Since the collection is not identical to any of the coins, it is a different object than any of the coins. — litewave

It's not identical to any one of the coins but it is identical to both of the coins. So you're duplicating entities when you count both coins individually in addition to the collection as a whole. This post really makes this point clear.

It's not identical to any one of the coins but it is identical to both of the coins. So you're duplicating entities when you count both coins individually in addition to the collection as a whole. This post really makes this point clear. — Michael

Well, physical properties like weight reflect the subsuming nature of a collection: a collection doesn't add weight additional to the weights of its parts; it subsumes their weights.

Well, physical properties like weight reflect the subsuming nature of a collection: a collection doesn't add weight additional to the weights of its parts; it subsumes their weights. — litewave

And the same when it comes to counting the things that exist. The existence of the collection subsumes the existence of its parts. Either you count the collection and say that 1 thing exists, and weighs 3g, or you count its parts and say that 2 things exist, and collectively weigh 3g. You can't count both the collection and its parts and say that 3 things exist, else you then have to say that they collectively weigh 6g.

Cantor's theorem must give way here, because it is not based on a self-referential model of logic. Logic has to be self-containing (self-justifying on its own) and has a set theory compliment.

P(E) = E. The universal set contains itself, just as logic contains itself.

And the same when it comes to counting the things that exist. The existence of the collection subsumes the existence of its parts. Either you count the collection and say that 1 thing exists, and weighs 3g, or you count its parts and say that 2 things exist, and collectively weigh 3g. You can't count both the collection and its parts and say that 3 things exist, else you then have to say that they collectively weigh 6g. — Michael

The problem may be in the fact that physical forces act only on elementary particles and not additionally on collections of elementary particles. So the weight of a collection of two elementary particles, which is determined by gravitational force, is only the sum of the weights of the two elementary particles (adding up of gravitational forces acting on elementary particles) because there is no gravitational force acting on the collection of the two elementary particles as an additional object. It doesn't mean that the collection doesn't exist as an additional object, only that gravitational force does not act on it as on an additional object.

But I'll try to be more precise in future if this is a misuse of the term. — Michael

I understood that you referred to the fallacy of reification, as used by AN Whitehead, referring to that error of reasoning in where abstract objects are treated as if they were concrete. Coincidentally, reification came to mean the same sense in ordinary language. Not sure if contemporary analytic philosophy continues to give the term the same meaning, & I'm with you on valuing precision, so I opt for the standard terminology where platonism denotes the position that abstract objects really exist (and are actually abstract, viz. they're not mind-dependant) and nominalism the contrary. I'm basically answering the request for clarity here by saying we can use 'platonism' instead of 'reificaiton' to refer to the position of sets existing as abstract objects.

The point I'm making is that if we have a red ball and a green ball and a blue ball, then even though we can consider them in various configurations, e.g. (1) a red ball and a green ball, (2) a red ball and a blue ball, (3) a green ball and a blue ball, etc., it's not the case that there are multiple balls of each colour, and it's not the case that each configuration is a distinct entity in its own right, additional to the red ball, the green ball, and the blue ball. That realist interpretation of sets (what I think of as reification) is, I believe, mistaken. — Michael

This is an inaccurate understanding of sets. Recall the axiom of extensionality. {a, b, c} and {c, b, a}, as well as {b, c, a} are all just the same set, because they have the exact same members and thus satisfy coextension. Sets, plainly as sets, are therefore invariant with respect to these configurations you use in your example, which are otherwise too fine-grained of a notion. There's a grain of truth here in that a realist interpretation of sets would indeed count {b} and b as separate, distinct objects and thus count two things, but this is unrelated to your configuration problem.

Though worry not, set theory has exactly the notion to capture what you're looking for (the beauty of mathematics at work) for your configuration problem, that is, the notion of ordered pairs, triples, etc and so on- generalized as ordered n-tuples. While {a, b} and {b, a} are exactly the same set in set-theory, ordered pairs like (a, b) and (b, a) are strictly non-identical when a & b are non-identical (had they been identical, it'd be a singleton satisfying "co"extensionality reflexively, hence why non-identity cases fail here).

