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.

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

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.

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.

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.

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.

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

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.

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

Amen. :love:

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.

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.

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

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?

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'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.

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"

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.

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.

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.

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.

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)

Why doesn't aporia lead to intellectual anarchy? — J

Sheer stubbornness of the philosopher :D

Oh, sorry. I thought that's what you were looking for in set theory. — frank

Nope.

The intuitive bit I can see is wanting to equate predicates with sets since we can quantify over both. The unintuitive bit is where I'm arguing there's a difference that maybe doesn't look like there is a difference between these logical objects.

Well it's not surprising that we see eye-to-eye, is it? :D

Do you think this a bad interpretation @Count Timothy von Icarus?

:) Thank you.

At this point I'm wondering if the difference in positions is that I think of "underdetermination" with respect to scientific theories, especially -- rather than applying to the radical skeptical position.

For thousands of years mathematicians would have said that set theory is illogical. It flies directly in the face of Aristotle's finitism, but it solves problems that are otherwise unsolvable. Don't look for an intuitive basis for set theory down in your noggin. It's not there. — frank

Oh yeah I don't think logic is intuitive at all. It's part of why it's interesting.


Just thinking out loud here about the stuff I like to think about here, I doubt I'm introducing you to anything:


One thing w/ Ari's deductive logic is that it can be reconstructed with the latter inventions of logic. Ari pretty much sets his definitions such that quantification, predication, and sets are all rolled into kinds of propositions and explores the validity between these kinds of propositions.

I find that interesting on many levels -- one, Ari's deductive logic is supportive of his inductive logic due to there being properties which things have that can be discovered. So it's not "innocent" in the sense that it fits into what we tend to interpret as his Philosophical Project: but it is "naive" in the sense that today we'd distinguish these things.

Well, in predicate logic you have individuals that have/satisfy a property/predicate. I propose that the property is the set of these individuals. — litewave

Cool. Let's look at this.

If the property is "is the set of these individuals", effectively F in F(x), what is the individual which satisfies this predicate?

It's interesting to try and think of sets in terms of predicate logic -- and I can see the analogy between a predicate and a set since we can quantify over both and make valid deductions between those quantifications. So the temptation is strong to equate a set with a predicate.

The way I was taught*, at least, sets are different from propositions are different from predicates, but they can have relations to one another. If I were to render set theory in terms of predicates I would say "Set theory is the study of validity of the "is a member of" relation", whereas predicate logic is the study** of validity between predicates. So they're kind of just asking after different things -- one is "how do we draw valid inferences between two propositions?" and the other is "how do we draw valid inferences between collections of individuals?"

Now, interestingly, I think we can mix these logics sometimes -- but usually we want to keep them distinct because they're hard enough to understand as it is that it's better to not overgeneralize :D

I'd argue what this shows is that logic is something we choose to utilize. I'm not sure there really is some universally true thing we can say about sets and predicates sans the rest of the logical system. We could choose, for whatever reason (perhaps because this question is interesting and we're interested in how logic works), to start with the equation "Properties are equivalent to sets" and then work out the validity of that identification.

But, at least if we're learning, these are generally treated somewhat separately (even though, yes, there's a lot of overlap between these ways of talking at the intuitive level)

**EDIT: OK, is the result of the study.... a theory is not an -ology

*EDIT: Also, "taught" was by a math instructor and the rest is self-study, so I could be missing something. I don't want to lead people astray but I do like thinking about this stuff.

This might make the point better --

Consider "The set whose elements consist of sets without properties which is a member of itself" --

Well, I'm trying to describe the concept of set in some intuitive terms. You may say that the concept of set is extra-logical but I wouldn't be able to make sense of logic without it. Like, why are the conclusions in syllogisms necessarily true if the premises are true?

The set is an object that somehow unifies different objects without negating their different identities. One over many. — litewave

The concept of a set could be extra-logical, yes. If I'm talking about a chess set, for instance, I'm not using "set" to talk about logical sets. So in a way what I'm asking here is to say "How does this notion of unification fit within a strict logical definition?"

Intuitively I understand what you mean -- I just think that we can drop this business about sets having properties at all if we can always substitute the members of a set for whatever the property picks out. The abstract property which picks out the nearest pebble and the first sentence I say five miles from now just is that these are in a set, and there is no more to it than that.

It's a set because we decided to treat these elements within a logical structure, not because there's some property which the set has that we pick the elements out with.

