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. — Kuro

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.

Of course. His accusations of investigations into him being politically motivated are him projecting his own corruption.

Eric Trump inadvertently made a similar claim a while back:

Speaking on Fox News, without providing any evidence, Eric Trump said: "I know the White House as well as anyone, I spent a lot of time there, I know the system, this did not happen without Joe Biden's explicit approval. The White House approved of this.

Because Trump directed the FBI, it must be that Biden does it too.

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, 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?

This is a category error namely in that sets never weigh anything — Kuro

I addressed that when I first brought up this example. If you’re saying that the set exists, in addition to its members, then you’re presumably saying that the set exists as some abstract thing, à la Platonism. That’s 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).

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

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

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

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

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

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

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

That's a misquote. What he said was:

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

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

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

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

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

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

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

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

Not sure what you mean.

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

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

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

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

Thanks.

It seems to be saying the same thing that I said:

... the equivalences of the form 'A' is true if and only if A ... define the conditions under which [my emphasis] a sentence is true.

Just as my example of "p" is foo iff p defines the conditions under which a sentence is foo. But the T-schema doesn't define "[is] true" and my F-schema doesn't define "[is] foo".

If we want a definition of truth (and not just a definition of truth conditions) we need some q where "[is] true" means q, or where "'p' is true" means q. Ramsey's redundancy theory is at least one attempt at this.

I don't have a jstor account so I can't read it.

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.

The Revision theory, discussed in some other posts, appears to offer a way to map out the circularity of the T-sentence definition of Truth. — Banno

It still has to be explained how the T-sentence is a definition of truth.

1. "p" is foo iff p

(1) isn't a definition of "foo". (1) only states the condition under which "p" is foo. And so too with the T-sentence: prima facie it only states the condition under which "p" is true; it doesn't define "[is] true".

As I mentioned before, Tarski didn't think of the T-sentence as being a definition of truth, only as something that must be entailed by the definition of truth. You've responded several times by saying that it is later authors who have taken the T-sentence as being a definition, so perhaps you could present their arguments to that effect?

I would say we're looking for some q where "[is] true" means q, or where "'p' is true" means q.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Here's another argument:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The first is a 1973 decision by the Justice Department's Office of Legal Counsel stating that a sitting president cannot be indicted. For that reason, Mueller said, charging Trump with a federal crime "is unconstitutional."

A federal judge addressed this memo a while back.

Judge tosses Trump's lawsuit to keep his taxes secret

Marrero’s most dramatic conclusion, however, may be his charged swipe at the Justice Department’s legal guidance that a president cannot be indicted while in office.

“The court is not persuaded that it should accord weight and legal force the president ascribes to the DOJ memos,” Marrero wrote.

He noted that the argument a sitting president cannot be indicted often relies entirely on these memos, which don’t carry the force of law or legal precedent.

"[T]he theory has gained a certain degree of axiomatic acceptance ... as though their conclusion were inscribed on constitutional tablets so-etched by the Supreme Court,” he said.

It might be DOJ policy not to indict a President, but that's all.

It could be documents about Chad for all you know. — NOS4A2

I don't think Chad has any nuclear capabilities. The document is likely to be regarding one of the nuclear powers.

Now it’s about a different country’s national security. — NOS4A2

This is one document amongst many, many more. That same report also says:

Among the 100-plus classified documents taken in August, some were marked “HCS,” a category of highly classified government information that refers to “HUMINT Control Systems,” which are systems used to protect intelligence gathered from secret human sources, according to a court filing.

We just don't know the details of these documents (obviously). We'll know more about the national security risk after the intelligence agencies have completed their damage assessment (assuming the results will be made public).

Either way, I wasn’t talking about you. — NOS4A2

Then who were you talking about? The news agencies reporting on the matter? Because, again, the original report specified that it wasn't clear whether or not it was concerning the United States or foreign nations.

But if you want to go ahead and say that other posters here were jumping the gun, then go ahead, but it seems like you're trying to deflect attention away from the fact that Trump was illegally in possession of, and insecurely storing, at least one document related to the defence and nuclear capabilities of some other nation. That's a huge deal. Your attempts to try to defend Trump and make it out to be some Democrat/FBI-led political hoax or whatever is absolutely ridiculous.

You also said this, linking to a report alleging the Trump administration was sharing nuclear tech with Saudi Arabia. — NOS4A2

Because I didn't rule out that it was about U.S. weapons. But neither did I say that it was about U.S. weapons. I'm actually capable of nuance.

We’re now going to pretend that we haven’t been speaking about US nuclear documents this whole time? — NOS4A2

I haven't. I don't know about you. You make a habit of misrepresenting whatever anyone else is saying. This is what I quoted in my original post on the matter:

FBI searched Trump’s home for classified material about nuclear weapons: report

FBI agents searched for classified material about nuclear weapons, among other items, when they served a warrant at former President Donald Trump’s home in Florida earlier this week, the Washington Post reported Thursday night.

Citing sources familiar with the investigation, the Post reported that government officials were deeply concerned that the nuclear documents believed to be stored at Trump’s Mar-a-Lago residence could fall into the wrong hands.

Separately, the New York Times reported the documents were related to some of the most highly classified U.S. programs, and that officials feared they were vulnerable to be stolen from Trump’s home by foreign adversaries.

The Post said their sources did not give details about the nuclear documents, such as whether it involved U.S. weapons or those of foreign countries.

Sensitive information about U.S. nuclear weapons is usually restricted to a small number of government officials, the Post reported, noting that material about U.S. weapons could be an intelligence coup for adversaries, and that other nations could see classified U.S. information about their nuclear programs as a threat.

First it was nuclear documents, now it’s a document describing a foreign government’s military defenses. — NOS4A2

"[A] foreign government's military defenses, including its nuclear capabilities."

The original report saying:

Classified documents relating to nuclear weapons were among the items FBI agents sought in a search of former president Donald Trump’s Florida residence on Monday, according to people familiar with the investigation.

...

They did not offer additional details about what type of information the agents were seeking, including whether it involved weapons belonging to the United States or some other nation.

So what sands have shifted? It seems to me that the new report simply confirms that they found what they were reportedly looking for, and clarifies that the documents were related to some other nation.

Material on foreign nation’s nuclear capabilities seized at Trump’s Mar-a-Lago

A document describing a foreign government’s military defenses, including its nuclear capabilities, was found by FBI agents who searched former president Donald Trump’s Mar-a-Lago residence and private club last month, according to people familiar with the matter, underscoring concerns among U.S. intelligence officials about classified material stashed in the Florida property.

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

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

Reification deals with treating abstract entities concretely — Kuro

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

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

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

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

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

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

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

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

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

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