Are you saying that we can't tell rutabegas from swedes? I thought they were the same. These things:

Taste like squashed bugs. Or at least, how I supose squashed bugs might taste, not having tasted them... to the best of my knowledge.

I suppose they might be OK with enough maple syrup. But even then, better just to eat the syrup.

Turnips, Purple Tops.

I think it's pretty easy to identify red things — Count Timothy von Icarus

Me too.

I don't think anyone here is suggesting there are no red things or no triangles.

"nothing has the property of being triangular" which would seem to imply that nothing is triangular. — Count Timothy von Icarus

The picture holds you. Can't we just say that there are triangles, and leave "there is a property of triangularity" or whatever as a slip into reification?

The slide from "there are red things" to "therefore redness must be a thing" and then to Platonic forms floating in metaphysical space and all the historic mess that followed. No, but there is that car and that sunset.

So are swedes and rutabegas and purple top turnips extensionally identical?

If so, we may substitute one for the other and achieve the same distasteful result.

No, rutabaga is a different plant, from the pictured turnips

Brassica napobrassica

https://en.wikipedia.org/wiki/Rutabaga

So are swedes and rutabegas and purple top turnips extensionally identical? — Banno

At my grocery store, a rutebega is a large round root with purple on all parts, for some reason with a very waxy exterior. A turnip is smaller, shown in the picture you provided. A Swede is a tall blonde specimen from up north.

Are they extensionally identical? Is water and H2O?

Brassica napobrassica — Banno

This makes the H2O - water point, right? The scientific term means something different from the common one, but they collapse under the same set, losing that distinction.

For example, the property of redness would be identified with the set of all red things, or the property of being a car would be identified with the set of all cars. — litewave

The op is messed up. You cannot identify the property of redness with the set of all red thing. The supposed conclusion would actually be inconclusive, requiring a definition. Further, "being a car" cannot be defined as a property.

Ah - Brassica rapa subsp. rapa. Ok. Neither is worth substituting for potato.

All a property, in the broadest sense, is just an attribute or quality possessed by something. So, Socrates is a man, a rose is red, etc. The rose is red, but the rose is not identical with the color red. And there is the idea that multiple things can instantiate the same properties.

For instance, a common simple version of Leibniz' Law is: "Necessarily, no two objects share all their properties." One need not assume redness is a "thing" to make use of this.

There are nominalist accounts of properties. One does not need to be a "Platonist" to make use of properties. It's a quite useful concept because:

If there are no properties, in virtue of what would some things be members of "the set of red things" but not others?

Or in virtue of what would different individual things he discernible?

From Wikipedia:

Rutabaga has a chromosome number of 2n = 38. It originated from a cross between turnip (Brassica rapa) and Brassica oleracea. The resulting cross doubled its chromosomes, becoming an allopolyploid. This relationship was first published by Woo Jang-choon in 1935 and is known as the Triangle of U.

Be aware of the mysterious "Triangle of U".

All a property, in the broadest sense, is just an attribute or quality possessed by something. — Count Timothy von Icarus

You say that with great certainty, as if it were an explanation of what a property is. But what is an attribute, if not what we attribute to something? Etymology: "assign, bestow," from Latin attributus, past participle of attribuere "assign to, allot, commit, entrust;" figuratively "to attribute, ascribe, impute," from assimilated form of ad "to" (see ad-) + tribuere "assign, give, bestow"

So it's whatever we say it is? Cool. But I doubt that's what you meant.

Otherwise, yep. Except that you shouldn't include the modal operator in Leibniz' Law without some clarification.

Identical" is defined extensionally by substitution. I hope we agree that there is nothing more to the set {a, b, c} than a and b and c, no additional "setness" in the way RussellA supposed by adding his box. — Banno

A set is not its elements. Imagine a club that all teachers automatically belong to, by virtue of being teachers. The set is this membership criteria, not the actual teachers. A set is an abstract object.

