All That Exists
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.
Thoughts?
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.
Thoughts?
Comments (140)
I question whether that is actually a set in ZFC. A neighbor of Russell's paradox perhaps.
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.
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.
Yeah, just as there is no biggest number. There is always something bigger.
That's not true. The power set includes repeated members. Taken from the Wikipedia article:
x, y, and z are repeated.
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.
Quoting 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.
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: its not the case that a red apple exists and a green apple exists and the abstract, Platonic set of both apples exists.
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.
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.
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.
What doesn't exist only in the mind then? Non-composite objects?
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.
But if the parts themselves are collections of parts, what exists then? Only non-composite objects?
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.
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.
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.
Quoting 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.
Quoting 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.
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.
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.
Is my wastebasket a set ?
But nominalism is the position that abstract objects don't exist?
Quoting 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.
Quoting 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).
Quoting 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.
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.
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.
Quoting RussellA
Infinity is merely a concept; it is not a construct. Statements such as from wiki:
[b]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.[/b]
This is just conceptual maths or propositional logic it need not be fact or truth to be useful in calculations.
Quoting 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.
Quoting RussellA
What problem?
Quoting RussellA
Being in combination is obviously not instantaneous. If the sun exploded right now, the Earth would not know for around 8 minutes.
Quoting 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.
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.
Quoting RussellA
Infinite number of objects doesn't seem sensible?
Quoting 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.
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.
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 ?
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 ?
Quoting 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.
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.
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.
Quoting 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?
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.
Quoting 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.
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.
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.
Quoting 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 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?
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.
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.
So is the collection a single object or not?
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.
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.
:up:
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.
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.
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.
In terms of function or use or conception, sure. But it terms of counting the number of things that exist, no.
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.
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.
"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.
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.
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.
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.
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.
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.
P(E) = E. The universal set contains itself, just as logic contains itself.
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.
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.
Quoting 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.
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.
Quoting 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
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.
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.
Quoting Michael
I think you may have misread. I was comparing {a, b}, {a, c}, and {b, c}. I think its 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 dont 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 isnt an entity thats additional to the husband and the wife, even though there are things we can say about the married couple that we cant 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, youre counting the husband and the wife twice (or rather, 1.5 times each).
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?
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.
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?
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.
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.
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.
Quoting 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.
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?
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.
That's a misquote. What he said was:
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.
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.
Right, so this shows that you clearly misunderstand what is being talked about.
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.
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.
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?
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.
No mathematician will ever tell you that discussing whether abstract objects, like mathematical objects, really exist in the world or are just contained within minds is a mathematical topic: this is what a substantive portion of this discussion led into, so this is more accurately construed as philosophy of mathematics informed by insights from mathematics (it's very difficult to do philosophy of any x without pretty extensive detail into x itself).
Quoting ThinkOfOne
Powersets do contain all subsets of their original set, this is a well-proven theorem by Cantor that any elementary introduction to set theory should teach you. The set of all exists, by its very definition, includes the cardinality of a set strictly larger than it is, and is therefore incoherent/a contradiction (the proof of this is a very trivial exercise: Suppose E, then there exists P(E), P(E) is cardinally larger than E, therefore there exists x's that are members of P(E) and not E and thus E isn't E).
This does not mean the things that exist, like my keyboard or this screen, do not actually exist, but rather they cannot all be collected into set. If your standards of "conceptual understanding" is mathematical inconsistency, this is a problem on your end.
Quoting Michael
This is a category error namely in that sets never weigh anything (you're confusing mereological complexes with sets, a confusion many participants of this discussion took on and continued to presume despite my clarification of the distinction in an earlier post right here) Sets lack the physical properties of mass or weight, even if all their members have this property (hint: they're distinct from their members).
What you're looking for is either 1. plural quantification (in plural logic) or 2. mereological composition - both of these are distinct formalisms that are not isomorphic to our set theoretic apparatus. In plural quantification, you are quantifying over nothing over and above the x's, and not any notion of collection they form treated as a single. In mereological composition, you're quantifying over parts bearing a particular relation and arrangement, often proper-parthood, to some whole, and in that venue it's under dispute whether the wholes are identical to just the collection of their parts (this is not at all isomorphic to whether sets are identical to their members, whose answer is a trivial no)
That said, mereological wholes may have physical properties: for instance, an apple is a complex composed of its atoms, and it has weigh, though I'm not sure if weigh would also be possessed by its super subatomic particles, I suppose that debate would be fleshed out more there
Plural logic would be the most natural formalism of asking about, say, the weigh of a collection which all have physical properties, because the plural quantifier quantifiers over nothing aside those x's in the collection itself and would simply be a summation of their properties. When comparing these collective relations, the plural ? would be the "less committal", the mereological P being in the middle and the set theoretic ? being the most committal.
