Sounds about right because the world of all that exists should not be a mathematically flat set ?

Suppose that all that exists forms a set. — Kuro

I question whether that is actually a set in ZFC. A neighbor of Russell's paradox perhaps.

Problem I have with an "All inclusive set" is that apparently I can imagine a larger and larger set, ad infinitum.
No matter how big I imagine "Everything" to be, the actual 'everything' may be a lot bigger. An ant's idea of "everything" may be a lot smaller than my conception. While God's or a much more advanced being's idea of "everything" may be much bigger than mine.

Suppose that all that exists forms a set — Kuro

In the world exists x, y and z.
Suppose x,y and z exist in the world.

This gives us 6 sets - (x) - ( y) - (z) - (x,y) - (x, z) - (y,z).

If sets exist in the world
You say that sets "exist".

Q1 - If sets exist in the world, we start off with 3 things that exist and end up with 6 things that exist. Where did the extra 3 things come from ?

But if sets do exist in the world it gets worse.
Let set F be (x), set G be ( y), set H be (z), set J be (x,y), set K be (x,z) and set L be (y,z).
This gives us the additional sets (F), (G), (H), (J), (K), ( L), (F,G), (F, H), (F, J), ((F,K), (F,L), (F,G,H), (F,G,J), (F,G,K), (F,G,L), (F,H,J), (F,H,K), (F,H,L), (F,J,K) - etc - a lot.
We can continue the same process and end up with the existence of an infinite number of possible sets.

Q2 - if sets do exist in the world, we start off with 3 things that exist and end up with an infinite number of things that exist. Where did the extra things come from ?

If sets don't exist in the word
If sets don't exist in the world, life is a lot simpler, and the only things that exist in the world are x,y,z.

The implication is, that as an object such as an apple is only a set of parts, as sets don't exist in the world, then apples don't exist in the world, which is my belief.

IE, set E (x,y,z) is the set of all that exists in the world.

This is to say that there does not exist a set of all that exists. — Kuro

Yeah, just as there is no biggest number. There is always something bigger.

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. — Kuro

That's not true. The power set includes repeated members. Taken from the Wikipedia article:

If S is the set {x, y, z}, then all the subsets of S are

• {}
• {x}
• {y}
• {z}
• {x, y}
• {x, z}
• {y, z}
• {x, y, z}

and hence the power set of S is {{}, {x}, {y}, {z}, {x, y}, {x, z}, {y, z}, {x, y, z}}

x, y, and z are repeated.

Right, so this is an issue of reification. Some people think of a set as being some abstract, Platonic entity that "exists" in some sense, distinct from its members? I'm not a mathematician but that just strikes me as nonsense.

The 'extra things' come from combination of the fundamentals, in your example, the fundamentals would be x, y and z.
If photons, electrons and quarks are fundamentals then all we can observe around us, or as a part of us or can instrumentally detect, are combinations of fundamentals.

Some people think of a set as being some abstract, Platonic entity that "exists" in some sense, distinct from its members? — Michael

The 'extra things' come from combination of the fundamentals — universeness

A set is a combination of things. I would accept that "combinations" exist in the mind. I would accept that the mind observes "combinations" in the world when observing the world. I would accept that there are forces between things in a mind-independent world, but the concept of force is different to the concept of "combination". A set of things does not require that there are forces between these things.

Q1 - do "combinations" exist in the world when the world is not being observed ?

Is there any persuasive argument that "combinations" do exist in a mind-independent world ? I have yet to come across one.

Right, so this is an issue of reification. Some people think of a set as being some abstract, Platonic entity that "exists" in some sense, distinct from its members? I'm not a mathematician but that just strikes me as nonsense. — Michael

All everyday concrete objects are sets, or collections, of other objects. Do those collections not exist? For example, does an apple, a collection of atoms or subatomic particles, not exist? What exists then?

If a red apple and a green apple exist then I wouldn't say that three things exist: it’s not the case that a red apple exists and a green apple exists and the abstract, Platonic set of both apples exists.

The apple exists as a set of parts in the mind. When the mind believes that it is observing an apple in the world, for the apple to also exist in this observed world as the same set of the same parts would be an example of overdetermination.

IE, an apple does not need to exist in the world in order for the mind to believe that it is observing an apple in the world.

Q1 - do "combinations" exist in the world when the world is not being observed ?