A set is a collection of individuals. They need not have anything related to one another, or share anything at all -- the individuals are the set and there's nothing else to it. The pebble on the ground and the sentence I say 5 miles away can form a set. — Moliere

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

Even the extravagant set that @Moliere has mentioned above is something in addition to the pebble and the sentence, and this something is a property that the pebble and the sentence share. It is an unimportant property for which we have no word, and being in that set means having that property.

The set is this membership criteria, not the actual teachers. — frank

Nuh. The set is the teachers. The criteria are not the set.

Even the extravagant set that Moliere has mentioned above is something in addition to the pebble and the sentence, and this something is a property that the pebble and the sentence share. It is an unimportant property for which we have no word, and being in that set means having that property. — litewave

Oops.

Nuh. The set is the teachers. The criteria are not the set. — Banno

nope. Read Mary Tiles' book on set theory. The club metaphor is from her book.

Then she is mistaken. Or has been misread.

It does not matter how we specify the set, or how we order its members, or indeed how many times we count its members. All that matters are what its members are. — Set Theory An Open Introduction

Then she is mistaken. Or has been misread.
It does not matter how we specify the set, or how we order its members, or indeed how many times we count its members. All that matters are what its members are.
— Set Theory An Open Introduction — Banno

I stumbled over this same issue, so you're not alone, but you're wrong. In the club metaphor, the set is the membership list, not the members themselves.

A set is an abstract object. That's the part you have to get in order to understand set theory.

OK. :grimace:

Keep reading.

Several quite different points, all of them muddled together.

First point. might be understood as saying that in addition to the set consisting of {book, car, apple} there is a fourth item, grouping these together, the box the set comes in, as it where. That's not right. There is nothing in addition to the elements.

Second point. What we mean by identity is when talking about sets is extensionality, that is, that if A and B are sets, then A=B iff every member of A is also a member of B , and vice versa. Read that as a definition of how to use "=". So we should read S={a,b,c} as an identity between S and {a,b,c} and we can say that they are identical. That is reply to .

Third point. {a,c,b} and {a,b,c} are the same set in that they are extensionally identical, but are not identical in that they are written down differently. That the description is different does not make them different sets.

Forth point. Similarly, The set of the first three letters of the alphabet is extensionally identical to {a,b,c}. Again, how the members are specified is not a part of the set. Only the elements are apart of the set.

Fifth point. Those first four points hopefully server to show that the members are what count in determining a set. Now from the other direction, the apple in hand is not the set {apple}. This difference is usually set out by saying the set is an abstract individual apart form the apple. The tricky part is realising that this does not contradict those first four points. We do not write apple={apple}. .

That might clear things up. Maybe. But the thread "An unintuitive logic puzzle" got to fifteen pages.

Properties, qualities, characteristics, and so on, are mental or linguistic abstractions of the things described, or even the descriptions themselves. Your morphological derivations “redness” and “carness” indicate this. They are derivations, not sets or properties.

A set is a different object than any of its elements. But if the box is black then it also contains instances of blackness, not just redness. For example the walls of the box may be black. Your example looks like the property of redness contained in a black box. — litewave

Consider the singleton set containing one element, such as Socrates = {Socrates}.

From Zermelo-Fraenkei set theory, no set can be an element of itself, meaning that a singleton set {Socrates} is distinct from the element it contains, Socrates. (Wikipedia - Singleton (mathematics))

Does this not mean that saying the box can only be black if it contains instances of blackness violates the Zermelo-Frankei set theory, in that the singleton set must be distinct from the element it contains?

IE, thinking about set theory, a black box would then not be distinct from the instances of blacknesses within it.

I don't know, but am curious to know.

I am proposing that we could plausibly identify a property with the set of all things that have this property. — litewave

This is exactly true, contingent on the what we mean by the word "plausible".

We can plausibly do a lot of things that on closer inspection can't be done, but finding that out does not negate that it was plausible to begin with.

In general, these kinds of ideas I would argue are best understood as naive set theory used in ordinary language.

What I mean by that, is that we have a bunch of mathematical concepts about sets that nothing stops us from using in ordinary discourse outside an axiomatic system to discuss various things; both to talk about axiomatic systems from a non-axiomatic point of view (such as to talk about what an axiomatic system "is like" and kinds of things that can be done with it using ordinary language to convey concepts to both experts and non-experts) as well as developing concepts that have nothing to do with axiomatic systems but the words and ideas of set theory are useful for the purpose.