Apples can be members of sets but are never themselves sets. There are no instances at all where a physical object like an apple is a set, especially if it's just because it has parts.
The reason you and some of the others are getting mixed up here on this non-issue is that you're employing the inappropriate apparatus (set theory instead of mereology) to think of this whole-parts identity problem, but you're thinking about it with presumptions that are already at fault, like supposing of complex objects as literally being sets or that they themselves have physical properties like weight which are both incoherent.
I addressed that when I first brought up this example. If youre saying that the set exists, in addition to its members, then youre presumably saying that the set exists as some abstract thing, à la Platonism. Thats a view I take issue with.
I should add that I wasn't using the word "set" in the mathematical sense here. I moved beyond maths and was considering physical collections in response to litewave's comments. A collection of two coins has a weight (the sum of each coin). My argument was that even though a collection of two coins is conceptually distinct from each of its individual coins it is wrong to say that three ontologically distinct things exist (the one coin, the other coin, and the collection).
I've read your other (later) post prior to this one, but coincidentally it indulges in the same error that I pointed out in my response to your later post: you default to the idea that all sorts of plural quantification are identical with the sets corresponding to the plurality being quantified.
The set of the married couple {husband, wife} is not identical to the plurality of the husband and wife, nor is the arbitrarily infinitely many sets including the husband and wife (i.e. {{husband}, wife} identical with each other nor that first plurality.
Physical things, be they collections or singular, are never themselves sets: they just can join sets. By talking about a set of objects where that object is a member of the set, you've not counted the object again. It isn't the case that the wife exists infinitely many times because there are infinitely many sets that contain the wife.
OK, as you like. :smile:
OK.
You seem to have missed the point. There is a distinction that needs to be made between the definition of a SET and the definition of a POWERSET. They are not one and the same.
There is a SET of "all that exists" and there is POWERSET of "all that exists".
Perhaps a thought experiment will help.
Let's say that there is a universe with a SET called Tegwar.
Tegwar contains two members {x, y}.
The POWERSET of Tegwar contains "all subsets, empty set and the original set itself".
If the only things that exist in this universe are x and y:
Does Tegwar contain "all that exists" in the universe?
Does the POWERSET of Tegwar contain "all that exists" + all subsets + empty set?
As I posted earlier:
But either way, both both the SET and the POWERSET exist. Whether the SET is called Tegwar or "all that exists", the result is the same.
I'll add that if "all subsets" are allowed to be members, then the set is infinite.
If "all subsets" are not allowed to be members, then the set is finite.
This, I think, shows a more fundamental problem. You appear to equivocate. When you say that there exist elements in P(E) that are not members of E you're actually just saying that there are members of P(E) that are not members of E. But that something is a member of a set isn't that it exists. For example, Santa doesn't exist and so isn't a member of E, but it is a member of the set {Santa}.
If E is {John, Jane} then P(E) is {{}, {John}, {Jane}, {John, Jane}}. No member of P(E) is a member of E and so no member of P(E) exists. Therefore, that P(E) has a greater cardinality than E doesn't entail that E is not the set of all that exists. Whether you consider E or P(E), the only things which exist are John and Jane.
The one issue I see with this line of reasoning is that if sets exist then if something is a member of P(E) then it must be a member of E, which given the fact that P(E) has a greater cardinality than E entails a contradiction. So do sets exist, and if so, in what sense? Platonism?
There's no point I missed: no where in any of my entries I equivocated sets with powersets. Tegwar does not contain all that exists in that universe, namely because the set cannot contain either itself or the powerset (and its powerset, ad infinitum). The notion of a set of all that exists is not possible. Similarly, you cannot exhaust all that exists in any universe obeying set-theoretic principles: some universe where all that exists is some ball in space denoted by a letter is a figment of the imagination.
