1. There is no set of all that exists
2. There is no powerset for every set — Kuro

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.

From what I can tell, what you have is a problem of conception.

Consider the definition of powerset:
"the collection of all subsets, empty set and the original set itself".

All that needs to be decided is whether or not to allow "all subsets" of "all that exists" to be members of "all that exists".

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

Doesn't this get to the essence of your "problem"?

BTW, allowing "all subsets" to be members of the set of "all that exists" doesn't seem to make logical sense. "All subsets" are merely derived from the "original set" and are not a part of the "original set" proper.

Tree's rocks and stars can exist as composites without the labels tree, rock, or star. — universeness

You write that trees, rocks, stars, solar systems, etc are combinatory systems that can exist
independently of a lifeform and can exist without the labels tree, rock, star, solar system, etc.
@Kuro wrote "Suppose that all that exists forms a set."

Taking the Milky Way Galaxy as an example, The Milky Way Galaxy is an object, a combinatory system, a collection of things. As an object it is a set of parts.

The Mathematical Platonist would argue that the Milky Way Galaxy as a set of parts exists as an abstract entity, independent of any mind. A Nominalist would disagree.

If the Milky Way Galaxy exists as an abstract entity, by what mechanism do you propose that the parts are connected, parts that could be 87,000 light years apart ?

I would say that all combinations exist regardless of the mind..........But who says that parts of an object need to be in causal contact? — litewave

In Mathematical Platonism, sets exist in the world as abstract entities. The parts don't need to be in causal contact. Yet the parts must be connected in some way in order for the set to exist. How exactly ? How are things in the world abstractly connected ? By what mechanism ?

Infinite number of objects doesn't seem sensible? — litewave

For a world to start off with 3 objects and end up with an infinite number of objects because of the ontological existence of sets doesn't seem sensible.

Suppose that all that exists forms a set. Call this set E. It follows from the powerset axiom that there'd exist a powerset of E, P(E). Recall that from Cantor's theorem, the cardinality of a powerset is strictly larger than its set. But the cardinality of P(E) can only be greater than E's if there exists elements in P(E) that are not members of E. Though if there exists things that are not members of the set all of that exists, then the set of all that exists is not the set of all that exists.

By proof from contradiction, we're allowed to suppose that our premises are at fault by entailing a contradiction. We're left with:
1. There is no set of all that exists
2. There is no powerset for every set

Since the powerset axiom is ubiquitous in various mathematical set theories, we're only left with (1). This is to say that there does not exist a set of all that exists. — Kuro

I'm sorry but I'm not familiar with the math you are talking about and what the meaning of "set", "powerset", or other terms you are using. Because of I don't know if it is proper for one to say something like "Suppose that all that exists forms a set" and then label a set "E".

I could be wrong but as far as I know there is no mathematical formula's or functions that can be used in such a way. If there is I would like to know them.

In Mathematical Platonism, sets exist in the world as abstract entities. The parts don't need to be in causal contact. Yet the parts must be connected in some way in order for the set to exist. How exactly ? How are things in the world abstractly connected ? By what mechanism ? — RussellA

If the parts exist, their collection necessarily exists too. There can be no parts without their collection and there can be no collection of parts without the parts. The parts and their collection are connected by necessity.

Some collections constitute spaces, as defined in point-set topology (the collections in a collection that is a space must have the property of continuity, as defined in point-set topology). Spacetime is a space where time is a special kind of space, as a separate spatial dimension, as defined in theory of relativity. Collections in a spacetime can have causal relations between them. Causal relations between collections can be seen as a special kind of relations between spatiotemporal collections in the presence of the arrow of time (rising entropy of spatial structures along the time dimension), where the "consequences" logically follow from the "causes", and the "causes" are initial conditions and spatiotemporal regularities known as the laws of physics.

For a world to start off with 3 objects and end up with an infinite number of objects because of the ontological existence of sets doesn't seem sensible. — RussellA

There are just all possible (logically consistent/self-identical) collections, from the empty ones to infinitely large ones. After all, what would be the difference between a possible collection and a "real" collection?

If the Milky Way Galaxy exists as an abstract entity, by what mechanism do you propose that the parts are connected, parts that could be 87,000 light years apart ? — RussellA

Ah, now I see where you are getting the 87,000 from. That is an isophotal (based on the brightest part of the milky way) approximation for the diameter of the milky way. The more probable approximations put's it somewhere between 100,000 and 120, 000 light years across, perhaps a lot more, depending on how much dark matter is also present.
I propose the parts are connected systemically and they are gravitationally bound, so, in its largest scale, all parts of the milky way, rotate around a central supermassive black hole.
To answer your question more directly, the 'mechanism' would be gravity which may not be a force but a consequence of the presence of clumped or bulk mass over a particular extent of spacetime.

I don't see much difference between a galaxy posited as an abstract entity and me as an actual entity. Both are collectives, both are systemic, and both are combinatorial.

If the parts exist, their collection necessarily exists too. There can be no parts without their collection and there can be no collection of parts without the parts. The parts and their collection are connected by necessity. — litewave

There are just all possible (logically consistent/self-identical) collections, from the empty ones to infinitely large ones. After all, what would be the difference between a possible collection and a "real" collection? — litewave

The point being made is that if I have two coins then it's not the case that I have the first coin and I have the second coin and I have a pair of coins, such that I can be said to have 3 things. Either I say that I have the first coin and I have the second coin or I say that I have a pair of coins.

That a pair of coins exist just is that the first and the second coin exist. The mistake made is to treat the existence of the pair of coins as being distinct from the existence of the first and of the second coin.