Now, a realist about sets presumably will be a realist about ordered n-tuples, so there you go, we've "fixed" the configuration problem on the technical level. This is still hardly a problem though, namely because of Leibniz's Law: there are predicates true of a set that are not true of its members. For instance, consider cardinality. The set {a, b, c} would be truly predicated of having the cardinality of 3, though none of its members have a cardinality of 3, in fact, it'd be a category error to speak of the cardinality of its members in any case where its members are urelements. There are probably even more obvious examples like sethood, but I wanted to appeal to purely mathematical properties. In other words, I think the problem you raise is completely artificial.

I think you don't even need the set of everything to generate the problem. You just need any set that includes its own cardinality and it will blow up incoherently to a meaningless version of infinity. — Cuthbert

Correct! Though it's unclear if it's meaningless- meaningless per (I think all?) all set theories, yes, but certain notions of infinities that are too large for any cardinal to have meaning in set theory have actually been captured with the use of plural logic.

Either way:
1. There is a set of "all that exists"
2. There is a powerset for "all that exists" — ThinkOfOne

The powerset will always strictly be cardinally larger than the set, and as you yourself understand, those subsets are not actually part of the original set (so there will exist members of the powerset not in the set, making the set not itself hence why the set doesn't exist) You've articulated precisely what I said in my post, so maybe this is a misreading? I'm not sure where exactly you're disagreeing with me or objecting

Is my wastebasket a set ? — magritte

It's mereologically complex, and thus a composite by having mereological parts (as in, the parthood relation P). It is not a set, though there exists a set that contains exactly your waterbasket, and this set is not identical to your waterbasket (neither is the set that contains the set which contains your waterbasket).

I'm writing this reply not to just you in particular but also to everyone else reading because a good chunk of various people in this thread have confused the notions of mereological composition with that of being a set. There is indeed a debate in metaphysics on whether composition-is-identity, i.e., whether an apple is identical to its atoms or a further thing, but this has nothing to do with whether a set is identical to its members because ordinary objects are not sets.

And the answer to the latter question, of whether a set is identical to its member/s, it's a mathematical consensus of no, namely in that {x} has the property of "is a set" and x doesn't. This consensus much unlike the mereological debate of whether a composite is identical to its parts, which is indeed a heatly debated topic in metaphysics, and what a good chunk of this thread mixed up with its set theory counterpart that does not yield any real dispute.

FWIW, there are many interesting similarities between mereology & set theory, though they're not analogous at all in this particular respect. Their differences stem from the fact that mereology is intended to formalize our general notion of 'parthood', whereas set theory a regimented notion of collection & infinity. There are fascinating intersections like mereotopology.

This is still hardly a problem though, namely because of Leibniz's Law: there are predicates true of a set that are not true of its members. For instance, consider cardinality. The set {a, b, c} would be truly predicated of having the cardinality of 3, though none of its members have a cardinality of 3 — Kuro

I address something like that here. The set of both metals weighs 3g but none of its members weigh 3g. It doesn't then follow that we should treat the existence of this set as being additional to the existence of each of its members, else the total weight of things which exist would be 6g, which is false in this example.

I have a piece of metal that weighs 1g and a piece of metal that weighs 2g. So the collection of metal weighs 3g. This is the only metal that exists.

What is the total weight of all the metal that exists? 3g or 6g? Obviously 3g. You don't add the weight of the collection to the weight of its parts. So you can't say that the collection exists in addition to each of its parts. Unless you want to be a Platonist and say that the collection exists as some abstract, weightless object, which I think is absurd. — Michael

This is an inaccurate understanding of sets. Recall the axiom of extensionality. {a, b, c} and {c, b, a}, as well as {b, c, a} are all just the same set, because they have the exact same members and thus satisfy coextension. Sets, plainly as sets, are therefore invariant with respect to these configurations you use in your example, which are otherwise too fine-grained of a notion. There's a grain of truth here in that a realist interpretation of sets would indeed count {b} and b as separate, distinct objects and thus count two things, but this is unrelated to your configuration problem. — Kuro

I think you may have misread. I was comparing {a, b}, {a, c}, and {b, c}. I think it’s a mistake to think of these as being things that exist distinctly from/in addition to one another, and distinctly from/in addition to a, b, and c.