Is there any persuasive argument that "combinations" do exist in a mind-independent world ? I have yet to come across one. — RussellA

Yes, a tree is a combination as is a grain of sand, a rock or a star.
They need no lifeform to exist as combinations of fundamentals.
For the vast majority of the existence of our solar system, our galaxy and even the universe (13.8 billion years), probably, no life existed anywhere. From the moment the 'singularity,' 'inflated' / 'expanded,' until the first 'mind' formed via combinatorial evolution, the universe was mind-independent.

If a red apple and a green apple exist then I wouldn't say that three things exist: it’s not the case that a red apple exists and a green apple exists and the abstract, Platonic set of both apples exists. — Michael

It seems arbitrary to say that some collections exist and some don't. If the constituent parts are there, then their collections are automatically there too. Some collections may be less interesting, like a collection of two apples, as opposed to a single apple, but what does existence care about interestingness.

The apple exists as a set of parts in the mind. When the mind believes that it is observing an apple in the world, for the apple to also exist in this observed world as the same set of the same parts would be an example of overdetermination. — RussellA

What doesn't exist only in the mind then? Non-composite objects?

If the constituent parts are there, then their collections are automatically there too. — litewave

Not as abstract, Platonic entities, distinct from and additional to their constituent parts.

The existence of each member of a set is the existence of that set.

Not as abstract, Platonic entities, distinct from and additional to their constituent parts. — Michael

But if the parts themselves are collections of parts, what exists then? Only non-composite objects?

Are you not reading what I'm saying? Sets don't exist as abstract, Platonic entities, distinct from and additional to their constituent parts.

An apple, for example, isn't an abstract, Platonic entity, distinct from and additional to the atoms that constitute it. It's not the case that the atoms exist and the apple exists, but rather the existence of the atoms is the existence of the apple.

An apple, for example, isn't an abstract, Platonic entity, distinct from and additional to the atoms that constitute it. — Michael

What is an apple then? Is it a single object? If it is a single object it is surely not identical to any of its atoms.

That's not true. The power set includes repeated members. Taken from the Wikipedia article: — Michael

You're confusing singletons with just the elements. x, {x}, {{x}}... so on are all not identical with each other, and for instance the singleton set {x} is a member of the powerset but not the set, whose member would be x.

Right, so this is an issue of reification. Some people think of a set as being some abstract, — Michael

Reification deals with treating abstract entities concretely, like asking where the average family of 2.4 children lives (that average family is an abstracted notion that is not concretely instantiated). Reification does not target merely the existence of abstract entities, otherwise it's simply another name for the philosophical position of nominalism- a substantive metaphysical viewpoint- and not a general error in reasoning.

Is there any persuasive argument that "combinations" do exist in a mind-independent world ? I have yet to come across one. — RussellA

This is really a response to both your first and second reply, but I'm quoting the second one so that this message is shorter. Taking sets to exist is the most natural interpretation of the existential quantifier in set theory without awkward paraphrases: it's unclear what we mean by that the set of natural numbers N exists but not the contradictory Russell set if neither sets exist (in fact, the very invention of ZF, ZFC, and later NF over naive Frege-Cantorian set theory is just to prevent the existence of contradictory sets). That means the standard reading of mathematical facts, like "there are prime numbers" is that there really are prime numbers, along other things. Here, I'm just presenting a standard Fregean argument in virtue of the fact that mathematical terms and statements are meaningful, and in being meaningful they refer to something: namely mathematical entities which are neither mental nor material (thus abstract).

The other influential line of argument for this view comes from Quine-Putnam indispensability arguments, which owe to the fascinating empirical success of mathematics in the natural sciences. By regimenting natural scientific theories into a canonical language like first order logic, we have to quantify over mathematical objects in our domain. But if by quantifying over, say, electrons and their properties in our domain, we take those things to exist, then in an analogous manner by quantifying over mathematical objects that are necessary for our scientific theories we take them to exist. Field famously objected to by attempting to formulate mathematics from spatial relationships, although the project was unsuccessful. The Quine-Putnam indispensability argument can be seen as a move from scientific realism to mathematical realism.

Suppose x,y and z exist in the world.

This gives us 6 sets - (x) - ( y) - (z) - (x,y) - (x, z) - (y,z). — RussellA

Additional reply: this is technically incorrect. The existence of any object in the world allows us to generate infinitely many sets by reiterating supersets as well as empty sets, but these cardinalities are not at all problematic with respect to their powerset in the same way the set of all which exists is (whose identity defines it to include its powerset).

On the other hand, we might say a universal set can trivially exist without this problem by just defining it as an NBG class, i.e. as not a member of anything else.