Quoting Michael
Sets do not have meontological members, because set-membership itself is a relation requiring that there are two relata of the set and the given member, yet the necessary condition can't be satisfied when one of the relata quite literally isn't there. Since Santa does not exist, {Santa} as a set doesn't exist in the real world (though there are possible, hypothetical universes out there where Santa does exist, and thus the singleton exists as well).
There actually is one set with no members, but it is a unique set. This is called the empty set, uniquely satisfying that ?x x?S. This does not mean that some spooky metaphysical concept of nothingness/nonexistence/emptiness/whatever is itself a member of the set, rather, literally that the set has no members.
No equivocation at all between "is a member of some set" and "exists", it's not a matter of conflating the concepts rather simply a matter of logical entailment. It's incoherent (and inconsistent) for anything to be a member of a set but also simultaneously not exist.
The "existence" of mathematical objects in mathematical anti-realism is different to the "existence" of mathematical objects in mathematical realism. I took the "set of all that exists" as referring to existence in the realist sense. If this is correct, and if mathematical anti-realism is true, then no member of the power set exists in the realist sense (every member of the power set is a set and sets don't exist in the realist sense), and so that the power set has a greater cardinality is not a proof that there isn't a set of all that exists.
So could you clarify what you mean by "exists" when you consider the set of all that exists, and whether or not you're arguing for mathematical realism.
Correct, hence why platonism and nominalism about mathematics here is far-reaching and beyond the closer phenomenon at hand, being just that universal set itself. Surely any ordinary set, even a coherent one like {1, 2, 3} does not exist independently for a nominalist about abstract objects (though, they might still insist it exists as a concept or, for Field, as a fiction abstracted from concrete reality) though it will for the platonist. This is not to say that there's some different theory of existence necessarily being employed by the platonist or the nominalist (of course, there can), but the nominalist and platonist can perfectly disagree in using the same sense of existence (say, as a second-order predicate of concepts a la Frege, or as an instantiation of properties a la Russell)
There's one thing that the platonist and nominalist would still agree on, in that contradictory sets, like the Russell set, or this universal set, do not exist because they're incoherent (and so would their existence). Certainly the nominalist needs not raise the issue of whether any sets exist at all to just say that this one set does not exist, which is the first point I made in this post: the fact that this universal set, the set of all that exists, is contradictory.
Quoting Kuro
This is why I think you need to clarify your argument. Is the set of all that exists the set of all that realist-exists or the set of all that antirealist-exists? Because if it's the former, and if mathematical anti-realism is true, then your set of all that realist-exists isn't a universal set, as the universal set contains members that don't realist-exist.
If you assume physicalism, the set of all that exists, let alone the set of anything, since sets are not physical objects neither identical to their physical members nor the collection of their physical members (the proof of this is simple: suppose it is the case, then submerge that same set under a further set, which is mathematically non-identical!)
So physicalism just entails mathematical anti-realism, and it goes back to what I said earlier here:
Quoting Kuro
The fixing to what sense of existence is unnecessary, namely because of the above, and further because the platonist/nominalist can perfectly disagree while holding to the same theory of existence (though they can differ if they want to, obviously), as I clarified earlier as well:
Quoting Kuro
Alas, I think your request for clarity was preemptively satisfied unless you hold to the presumption that realists and anti-realists must be using different senses or theories of existence. I don't take you to be claiming this, since you didn't object to it when I asserted its negation, but just on the possibility this is a claim of dispute it can be falsified using a trivial example (i.e. Quine vs Field), a pair of platonist / nominalist(fictionalist) who disagree on the existence of mathematical objects while using 'existence' in the same sense. Obviously though, the reverse claim of them having to use the same sense is not true either, as there are indeed platonists/nominalists who disagree while using different senses, examples being along the lines of Meinong
I don't understand how this addresses my argument. Can you specify which step you disagree with?
1. Physicalism is true (assumption)
2. Everything that exists is a physical object (from 1)
3. The set of all that exists is the set of all physical objects (from 2)
4. The power set has more members than the set of all that exists
5. No member of the power set exists (from 2)
6. Therefore, that the power set has more members than the set of all that exists does not prove that some things exist which are not in the set of all that exists (from 5)
(1) entails that no sets exist, including that set in (3) regardless of its incoherent status. It could be any ordinary set, like a set of an apple, someone's toenail & an ant. A set whose members are physical objects is not itself, as a set, physical (for obvious reasons: it'd entail infinite interpenetration)
The other issue is that, obviously, this still instantiates the same contradiction (4-5) in my initial argument: the powerset is either not larger than its set because there aren't more members in it than in the set, falsifying either the powerset axiom or this set's status as a powerset, or there is no set of all that exists.