In Mathematical Platonism — RussellA

From the on-line philosophical encyclopaedia:
Traditionally, mathematical platonism has referred to a collection of metaphysical accounts of mathematics, where a metaphysical account of mathematics is one that entails theses concerning the existence and fundamental nature of mathematical ontology.

I give much more credence to mathematics than I do to metaphysical accounts of mathematics.
I don't think much of Plato either.

The point being made is that if I have two coins then it's not the case that I have the first coin and I have the second coin and I have a pair of coins, such that I can be said to have 3 things. — Michael

You also have the collection of the two coins, which is a third collection (the two coins being the first two collections); it's just a different kind of collection and it is not a coin.

That a pair of coins exist just is that the first and the second coin exist. The mistake made is to treat the existence of the pair of coins as being distinct from the existence of the first and of the second coin. — Michael

But is the pair a single object? If not, is a coin a single object? If not, is there any single object at all?

You also have the collection of the two coins, which is a third collection (the two coins being the first two collections); it's just a different kind of collection and it is not a coin. — litewave

You don't have the collection in addition to each of the two coins. It's really a very simple point, what's hard to understand?

You don't have the collection in addition to each of the two coins. It's really a very simple point, what's hard to understand? — Michael

If you don't have the collection in addition to each of the two coins, what is the collection then? Is it not an object? You keep avoiding this question.

If you don't have the collection in addition to each of the two coins, what is the collection then? — litewave

The collection is the two coins. You either think and talk about them as being two coins or you think and talk about them as being a collection of coins. They're different modes of speaking.

The collection is the two coins. You either think and talk about them as being two coins or you think and talk about them as being a collection of coins. They're different modes of speaking. — Michael

So is the collection a single object or not?

That depends on your mode of speaking. You can talk about a collection as being a single object if you want, but you can't then say that because the collection is a single object and because each of its members is a single object then there are three objects. That would be like saying Joe Biden exists and the President of the United States exists, therefore at least two people exist, which is an invalid inference.

That would be like saying Joe Biden exists and the President of the United States exists, therefore at least two people exist, which is an invalid inference. — Michael

No, it's not, you made two references to the same object. You referenced the same object by its name and its job title. That's like two ways of describing a single coin. Two coins can make a collection, one coin cannot. One atom cannot make a human, but many atoms can. A single computer has a certain processor speed and throughput, if you network identical computers together then the processor power and throughput is vastly increased, and you can also manipulate the network of computers in ways that you cannot with a single computer. For example, you can perform parallel processing rather than serial processing. Combination results in new functionality.
A coin collection is a set that means more that a number of individual coins as the collective can be related in many different ways compared to treating the coins as unrelated units. The sum becomes more than its parts.

That depends on your mode of speaking. You can talk about a collection as being a single object if you want, but you can't then say that because the collection is a single object and because each of its members is a single object then there are three objects. That would be like saying Joe Biden exists and the President of the United States exists, therefore at least two people exist. — Michael

Joe Biden is identical to the current President of the United States - it is the same object. But if a collection is an object, what is it identical to? It is obviously not identical to any of its parts. So it must be a different object than any of its parts. Hence, a collection of two coins is a different object than any of the two coins.

Sorry, I didn't mean to 'butt in' to your exchange with Michael, with similar points.

Ah, I see. No problem.

:up:

No, it's not, you made two references to the same object. — universeness

And referring to a collection of coins refers to each of the coins in the collection. So you refer to the same coins twice when you say that the collection exists and each coin exists.

I notice the set theorists on TPF are keeping their distance from this thread. :cool:

But if a collection is an object, what is it identical to? It is obviously not identical to any of its parts. — litewave

It's identical to the sum of its parts. If you say that the collection exists in addition to each of its parts then you count each of its parts twice; once when counting the parts themselves and once when counting the collection. This really is such a simple point, I don't understand the objection.

I notice the set theorists on TPF are keeping their distance from this thread. :cool: — jgill

Is it even a matter of set theory? Seems to me that it's more to do with the philosophy of mathematics: mathematical realism or anti-realism? I'm clearly on the side of anti-realism.

A coin collection is a set that means more that a number of individual coins as the collective can be related in many different ways compared to treating the coins as unrelated units. The sum becomes more than its parts. — universeness

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

Here's another argument:

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.

Is it even a matter of set theory? — Michael

Starting off with a proposed "set of everything" I would say yes. Beyond that the discussion is mostly the typical banter about the definitions of words seen on the site. No big deal. Philosophy of mathematics? Questionable.

It's identical to the sum of its parts. If you say that the collection exists in addition to each of its parts then you count each of its parts twice; once when counting the parts themselves and once when counting the collection. This really is such a simple point, I don't understand the objection. — Michael

"Sum" is just a different name for "collection". If the collection is an object that is not identical to any of its parts then it is a different object than any of its parts - simple, isn't it? The collection is an object in addition to its parts. You dismiss this object because it coincides with the parts but it is something else than any of the parts.

"Sum" is just a different name for "collection". If the collection is an object that is not identical to any of its parts then it is a different object than any of its parts - simple, isn't it? The collection is an object in addition to its parts. You dismiss this object because it coincides with the parts but it is something else than any of the parts. — litewave

A collection of two coins has two parts; each of the coins. I am saying that the existence of the collection is identical to the existence of each of the coins; you're saying that it's additional to the existence of each of the coins. So you're saying that the existence of a collection of two parts is additional to the existence of its two parts. That's nonsense.

The existence of a collection of two parts is identical to the existence of its two parts. That's common sense.

I am saying that the existence of the collection is identical to the existence of each of the coins — Michael

The fact that the collection necessarily exists when the two coins exist doesn't mean that there are only the two coins. Since the collection is not identical to any of the coins, it is a different object than any of the coins. You conflate necessary coexistence of objects with a reduced number of objects.