I’m not talking about the universal set. I’m talking about the set of all that physically exists. These are not the same thing. — Michael

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.

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! — Kuro

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. — Kuro

We can assume, when doing maths, that sets exist even if sets do not exist. A physicalist, who doesn’t 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 isn’t contradictory because it isn’t 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.

We can assume, when doing maths, that sets exists even if sets do not exist. A physicalist, who doesn’t 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 isn’t contradictory because it isn’t a universal set. — Michael

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.

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. — Kuro

The set theory we’re operating on is the one in which sets exist, and so the set of all that physically exists isn’t a universal set.

The set theory we’re operating on is the one in which sets exist, and so the set of all that physically exists isn’t a universal set. — Michael

Reread: FOL and set theory is extensional. Why is it the case that so many of these queries are pre-emptively addressed?

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. — Kuro

So how is it that mathematical anti-realists, like physicalists, can use set theory?

So how is it that mathematical anti-realists, like physicalists, can use set theory? — Michael

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.

Then the mathematical anti-realist can use set theory to define the set of all that physically exists in my cupboard, or all that physically exists in England, or all that physically exists in the one and only universe. None of these are a universal set within ZFC.

hen the mathematical anti-realist can use set theory to define the set of all that physically exists in my cupboard, or all that physically exists in England, or all that physically exists in the one and only universe. None of these are a universal set within ZFC. — Michael

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)

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. — Kuro

But sets don’t 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 isn’t a universal set (as the universal set includes sets, which don’t physically exist).

But sets don’t exist if physicalism is true, and so following this 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 isn’t a universal set (as the universal set includes sets which don’t physically exist). — Michael

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.

Then I think the problem is with the wording of the discussion. In ZFC, urelements are not allowed. Everything is a set. But in the real world things exist which aren’t a set, e.g apples. If you had worded this as saying that the universal set in ZFC is impossible then I wouldn’t have even bothered replying. I thought this was talking about more than just pure maths.

Then I think the problem is with the wording of the discussion. In ZFC, urelements are not allowed. Everything is a set. But in the real world things exist which aren’t a set, e.g apples. If you had worded this as saying that the universal set in ZFC is impossible then I wouldn’t have even bothered replying. I thought this was talking about more than just pure maths. — Michael

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

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 — Kuro

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 we’re not using ZFC, or only sets, in which case it has no relevance to real life where non-sets exist.

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 we’re not using ZFC, or only sets, in which case it has no relevance to real life where non-sets exist. — Michael

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.

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. — Kuro

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.

But in the real world things exist which aren’t a set, e.g apples. — Michael

Apple has parts, so it is by definition a collection (set).

But no theorem will say that something is bigger than itself. — god must be atheist

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).

What does the power set axiom state precisely? What IS a powerset? — 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.

So yes, there could be a set that has everything in it. — god must be atheist

Nope.

A powerset of some set X is composed precisely of itself and all its subsets — Kuro

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

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.

but pray tell what the cardinality of a set is. — god must be atheist

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 :)