FWIW, this is technically not a valid argument since 6 seems to only follow from (what I assume are crudely skipped steps for brevity's sake) the assumption in 5, that the powerset is literally empty, which not only is an issue in the terms I explained earlier (falsifying the powerset axiom or just giving up the nonexistence of that set), this still never means that the powerset is empty. That falsifies the axioms we use for set building, namely in that we can join any urelement or set into a further set containing just that set or urelement (being a singleton), but if the members of that set can't be joined into singleton supersets, let alone the powerset itself, then we've falsified several of our basic set theoretic apparatus just to suppose the existence of this set (which still manages to be incoherent, regardless: this doesn't actually make a powerset of a non-empty set empty!)
I think there's a difference between saying "there is a set of all that exists" and saying "the set of all that exists, exists". The mathematical anti-realist will assert the former but reject the latter.
Quoting Kuro
I didn't say that it's empty. Similar to the above, there's a difference between saying "the set has members" and saying "the members of this set exist". The mathematical anti-realist will assert the former but reject the latter.
So, the power set isn't empty, but as all of its members are sets, and as sets don't exist, none of its members exist. As such, it doesn't follow from the fact that the power set has more members that there exist things which aren't in the set of all that exists.
As I said before, I think you're equivocating on the word "exists". Being a member of a set isn't the same thing as existing (if physicalism and mathematical anti-realism are true).
Incorrect. This is a Meinongian there-is/exists distinction which has been proven inconsistent by Russell and largely abandoned ever since (I can elaborate on that further if you'd like), and like I said earlier, the presumption that an anti-realist is forced to use this outdated theory of existence is nonsense: anti-realists and platonists can perfectly disagree using the same theory of existence, as I said about 3-4 times earlier (and even gave examples, in case you don't believe me). Let's not be silly.
Furthermore, it's widely accepted by empirical linguistics that "is" has three senses, (1) predicative in the form of "x is an F", (2) identity in the sense of "x is y", and (3) existential in the sense of "there is x". This usage is also standard by logicians and mathematicians, in that the particular quantifier ? is understood as the existential quantifier and translated as 'there is' beyond the domain of heterodox systems like free & inclusive logics.
Quoting Michael
Nonsense: I explained earlier why there's no such thing as "this set has members such that they do not exist". If you plan to assert negations of some of the claims I've formally elaborated on to explain them to you, you need to either substantiate them of a similar level or at the very least address what I said:
Quoting Kuro
Quoting Michael
Nope! When we say "Pegasus does not exist", we're not referencing that there is a Pegasus such that it does not exist, as a Meinongian would, for that would be a contradiction, instead, we're quantifying over our most general domain to say that there is no x such that x is identical to Pegasus. It's a trivial inference in first-order logic, the language of set theory, to infer from being predicated to existing (to be predicated is just such that the constant is a member of some predicate F's extension, or that the extension of F satisfies the existential quantifier in that it has at least one member, and so on.) Deviant logics like free/inclusive logics obviously preclude this result, but these are not the languages which set theory is built on.
If you wish to protest these logical results in virtue of a distinct metaphysical theory (one that has been almost universally done away with and proven inconsistent several times), that alone requires substantial motivation on your part which you have not provided whatsoever.
Quoting Michael
I've already explained that "being a member" and "existing" is not the same thing, in the same way that (1) "P -> Q, P" and (2) "Q" are not "the same thing", but (1) logically entails (2) in the same way the former logically entails the latter (being predicated of anything entails existing). Just in case you really don't believe me and don't want to take me at my word (and I hate to use this, especially for really basic inferences, but this is about the second time I had to do this on this site), you can run this inference in any truth tree generator of your choice:
Also, the issue with this being a strawman is because it's after I've already clarified to you that my position, (I hate to call this a 'position' since it's literally a well-understood and universally uncontroversial logical inference), is of logical entailment and not of identity between the antecedent (being a member) and the consequent (existing), I've spelled this out for you here:
Quoting Kuro
So I find this quite bad-faith on your end, especially in the sense that I've been more than happy to steelman your arguments and fix various technical or mathematical issues in some of them where I eagerly addressed the corrected versions to make for a fruitful discussion.