Well, to my simple and untrained mind, the "solution" is quite simple - some sets should be banned! Oui, mods? That's what Bertrand Russell did I believe.

I don't think that attitude is wrong at all. I just have suspicions it's somewhat ad hoc, much like the charge that proponents of inconsistent mathematics complain of post-ZF mathematics.

The powerset axiom is clearly an intuitive (and correct) principle, but generalizing it completely (along with existential assumptions) entails some very exotic and "naughty" entities that we may want to ban! The question here may be a question of whether we should do this, or what intuitions do we prioritize? I'm not sure.

Like Jim al-Khalili writes in his book 9 enigmas in science, resolving a paradox/dilemma can be done by seeking & subsequently finding the right angle with which to view such cases. For example, apologies if it fails to get the point across, what's impossible in 2D space (flipping chirality of 2D objects) is possible in 3D space.

Thoughts? — Kuro

Is my wastebasket a set ?

Reification does not target merely the existence of abstract entities, otherwise it's simply another name for the philosophical position of nominalism — Kuro

But nominalism is the position that abstract objects don't exist?

Reification deals with treating abstract entities concretely — Kuro

Maybe I'm being imprecise. I usually think of reification as taking a realist approach to abstractions, and so would include Platonism, not just as saying that abstract entities are "concrete" (which I assume by this you mean "physical"?). But I'll try to be more precise in future if this is a misuse of the term.

You're confusing singletons with just the elements. x, {x}, {{x}}... so on are all not identical with each other, and for instance the singleton set {x} is a member of the powerset but not the set, whose member would be x. — Kuro

I think this is just an issue of terminology. 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.

The existence of the set {red ball, green ball}, if anything, is the existence of the red ball and the existence of the green ball. It's something of a category mistake to treat them as separate (à la Ryle's example in The Concept of Mind of the person who, after being shown the various colleges of Oxford University, asks where the actual University is).

What doesn't exist only in the mind then? Non-composite objects? — litewave

Yes, a tree is a combination as is a grain of sand, a rock or a star. They need no lifeform to exist as combinations of fundamentals. — universeness

If combinations don't ontologically exist in a mind-independent world (aka relations) but do exist in the mind, then:
i) what exists in the mind-independent world are fundamental forces and fundamental particles. These fundamental particles may be called "objects", and are non-composite.
ii) a tree, which is a combination of parts, can only exist in the mind.

Argument One against sets as combinations existing in the world
From before, if only 3 things were introduced into a world, and if sets as combinations did exist, then an infinite number of other things would automatically be created. This doesn't seem sensible.

Argument Two against sets as combinations existing in the world
If combinations exist in the world, then an object such as an apple would exist as a set of parts. It would follow that one part 8cm distant from another part would be in combination.

The Earth would exist as an object, meaning that one part 12,000 km from another part would be in combination.

The Milky Way Galaxy would exist as an object, meaning that one part 87,000 lights years from another part would be in combination.

If being in combination was instantaneous, then the combination between two parts of the Milky Way Galaxy 87,000 light years apart would be instantaneous. But this would break the physical laws of nature as we know them, and would need to be justified.

If being in combination followed the physical laws of nature as we know them, then two parts could only be in combination once information had travelled between them at the speed of light. This raises a further problem.

If, during the 87,000 years it took for the two parts to become in combination, one or both of the parts ceased to exist, then a combination would come into existence without any parts. This doesn't seem sensible.

IE, Platonic Sets existing in a mind-independent world sounds fine until one considers the real world implications.

Taking sets to exist is the most natural interpretation of the existential quantifier in set theory without awkward paraphrases — Kuro

I am sure that both Platonists and Nominalists agree that sets exist. The question is where, in the mind or mind-independent.

Frege argued for mathematical Platonism as the only tenable view of mathematics, yet objectors include Psychologists, Physicalists and Nominalists.

Quine-Putnam's Indispensability Argument argued for the existence of abstract mathematical objects, such as numbers and sets, yet persuasive objectors include Harty Field.

I agree that the Existential Quantifier, having the meaning "there exists", "there is at least one", "for some", is invaluable in logic. For me, however, the most natural interpretation of "exist" means within the mind.

IE, the most natural interpretation for one person may be different to another person's.