There may indeed be different things that can be said about each, and they may have a different use in mathematics, but I think the ontological interpretation of that as involving the existence of additional entities is mistaken, which I think my example of the weight of the metals shows, and also the following example:

When I meet a married couple I don’t meet a married couple and the husband and the wife. Meeting the married couple is meeting the husband and the wife, and vice versa. The married couple isn’t an entity that’s additional to the husband and the wife, even though there are things we can say about the married couple that we can’t say about the husband or the wife individually.

If you try to say that the married couple and the husband and the wife all exist, and so 3 things exist, you’re counting the husband and the wife twice (or rather, 1.5 times each).

In terms of function or use or conception, sure. But it terms of counting the number of things that exist, no. — Michael

But this is the point being made. Algebraic fundamentals like x and y create a new object when combined. Let's say an instantiation is x=2 and y=3 then xy = 23. (or 6 for those who insist xy means x multiplied by y) In the rules of maths 23 is a much larger quantity than 2 or 3. So 2,3 and 23 are three separate objects with one being a combination of the other 2.
Under the logic you are suggesting, there could be no valid numerical sets such as the set of prime numbers as you would suggest but they are all just multiples of 1. So, 1 is the only true member of the set of primes, or integers etc? Is that a consequence of the logic you are applying?

Under the logic you are suggesting, there could be no valid numerical sets such as the set of prime numbers as you would suggest but they are all just multiples of 1. So, 1 is the only true member of the set of primes, or integers etc? Is that a consequence of the logic you are applying? — universeness

I'm not a mathematical realist. I don't believe that mathematical "objects" exist. But this topic isn't just about mathematics, it's about the set of all that exists, and so presumably (at least some of) its members are physical objects. It's this that allows us to see the problem with the realist approach, as shown with my example of the weighted metals.

I have a piece of metal that weighs 1g and a piece of metal that weighs 2g. So the collection of metal weighs 3g. This is the only metal that exists.

What is the total weight of all the metal that exists? 3g or 6g? — Michael

6g would not be a member based on the concept of weight if your fundamentals are 1g and 2g weights.
All you can have is 1g, 2g and 1g+2g or 2g+1g which is 1g, 2g and 3g.
There is no 6g, unless you are creating a numerical sequence based on addition, rather than a set based on weight. If it's a numerical sequence based on adding 1 then including 6g is logical.
You would just end up with the set of integers with g in front of each number indicating weight.
Btw. I am not sure if I would refer to myself as a mathematical realist either, based on the concept of objective truths, which I am not sure exists. A mathematical realist is described as:
Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities.

You seem to be missing the point.

If there are two pieces of metal that weigh 1g each then the collection that contains just these two pieces of metal weighs 2g.

If the collection that contains just these two pieces of metal exists as its own entity, distinct from/separate to each individual piece of metal, then we have one entity (the first piece of metal) that weighs 1g, another entity (the second piece of metal) that weighs 1g, and a third entity (the collection) that weighs 2g. The total weight of all the entities that exist is 4g.

Obviously this is wrong. So how do we avoid the absurd conclusion? By rejecting the premise that the collection that contains the two pieces of metal exists as its own entity, distinct from/separate to each individual piece of metal.

I don't think I have missed the point; you are taking an illogical step. You cannot create this extra member of the collection in the real world of having a physical 1g and 2g weight. You can create your 6g value based on the rules of arithmetic but if you do that then you must give a rule for your series or sequence and if the rule is adding previous weights together then you must include 4g and 5g and continue past 6g.

Its like you are playing arithmetic tricks. You imply the weights are real for your 1g, 2g and 3g posit and then notional for your 6g step. This reminds me of the old arithmetic trick:
Three men decide to buy an old tv costing £30 pounds. They pay £10 each. The salesperson then finds out that the tv was part of their sale and should have cost £25. He gives £5 to an assistant to give back to the three men. For simplicity, the assistant keeps £2 pounds and gives each man back £1. So, each man has now paid £9 each, 3x£9 = £27 + the £2 the assistant has, which adds to £29. What happened to the other £1?

You imply the weights are real for your 1g, 2g and 3g posit and then notional for your 6g step. — universeness

A piece of metal that weighs 1g does in fact weigh 1g, and a piece of metal that weighs 2g does in fact weigh 2g, and a collection that contains these two pieces of metal does in fact weigh 3g.