1. Physicalism is true (assumption)
2. The set of all that physically exists is {apple, pear, ...}
3. The power set of this is {{}, {apple}, {pear}, {apple, pear}, ...}
4. No member of the power set physically exists
5. Therefore, the power set is not proof that there are things which physically exist and are not members of the set of all that physically exists
(2) is no different than (3) or a contradictory Russell set or some ordinary {a, b, c} set. The members of the set in (2) physically exist, but the set itself doesn't per physicalism. Recall what I said earlier:
Quoting Kuro
In even asserting that the set is anything, like having the property of "containing apple as a member", you get back to existential commitments. There's not that much middle ground between accepting the full force of the claim that sets, even consistent ones, do not exist, because sets, even impure ones, are not themselves physical objects and physicalism entails that all that exists is physical.
Your alternative is a type of Quinean physicalism, though I'm not sure if you'd actually want to take a position like this. Quine's view was that mathematics was indispensable to the natural sciences, and his naturalism in that we should commit to whatever our best theories committed to (thus commit to mathematics). In that sense, Quine is a physicalist platonist.
I'm aware. What is the relevance of that? I'm not saying that the set physically exists. I'm only saying that the power set doesn't prove that some things physically exist which are not in the set of all that physically exists.
Quoting Kuro
Mathematical anti-realists and physicalists are quite capable of doing maths with sets.
Of course, so is Hartry Field, and so is my capability to pray while not assenting to "God exists". Reasoning about mathematics and believing in a philosophical claim are distinct.
Quoting Michael
The set itself asserted by that premise doesn't exist. Not its powerset: just the set of all that physically exists.
So you're saying that if mathematical anti-realism is true then there is no set of all that exists, because there are no sets? And so your very argument, which uses sets, depends on mathematical realism being true?
No? I'm saying that the non-existence of the set of all that exists is an issue far prior to the philosophy of mathematics (namely because it's an issue provable in mathematics): the existence of the set is contradictory, so both platonists, who are realists about other sets, and nominalists, realists about no sets whatsoever, would agree it doesn't exist.
Arguing that it doesn't exist because no sets exist, while not invalid (in the sense that it follows), is an odd choice of argument because it relies on controversial philosophical premises in comparison to the well-accepted mathematical reasons for why the set of all that exists doesn't exist (which I explored in the OP).
By the way, surprise surprise, this was also all stated earlier:
Quoting Kuro
If you look back to my reply with AgentSmith, I explored the mathematical and philosophical considerations of prioritizing competing intuitions (the axioms of set theory, and the result that a universal set doesn't exist) based on this result.
While the issue of mathematical realism and nominalism is interesting in its own right, certainly even so in relation to this very topic, it's not at all necessary for just this theorem being the universal set not existing (hence why no mathematician cites it)
Wheres the contradiction here?
Quoting Michael
Your response before was just to say that if physicalism is true then there are no sets.
The set of all that exists is contradictory.
Quoting Michael
Indeed. No sets exist if physicalism is true.
The set of all that physically exists isnt contradictory (at least with respect to the power set), which is what that argument shows.
Nope. This has already been addressed:
Quoting Kuro
This "physicalism-constrained set theory" fails not only the powerset axiom but basic set-builders like joining sets into supersets, subsets, unions, and so on (which would not all exist per physicalism, and in your interpretation, in the literal sense of not existing in the same way the powerset's members don't). So in the same way my initial argument shows why the set of all that exists is inconsistent with set-theoretic axioms but particularly the powerset axiom, yours only scores more axioms!
This is not even considering the empty set, which, per whatever this "physicalism-constrained set theory" is, either (1) wouldn't exist, as it's a set, going back to what I said earlier (2) you might say it exists, if it does, you can generate N with set-builders and suppose E, then ask P(E) - same thing as what I said earlier
Im not taking about that though. Use normal set theory. The set of all that physically exists is not contradictory. It might not be, within set theory, the set of all that exists, but if physicalism is true then everything that actually exists is a member of the set of all that physically exists.