Therefore, such informal use of set-theory language is going to have all the problems of ordinary language. We have zero problem with the fact ordinary language can be used to express all sorts of contradictions, paradox and nonsense, as well as having fundamental unresolvable problems such as delineation, universals, and so on, and throwing in some set theory words isn't going to change the situation.

That does not make it invalid to talk about sets of "everything red" for example, but we can know ahead of time that such a concept cannot be developed into something rigorous without axiomatization.

Once you axiomatize, if all goes well, you can have rigorously defined symbols, rigorously defined acceptable grammar (how you may put those symbols together), and rigorously defined permitted manipulation of those symbols (how you may move those symbols around), but in so doing we know ahead of time that we lose the flexibility of ordinary language; you can no longer just "say things" and hope to express meaning, but rather statements proposed to be true require rigorous proof.

What makes sense depends on what is being talked about.

For example, it makes sense to propose dividing the class into a set of short and a set of tall students. The meaning is clear that the goal is to either by symbolic representation of the students or then physically move them around, to define two groups of students based on height. The meaning can be clear and it can be equally clear that once we have our sets of students we can perform further set operations, such as creating subsets of those sets based on hats or whatever we please, and going on to create unions and bijections and so on.

What is of course not clear is exactly the difference between short and tall, how to handle a new student showing up (do the sets represent the students at the time of creating the sets, or then sets that represent students that may come and go, either by joining the class or then being expelled), and so on. Trying to resolve all these problems will run into all the problems of ordinary language and naive set theory, but the use of such language can be entirely sufficient to accomplish whatever the task was (forming teams by height and hat wearing for some purpose).

Second point. What we mean by identity is when talking about sets is extensionality, that is, that if A and B are sets, then A=B iff every member of A is also a member of B , and vice versa. Read that as a definition of how to use "=". So we should read S={a,b,c} as an identity between S and {a,b,c} and we can say that they are identical. That is reply to ↪litewave. — Banno

So litewave wants the property P to be equal to the set of all things that have P, we'll call it set Q. But we can't say that P=Q? Because they're different types of things?

First point. ↪RussellA
might be understood as saying that in addition to the set consisting of {book, car, apple} there is a fourth item, grouping these together, the box the set comes in, as it where. That's not right. There is nothing in addition to the elements. — Banno

But that's exactly what I am arguing - there is a fourth object and this fourth object is identical to the set of the three objects. The set as a single object cannot be identical to three objects, so it is identical to a fourth object. The identity of the fourth object is fixed by the three objects because there can only be one set of the three objects, but the set itself is not identical to the three objects.

Properties, qualities, characteristics, and so on, are mental or linguistic abstractions of the things described, or even the descriptions themselves. Your morphological derivations “redness” and “carness” indicate this. They are derivations, not sets or properties. — NOS4A2

Collections or sets are not just mental or linguistic abstractions though. As I am typing this I am actually holding a collection called "smartphone".

Does this not mean that saying the box can only be black if it contains instances of blackness violates the Zermelo-Frankei set theory, in that the singleton set must be distinct from the element it contains? — RussellA

Aha, I think I see what you mean. The singleton set is distinct from the element it contains and so it is something additional to the element. The element is red and the singleton set is something else (though probably not black, because that would require some more complex structure that can absorb light). I propose that the set of all red objects is the property "redness" but this property probably does not look red, in fact it probably does not look like anything that could be visualized because it is not an object that is contiguous in space or time.

That does not make it invalid to talk about sets of "everything red" for example, but we can know ahead of time that such a concept cannot be developed into something rigorous without axiomatization. — boethius

When I say that a property is identical to the set of all objects that have this property, I mean that the property is completely specified and thus the set is completely specified. In practice we usually don't have such complete specifications and we talk about approximately specified properties like "redness", but that doesn't refute my claim that a property (completely specified) is identical to the set of all objects that have this property.