Obviously it's wrong to say that 6g of metal exists, but this is what follows if you say that the collection exists as its own entity, distinct from the existence of the two individual pieces. Therefore to avoid the absurd conclusion you reject this premise. The collection doesn't exist as its own entity, distinct from the existence of the two individual pieces. Rather, the existence of the collection is identical to the existence of the two individual pieces. Only 3g of metal exists.

So it is wrong to say that three distinct entities exist. You're effectively double-counting the two distinct pieces of metal. And I think this is what happens when the OP considers the power set.

Either way:
1. There is a set of "all that exists"
2. There is a powerset for "all that exists"
— ThinkOfOne

The powerset will always strictly be cardinally larger than the set, and as you yourself understand, those subsets are not actually part of the original set (so there will exist members of the powerset not in the set, making the set not itself hence why the set doesn't exist) You've articulated precisely what I said in my post, so maybe this is a misreading? I'm not sure where exactly you're disagreeing with me or objecting — Kuro

The point that you seem to be missing is that it's a simply a matter of definition -and definition alone- that powersets don't contain "all subsets" of the "original set". The original set IS the "set of all that exists". To conclude that the original set does not exist is nonsensical. It is borne of a failure of conceptual understanding on your part.

Obviously it's wrong to say that 6g of metal exists, but this is what follows if you say that the collection exists as its own entity, distinct from the existence of the two individual pieces. — Michael

By what rule or logic do you claim that IT FOLLOWS, that 6g of metal exists (or any weight of metal > the 3g that actually exists in total). You already admit it does not, OBVIOUSLY. The weight 3g DOES exist as its own entity by combination. You attempt to combine the combination of 1g and 2g to make 3g with another non-existent 1g and 2g weight. ALL three entities 1g, 2g and 3g can be physically demonstrated separately using a weighting machine.
You cannot demonstrate all three physical quantities of weight at the same instant of time.
If you had two weighing machines then you could demonstrate two of the weights at the same instant of time, but not all three, even with three weighing machines, but you can demonstrate the existence of all three quantities over a time interval/duration. BUT, no matter how much time you have you can never demonstrate a weight of 4, 5 or 6g with two source weights of 1g and 2g.
IT DOES NOT FOLLOW that 4g, 5g or 6g are valid due to the combination of 1g and 2g being a separate REAL entity. As I already typed, you are just, in my opinion, employing smoke and mirrors.

If the parts exist, their collection necessarily exists too.........Collections in a spacetime can have causal relations between them — litewave

I don't see much difference between a galaxy posited as an abstract entity and me as an actual entity — universeness

Convention of quotation marks
Using the convention of Davidson's T-sentence "snow is white" is true IFF snow is white, where with quotation marks refer to language and the mind and without quotation marks refers to a world.

Sets (to my understanding)
A set is a collection of elements. A set with no elements is "empty", a set with a single element is a "singleton", elements can be numbers, symbols, variables, objects, people and even other sets. A set is an abstract, such that its elements don't have to be physically connected for them to constitute a set. An object is not a set, though it can be a set of objects.

Platonists vs Nominalists
A Platonist would argue that "galaxies" exist in a mind-independent world, whereas a Nominalist would argue that they don't. For the Nominalist, an apple in the world is a projection of the concept "apple" in the mind onto the world.

See SEP - Abstract Objects - https://plato.stanford.edu/entries/abstract-objects

I agree that galaxies exist in a mind-independent world, I agree that "galaxies" exist in the mind, but I don't agree that "galaxies" exist in a mind-independent world.

Argument Three against Platonism
A Platonist would argue that "apples" exist in a mind-independent world, a Nominalist would argue that apples exist in a mind-independent world.

It is argued that if two people observe the same world, and both independently perceive an "apple" then an apple exists in the world. However, I may observe the world and perceive a "duck", whilst someone else perceives a "rabbit".

IE, it does not necessarily follow that because we both perceive the same "object", then that object exists in the world.

Sets and Galaxies
@Kuro started the thread by asking about a set of all that exists. The word "exist" needs to be defined.

A Nominalist would argue that as sets are abstract, and as abstracts don't exist in a mind-independent world, neither do sets. Therefore, sets can only exist in the mind. A set of stars exists in the mind as a "galaxy". Galaxies exist in a mind-independent world.