If combinations don't ontologically exist in a mind-independent world (aka relations) but do exist in the mind, then:
i) what exists in the mind-independent world are fundamental forces and fundamental particles. These fundamental particles may be called "objects", and are non-composite.
ii) a tree, which is a combination of parts, can only exist in the mind. — RussellA

It's true that lifeforms like humans create categories or even convenient ontological groupings.
Every time I wrote a substantial computer program, I created such namespaces, data types, hierarchical storage structures, all of which could be called ontological, but this has nothing to do with what exists in a universe devoid of lifeforms which can ask questions and query their surroundings.
Tree's rocks and stars can exist as composites without the labels tree, rock, or star.
So, I think your part i) above does not hold, your part ii) also does not hold and such statements belong to human delusions of how vital they are to the existence of the universe. I think they are vital to assigning PURPOSE to the universe but not its existence, either in its fundamental constituents or in its ability to combine through random happenstance and end up with objects WE happen to have labelled tree, rock, star etc.

Argument One against sets as combinations existing in the world
From before, if only 3 things were introduced into a world, and if sets as combinations did exist, then an infinite number of other things would automatically be created. This doesn't seem sensible. — RussellA

Infinity is merely a concept; it is not a construct. Statements such as from wiki:
Paradoxes of the Supertask
In set theory, an infinite set is not considered to be created by some mathematical process such as "adding one element" that is then carried out "an infinite number of times". Instead, a particular infinite set (such as the set of all natural numbers) is said to already exist, "by fiat", as an assumption or an axiom. Given this infinite set, other infinite sets are then proven to exist as well, as a logical consequence. But it is still a natural philosophical question to contemplate some physical action that actually completes after an infinite number of discrete steps; and the interpretation of this question using set theory gives rise to the paradoxes of the supertask.

This is just conceptual maths or propositional logic it need not be fact or truth to be useful in calculations.

Argument Two against sets as combinations existing in the world
If combinations exist in the world, then an object such as an apple would exist as a set of parts. It would follow that one part 8cm distant from another part would be in combination — RussellA

An atom is mainly empty space, but an atom is also a functioning system which functions as a combination. A solar system is a combinatorial system. If you notionally want to label the entire universe a set of fundamentals containing all currently known mass, energy/force, fundamentals they you could at least start it with U= {quark, electron, photon, w boson, z boson etc...}
You could also start a set of everything that can be created from random happenstance or the actions of lifeforms and include E = {universe, galaxy, star, rock, human....... pencil, space rocket......} but it would be a very big set, not necessarily infinite, just vast. I see no important point in your argument two above that supports the idea that the members of sets cannot be considered as combinations of fundamentals whether or not those fundamentals are natural or algebraic.

If being in combination followed the physical laws of nature as we know them, then two parts could only be in combination once information had travelled between them at the speed of light. This raises a further problem. — RussellA

What problem?

If being in combination was instantaneous, then the combination between two parts of the Milky Way Galaxy 87,000 light years apart would be instantaneous. But this would break the physical laws of nature as we know them, and would need to be justified. — RussellA

Being in combination is obviously not instantaneous. If the sun exploded right now, the Earth would not know for around 8 minutes.

If, during the 87,000 years it took for the two parts to become in combination, one or both of the parts ceased to exist, then a combination would come into existence without any parts. This doesn't seem sensible. — RussellA

What? Where are you getting this 87,000 years from? The Milky Way started to form around 13 billion years ago! It didn't form as two halves that then joined together! Much of what you are typing makes little sense to me.

If combinations don't ontologically exist in a mind-independent world (aka relations) but do exist in the mind, then:
i) what exists in the mind-independent world are fundamental forces and fundamental particles. These fundamental particles may be called "objects", and are non-composite.
ii) a tree, which is a combination of parts, can only exist in the mind. — RussellA

But a non-composite object is a combination too - a special kind of combination: a combination of zero objects. It seems arbitrary to state that some combinations exist only in the mind and others also outside the mind. I would say that all combinations exist regardless of the mind because I don't see why a mind or consciousness would be necessary for a combination or a collection to exist.

Argument One against sets as combinations existing in the world
From before, if only 3 things were introduced into a world, and if sets as combinations did exist, then an infinite number of other things would automatically be created. This doesn't seem sensible. — RussellA

Infinite number of objects doesn't seem sensible?

If being in combination was instantaneous, then the combination between two parts of the Milky Way Galaxy 87,000 light years apart would be instantaneous. But this would break the physical laws of nature as we know them, and would need to be justified. — RussellA

Theory of relativity says that the two parts would not be in instantaneous causal contact. But who says that parts of an object need to be in causal contact? Spatiotemporal objects are structurally a special kind of mathematical objects (sets) and mathematical objects need not be in causal contact.