. The validity that's being explored are the inferences one may draw about sets regardless of their contents.

So supposing two sets, supposing B is a subset of A, we can infer that -- no matter what elements are in B, if they are in B then they must be in A.


Right. That's not at odds with the theory that a set is a collection of objects -- as in, any collection of objects, regardless of what those objects are, even if the set does not have any objects in it or some of the objects are infinite.

I don't think sets exist as much as are ways to think about things.

@litewave seemed to be suggesting that people know firsthand about sets because they can see them. That is incorrect. You can't see a set.

Yeah, I tried to address that in the reply to @litewave -- waiting to hear back.

I'd call that a hypostization, which is an easy thing to do. Similarly so with treating sets like predicates.

Though, if we're Kantians, it'd seem like you couldn't help but to see the world through categories, so maybe there's a position wherein one could see sets -- but treat them in a logical way.

I prefer to think of sets as abstractions which we stipulate, though.

So here:

A set is a collection of objects. An average person surely knows what a collection is. Not so surely a universal. — litewave

A set is any given collection of objects.

An average person knows what a collection is and so you can start from there.

But the abstraction begins when we stop considering what is in the collections and consider the relationships between collections and the inferences we can draw given any collection whatsoever.

So we name sets things like A and B to signify that we're not talking about particular individuals, or even particular sets -- but rather the valid inferences one can make given any set whatsoever unspecified beyond being a set.

That jump to the "any set whatsoever" is the part the average person has to learn when we're talking about when learning set theory. Not just a collection, but the very concept of collection and how we can draw inferences from that.

I'm not sure what you're trying to say. The mainstream view among mathematicians is that sets are abstract objects. You can see them with the mind's eye, but not physical eyes.

The mainstream view among mathematicians is that sets are abstract objects. You can see them with the mind's eye, but not physical eyes. — frank

I'm trying to say that you're correct about sets abstractness (at least, in my view, while acknowledging possibilities), and that @litewave is correct about the definition of a set with a little tweaking.

The words are right, the interpretation isn't quite there.

Would that mean that "being in that collection of objects [or individuals, per Banno]" is a shared property? Can an object "wander in," so to speak, and partake of that property? This may not be a question about your definition so much as an expression of uncertainty about "property". — J

Yes. So what, if anything, would we want to say about identifying such a set with some property? I take it you don't want "being in set X" to count as a property -- nor could it, on the OP's proposal. — J

I'm coming to notice that I'm pretty much avoiding "property" all together and relying upon "predicate" (to circle back to where I left you off and rethink)

And where I've been reflecting from is the logical side, rather than the metaphysical side. I more or less took "property" to be substitutable with predicate, but if the conversation is going towards the perception of wholes then "property" may be the better term over the logical quandaries I've been raising.

In which case I'm on the side that "property" is something we distinguish within a metaphysical context rather than something anything "has" outside of that context. "Property" is an abstraction, too -- a word which can be used in various ways within a particular metaphysical expression. I prefer "affordance" to "property"

I'd put it to you that the collection of individuals is an abstract object. To use your cell phone example -- we can think about the cell phone as a collection of particular objects and then name this in accord with set-theory. I.e. we can make sets which refer to concrete individuals, but to treat something as a set is still an abstraction. — Moliere

I'm not sure what you mean by "abstract" or "abstraction" here. Is the phone a concrete or an abstract object? Is it a collection of other objects or not?

I'm not sure what you mean by "abstract" or "abstraction" here. Is the phone a concrete or an abstract object? Is it a collection of other objects or not? — litewave

What I mean by "abstraction" is that you can treat the phone in either way without changing anything real.

You can treat the phone as an element -- which is that which is a member of a set -- or you can treat it like a collection -- such that its elements are members of the set "my phone".

It's how you think about it that makes the difference in terms of perceiving the phone as an set element or a collection of individuals. It is both.

What I mean by "abstraction" is that you can treat the phone in either way without changing anything real. — Moliere

It depends on what we mean by "abstract" and "concrete". It is often said that concrete objects are located in space or in spacetime. Then a collection like a particular phone would be a concrete object. A collection consisting of a phone located in my house and another phone located in my friend's house would be a concrete object too, although some might resist that because the two phones are separated "too much". A collection consisting of my phone and of another phone in a different universe that is in a different spacetime might be regarded as a concrete object because its elements are located in a spacetime but then again, they are in different spacetimes, so this collection transcends a single spacetime. And then there is the general property/universal "phone" (or "phoneness") - that which all particular phones have in common - and I guess this would be regarded as an abstract object by almost anyone because unless we identify it with the set of all phones, it seems to transcend spacetime or be located in spacetime in an especially weird way.

So, to read you here, I'm taking your ideas about each to be:

Concrete:

A collection consisting of a phone located in my house and another phone located in my friend's house would be a concrete object too, although some might resist that because the two phones are separated "too much". A collection consisting of my phone and of another phone in a different universe that is in a different spacetime might be regarded as a concrete object because its elements are located in a spacetime but then again, they are in different spacetimes, so this collection transcends a single spacetime. — litewave

Abstract:

And then there is the general property/universal "phone" (or "phoneness") - that which all particular phones have in common - and I guess this would be regarded as an abstract object by almost anyone because unless we identify it with the set of all phones, it seems to transcend spacetime or be located in spacetime in an especially weird way. — litewave

And you're noting the weird part where it seems they come together.

yes? No?

I would classify objects as concrete and abstract as you have indicated. The weirdness part would fall under abstract.

So what's your belief with respect to "Identification of properties with sets" now?

I've tried to dissuade you, but are you still committed?

It seems that we have found genuinely different yet coextensive properties (like the property of redness and the property of being an instance of redness), so now I lean toward the view that properties cannot be identified with sets.

heh, well, once you flip your opinion I'll return -- but my work is done :D

heh, well, once you flip your opinion I'll return -- but my work is done :D — Moliere

I see what you wrote early on:

But, no, they're different -- a set is its' elements, rather than a property which all the elements share. — Moliere

But I still don't think that a set is identical to its elements because a single object cannot be identical to multiple objects. So a set is another object, additional to its elements.

But I still don't think that a set is identical to its elements because a single object cannot be identical to multiple of object. So a set is another object, additional to its elements. — litewave

Looking at that rendition I agree.

A set is any collection of elements is a better rendition. It's another (logical) object, to the point that its elements aren't a part of how we infer validity between sets.

To go back to the subset relation: if any element is a member of B, and B is a subset of A, then any element of B is an element of A.

To go back to the subset relation: if any element is a member of B, and B is a subset of A, then any element of B is an element of A. — Moliere

Right. There is a difference between "element" ("member") and "subset". Outside of set theory they may be both conflated with the concept of "part" but they are parts in different senses.

Amen. :love:

Still probably things to talk about wrt set theory, but glad you understood me.