A Platonist would argue that although abstracts exist in a mind-independent world, they are independent of any physical world. As abstracts exist in a mind-independent world, and as sets are abstract, then sets can exist in a mind-independent world. Sets can also exist in the mind. Therefore, "galaxies" exist both in the mind and in a mind-independent world. Galaxies also exist in a mind-independent world.

A galaxy is a gravitationally bound system of stars, stellar remnants, interstellar gas, dust, and dark matter. These physical parts are connected by physical forces, such as gravity.

A "galaxy" is an abstract entity of physical parts. These physical parts are connected, but not physically. If an object, such as a "galaxy", is a collection of parts, such as stars, there must be some kind of connection between the parts, otherwise it wouldn't be an object.

For the nature of connections see SEP - Relations - https://plato.stanford.edu/entries/relations

In summary, "galaxies" must be distinguished from galaxies.

You cannot demonstrate all three physical quantities of weight at the same instant of time — universeness

I know, which is why the claim that a set has its own independent existence, distinct from its members is false. What is so hard to understand about this?

What is so hard to understand about this? — Michael

Your claim is easy to understand, but it is also wrong.
Why would the fact that a time duration is needed to join or separate fundamentals mean that a combination does not have an independent existence which is not the same as the existence of its constituent parts? I consider most systems/combinations to be described as is attributed to Aristotle:
“The whole is greater than the sum of the parts.”

“The whole is greater than the sum of the parts.” — universeness

That's a misquote. What he said was:

For however many things have a plurality of parts and are not merely a complete aggregate but instead some kind of a whole beyond its parts...

Some things which have a plurality of parts are "merely a complete aggregate" and some things which have a plurality of parts are "some kind of a whole beyond its parts".

In the case of the set {apple, pear} we just have an aggregate.

The aggregate {apple, pear} may be conceptually distinct from the apple and the pear but it is not ontologically distinct from the apple and the pear.

If a and b are ontologically distinct then the weight of {a, b} is equal to the weight of a plus the weight of b. If the weight of {a, b} is not equal to the weight of a plus the weight of b then a and b are not ontologically distinct.

The weight of {apple, pear} is equal to the weight of the apple plus the weight of the pear, and so the apple is ontologically distinct from the pear. The weight of {apple, {apple, pear}} is not equal to the weight of the apple plus the weight of {apple, pear}, and so {apple, pear} is not ontologically distinct from the apple (and nor from the pear for the same reason).

What does this have to do with philosophy? It's pure Math.

BTW, I just read a topic that reminds of quizes one can encounter in a college class. This one could be a test even in a high school Math class.

In the case of the set {apple, pear} we just have an aggregate. — Michael

Combining an apple and a pear will have a quite distinct taste, compared to tasting an apple or tasting a pear. So, the combination produces a new entity of taste.

Combining an apple and a pear will have a quite distinct taste, compared to tasting an apple or tasting a pear. So, the combination produces a new entity of taste. — universeness

Right, so this shows that you clearly misunderstand what is being talked about.

Right, so this shows that you clearly misunderstand what is being talked about. — Michael

Your last argument was based on metal weights in the real world, I see no difference between separation based on physical weight and separation based on physical taste. Your claim that because I don't agree with you, it then follows that I just don't understand your logic is a matter for your own measure of your own arrogance.

Your claim that because I don't agree with you, it then follows that I just don't understand your logic is a matter for your own measure of your own arrogance. — universeness

If you think that the set {apple, pear} means that we've combined an apple and pair into some new hybrid fruit then you don't understand what sets are.

What does this have to do with philosophy? It's pure Math. — Alkis Piskas

I would say that it's actually not maths. It's making claims about things that exist. At the very least it concerns the philosophical interpretation of maths; do mathematical objects like sets exist, and if so is their existence distinct from the existence of their members?

If you think that the set {apple, pear} means that we've combined an apple and pair into some new hybrid fruit then you don't understand what sets are. — Michael

When you combine two separate entities then the attributes of both entities are combined in every way possible. If x=2 and y=3 then xy can have any operator/function applied to it just like you can have any function applied to every member of a set. You can use operators such as +, -, x, / or any function such as putting each element of the set into a blender! I think it's you that does not understand that you can perform any action you like on the members of a set as long as it's the same action performed on each one.