Anti-realists can pretend that sets exist for the sake of the maths to then talk about everything that actually exists.
Using "normal set theory" (suppose, it is ZF), then your argument is not even tenable
This set would violate the pairing axiom by being subsumed through a superset as well as the foundation axiom by there being sets disjoint from A. The axiom of empty set would also be false. This is all very trivial!
I've suggested using a very nonstandard "physical-constrained set theory" just to make your argument somewhat tenable to be charitable to you: if you agree on using ZF, ZFC, or other set theories that count as 'normal' then this is not even a topic for debate whatsoever. The non-existence of the universal set in ZF, ZFC and so on is a well established mathematical theorem in those set theories.
Im not talking about the universal set. Im talking about the set of all that physically exists. These are not the same thing.
So, in ZFC, why is the set of all that physically exists contradictory?
The universal set is the same as "the set of all that exists". In physicalism, "all that exists" is just physical stuff (though this does not mean "existing" and "being physical" having the same meaning, just that they coextend)
This is contradictory because, as explained to you several times, it would violate the pairing axiom, the foundation axiom as well as Cantor's theorem which I spoke of earlier.
Quoting Kuro
We can assume, when doing maths, that sets exist even if sets do not exist. A physicalist, who doesnt believe that sets exists, can make use of ZFC set theory.
In using ZFC set theory, this physicalist can define the set of all that physically exists. Within ZFC this isnt contradictory because it isnt a universal set.
But if physicalism is true then everything that exists in the real world is a member of the set of all that physically exists within ZFC set theory.
First-order logic, including set theory (a theory in the language of FOL), are extensional, so in the case of "everything" and "physical stuff" then if they coextend they're salva veritate substitutable. If not, it's just to say that either physicalism is false, or FOL+set-theory is false, regardless of whether the physicalist wants to pretend that sets exist or literally say they exist, or however else they want to reconcile their attitude.
After obtaining this set, it is, once again, inconsistent because it violates the three (or four) axioms I named for redundantly many times now. This is not even mathematically controversial nor is it the philosophy of mathematics anymore. This is an explanation of a relatively simple set-theoretic result, since we've already fixed the set theory we're operating in. I'm quite appalled that it requires this many replies and/or elaborations.
The set theory were operating on is the one in which sets exist, and so the set of all that physically exists isnt a universal set.
Reread: FOL and set theory is extensional. Why is it the case that so many of these queries are pre-emptively addressed?
Quoting Kuro
By moving their fingers to put down ink or press keys on a keyboard, in the same way anyone else can use set theory. In the same way atheists and Christians alike can go to Church on sundays and read the gospel if they wanted to.
The physicalist takes that all that exists is physical. In set theory, the universal set is the set of all that exists. Therefore, per extensionality, these are substitutable salva veritate and, per the axiom, the same.
There are far more pressing concerns of contradiction like the fact that neither ZF nor ZFC admit urelements (elements that are not themselves sets), but we'll just conveniently continue to ignore that as you insist on the "normal set theory" and not my previous suggestion. I'm honestly too tired to explain any other concepts right now aside the ones I've been repeatedly trying to communicate to you (and failing at doing so, anyway)
But sets dont exist if physicalism is true, and so following your reasoning the physicalist cannot define any set. Given that the physicalist does define sets when using set theory, his physicalism plays no role, and so, when using set theory, sets exist and the set of all that physically exists isnt a universal set (as the universal set includes sets, which dont physically exist).
He can pretend sets exist while using ZF/C, it's not a problem. This is not the problem. "The set of all that physically exists" is not a set in ZF/C. We'll pretend it is, though this is an actual problem, but fine. The pairing, fundamental, extensionality, powerset and empty set axioms are problems, and the set theory stops being the same set theory when you revise not one but several of its characterizing axioms to allow a set (on a purely technical level, this stopped being ZFC the moment we allowed a set of urelements, but forget that, we still have several other problems here. Recall my earlier posts.)
Genuinely speaking, every single thing I said not only in this post but about 3-4 of my earlier posts I've already said before. I love discussions, but not ones where I'm teaching or explaining things ad nauseam. I understand that you may still continue to have questions, queries or arguments, but I have a reasonable induction that, as has been the case with virtually all the previous ones, they will have had already been addressed by something I wrote here. If not, I'm sure Google or a textbook is a friend.