It's very unintuitive, I'll grant. But the intuitive statements can easily run into paradoxes which is kind of where I've been coming at the question from: with Russell's Paradox in mind which I tried, in my lasts reply, to reframe in terms of your use of "unificiation" -- it doesn't quite work because to be more precise the set would have to both contain itself and not contain itself, hence the paradox -- I was going for something like "Here's a set which has a property which is unification, and that property for this set is that they are all not unified, in which case the set is both unified and not unified".

Does that make any kind of sense, or is it just boring and not worth investigating?

"unification" -- I'd say this is an extra-logical notion. We may posit the set consisting of ununified elements, for instance -- is this then not a set because the elements are ununified? Is it possible to posit such a 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

If there are abstract properties which define sets without a known common property -- such as the set I proposed grouping an abstract and a concrete individual -- just how does this unknown property come to define the set? What is it that this property does that makes sense of saying sets are defined by properties if there are an infinite amount of abstract properties (considering there's at least an infinite amount of abstract objects this shouldn't be a stretch)?

Is it to say anything more than this set is a collection of these two members? What is this extra "unspoken property" doing for us in understanding what a set is?

Good. I like Sartre as an "in" to this approach to consciousness and I'm not particularly bothered by the critique in this context (also, my relative lack of knowledge of Husserl means I can't effectively argue the point anyway). — Baden

Same for me.

I now wish I had more patience for Husserl -- but I do need to recognize him as a giant that could provide interesting input if @Joshs is willing to give his take.

I'm trying to fill out above a context (more to come) that I'll try to loop back into a fuller application to body image disorders (including body dysmorphic disorder and eating disorders). — Baden

Cool :)

One thing that comes to mind for myself is something less intense: even the idea of "I need to lose weight" is a (mild)* form of wanting your body to be elsewise, and in a literal sense it's something that goes on in the mirror -- not just the mind's eye of the self, but the reflection one sees and judges of oneself.

I don't know Husserl well enough to say, but what I want to say is that this relates to the pluperfect tense of consciousness -- this is something I only recently picked up from Sartre so I could just be throwing it out there as the thing I've been thinking about -- but his notion of time seems to require a pluperfect tense in language which seems to indicate a structure of consciousness.

To use the mild example above: One cannot say "I need to lose weight" without being able to refer to a before-during-after that is not the current before-during-after -- more specifically the "pluperfect" tense is the tense one takes after having referenced either a past or future point of time and then, with that reference assumed, refers to another point of time relative to that reference.

In terms of "wanting to lose weight" it seems to me that right now I have to think about what my body once looked like and imagine what it could look like which requires a double pluperfect scenario to make sense of the desire.

Which kind of goes into the slogan, at least, of "consciousness is what it is not" -- not strictly in terms of being-in-itself/being-for-itself, but at the level of meaningful sentences.



*Started to think and realized that this isn't quite true -- it could be mild or intense, depending on the person, but it's mild in the sense that is relatable to lots of people and possible to conceive as a mild disappointment.

:up: Thanks.

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 open to 'being in set X' because I think Russell's paradox is legitimate, and generally I like the paradoxes of self-reference as a point of thought -- stuff like the liar's paradox seem to sit here.

But, yes, it could not count on the OP's proposal which is why the paradoxes of self-reference came to mind.

I don't want to say anything about identifying a set with a property for this very reason :D

What was the question? — Banno

" My understanding is that "classes" can include rules, but I don't understand how to do that formally while I do understand naive set theory at least. "

I suppose I was looking for reassurance of this distinction between sets and classes -- either in the right or wrong way. I've thought of sets as any collection of individuals whatsoever and classes as collections of individuals with inclusion rules. Is that bollox in my head that I'd need to defend or let go of, or something sensible from your perspective?

Does that avoid Russel's paradox?

If so, does it do so by delaying the question? :D

I'm good with forming sets in stages either way. Defining sets in a technical manner is something I still think on and think I don't understand, really.

A set is a single object. — litewave

That single object is the collection, but the thought here is that there's nothing more to that than being the collection of the individuals in the set.

We can name a set, so it can be an element -- and is an element of the set that is itself -- but a set need not be a single object (or name) at all. The empty set comes to mind here.

Relations are different from sets in that they are somehow connecting one set to another, but sets have no "rules" for inclusion.

My understanding is that "classes" can include rules, but I don't understand how to do that formally while I do understand naive set theory at least.

I think covered this well in that "object" is ontologically loaded. I'd include "property" there.

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.

I've expressed it as "a set is a collection of objects -- where objects are logical objects (any name whatsoever) -- that need not share anything in common other than being in that collection of objects"

Not sure how right I am as I still think on these things.

That's my understanding, at least given Russell's paradox. (which the OP reminds me of)

It feels plausible that a set can be identified by a property or even a set of properties.

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