It was. Using "normal set theories" like ZF or ZFC was your suggestion, not mine. I was evaluating your proposal from a much more generalized perspective and showing how it's untenable even with a set theory tamed to be "physicalist-friendly"- this was discarded per your call
Then you need to be more explicit with your argument. What sort of things are members of your premised set of all that exists? Urelements like apples, in which case were not using ZFC, or only sets, in which case it has no relevance to real life where non-sets exist.
Any & all satisfy my argument, whether you allow urelements like in NF or stick purely to sets like in ZF, ZFC or FST. There is absolutely no need for any further specification or "being more explicit" beyond the fact of using the word 'set', since that's the degree of specification necessary to make the argument true. There are universal classes, like in NBG, but those are not sets in being proper classes.
What does the power set axiom state precisely? What IS a powerset?
My refutation of the argument does not need the knowledge of the above, it is strictly for my curiosity that I asked the two questions.
It is the CARDINALITY of the power set that is larger than its set. It is not larger than itself. You make this out as if it were true that the cardinality of the set makes it bigger than itself. That is not true. (Maybe. I am going out on a limb here, because I don't know set theory.) But no theorem will say that something is bigger than itself. It is not itself, that is bigger than its set, but its CARDINALITY. Which is not the set itself.
So yes, there could be a set that has everything in it.
Apple has parts, so it is by definition a collection (set).
Cantor's theorem never says that a set is larger than itself, rather, it says that a set's powerset is larger than itself.
Obviously, Cantor's theorem would mean that if we define some set X to just be identical with its powerset, then this set could not possibly exist. The set of all things, by including all sets as well, would necessarily include its powerset within it, but to do is a contradiction. Yet, if it doesn't contain all sets, including its powerset, then it's not the set of all things, let alone of all sets, which is another contradiction, meaning this set cannot exist. (I opted for a simpler explanation here, let me know if you follow).
Quoting god must be atheist
Alright. Here's the easiest understanding of a powerset. Suppose there's a set {a, b, c}. It will have subsets, which are a set such that all of its elements are also elements of the set that contains it. These include {a, b}, {a}, {b, c}, so on.
Now that we have the concept of subset, defining powersets is simple. A powerset of some set X is composed precisely of itself and all its subsets: this means that, for the earlier defined set {a, b, c}, its powerset will be {{}, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}} - notice how much cardinally larger this powerset is. Cantor's theorem basically proves that, for all sets, including infinite sets, the powerset is strictly larger than the set.
Quoting god must be atheist
Nope.
Quoting Kuro
This is not true. Or else I don't understand what "cardinality" means. I know now what Powerset is (thank you very much, by the way, for the clear and succinct explanation); but pray tell what the cardinality of a set is. The count of subsets a set has?
If indeed the cardinality of a set is the number of its subsets, then your statement fails. Because the powerset of E will by definition overlap all its elements with the elements of E in the sets contained in either of them.
Cardinality of a powerset of E does not increase the elements that form the set E. Therefore cardinality is of no consequence when counting the unique elements in the subsets of both E and P(E).
Therefore the cardinality of P(E) can be larger than the cardinality of (E) without additional elements in any of the subsets of P(E) which are not to be found in E.
The notion of cardinality is much simpler, in fact, I'd wager you're already familiar with it (save for its technical term): it's just the amount of members a set has! {a, b, c} has three elements, so its cardinality is 3. {a, b} has two elements, so its cardinality is 2, and so on.
This will two-foldedly answer you second concern, the relation of cardinality between powerset & set. We can use an ordinary set like {a, b, c} and compare it to its powerset, {{}, {a}, {b}, {a, b}, {c}, {a, c}, {b, c}, {a, b, c}}, to observe that the powerset's cardinality is 8. (Note that, x and {x} are not the same thing, because {x} is the set containing it, similarly {{x}} isn't {x} and so on.)
In fact, if you insert any other set, suppose it's {a, b}, you'll also notice that its powerset {{}, {a}, {b}, {a, b}} is cardinally larger. In this case, the original set's cardinality was 2, whereas its powerset's was 4. If you already noticed the pattern, the cardinality of a powerset is 2^n the cardinality of its set (where n is the cardinality of the original set). Try this for any set you like and see for yourself :)