You might be implicitly presenting a science-philosophy duality in your post, which is really a modern artifact. Adam Smith was a social philosopher, for instance.

The effect of this duality is that in the present, scientists are untrained in philosophy, and philosophers too are largely untrained in the sciences. This makes certain problems in both philosophy & science difficult to deal with, namely because in the past the sciences and philosophy were one and the same, and their practitioners had just the same training, so cross-assistance was very common. Avicenna reached metaphysical conclusions in virtue of his physical investigations, and Leibniz reached the correct conclusion that spacetime is relative through a priori metaphysical investigation, which was at the time "empirically false" (according to the Newtonians), but was later found to be the correct theory through the works of Mach and Einstein.

On the other hand, it seems that the lack of cross-training is a natural evolution of the fact that our fields of inquiry are getting increasingly complex and advanced, and no one can really be a true polymath anymore. But this doesn't mean philosophers shouldn't learn the basic sciences, or that scientists shouldn't learn basic philosophy. Certainly the basics can still be learned and be part of any inquirer's essential training, just as both philosophers and scientists learn algebra and calculus while not needing to know the most avant garde mathematics.

It's been a while since I studied this, but, if I recall correctly, Hughes & Cresswell and L.T.F. Gamut do define an existence predicate in modal logic that is an extension of classical FOL=. (I'll happily stand corrected though if I my memory is incorrect.) — TonesInDeepFreeze

Ohh I'm actually familiar with this, I've recently read chapter 16 of Hughes & Cresswell after your recommendation. I fully understand: by relativizing formulas that lack modal quantifiers to quantify just over the actual world, we can coherently define an "existence predicate" whose extension is just those set of things that actually exist (i.e. exist in the actual world), i.e., it doesn't falsify the fact that Santa does not exist that he exists in other possible worlds, mainly because 'existence' simpliciter is relativized to the actual world. And we can generalize this such that for any world/node we can define an existential predicate just over the domain of that node.

This is moreso along the lines of the actualism vs possibilism issue and univocity of existence versus the distinct metaontological/logical debate about the nature of existence itself, though this is still nonetheless quite relevant because it is important to the related topic of modal realism. Someone like David Lewis, for instance, takes the proper existence simpliciter to be an unrestricted quantifier, quantifying over all possible worlds, so he takes "Santa exists" simpliciter to literally be true, and interprets our ordinary discourse as implicitly nested under actuality quantifiers. Of course, this will depend on our background theory of the metaphysics of modality and is highly controversial, but I digress.

I get what you're saying now. I suppose I have to make a three-fold distinction on the types of 'existence predicates' that philosophers and logicians consider:

• The trivial E formula, defining the predicate in terms of the existential quantifier. Uninteresting.
• The existential predicate defined in terms of the domain of a modal node / world
• The existential predicate simpliciter, such that ∃x¬Ex

(1) is uninteresting in all regards, (2) is relevant to the disputes about existence in relation to quantification & modality, i.e. relevant to Quine, Lewis and the like, and (3) is relevant to the disputes about the existence-being distinction, i.e. those between Frege & Russell against Meinong which preceded the formal regimentation of modality.

I did not consider the possibility that the term 'existence predicate' could be in reference to (2), which makes complete sense now in retrospect. I'm familiar with the topic of (2) though, so this is certainly a fault on my end.

But modal logic is the more common one to study than all the others combined. (That's not an argument that modal logic is "better" or anything like that, just that it's natural enough to first turn to modal logic, as a common subject, to see what it offers, while not precluding that the number of other approaches is potentially inexhaustible too.) — TonesInDeepFreeze

My issue isn't with modal logic here. I'm just unsure why you're characterizing modal logic as ones that deal with existence predicates: most modal logics are standardly extensions of FOL with K and some of the additional modal axioms, and therefore do not express nontrivial existence predicates.

Surely, certain modal logics can express existence predicates but these aren't extensions of a classical base, and by that point, there are similarly non-classical logics that express existence predicates.

So I'm just wholly confused why it is that we turn to modal logics to talk about existence predicates.

(1) AxEy y=x is a theorem — TonesInDeepFreeze

Correct.

Ey y=x in FOL= as a definiens for Exists(x). It would be pointless — TonesInDeepFreeze

Correct. I was just making sure, because this formula translates to FOL= extended modal systems like FOL + S5, but it's obviously trivial and not the controversial existence predicate that logicians (or metaphysicians) are interested in.

(2) My point is the opposite: FOL= does not have an existence predicate. — TonesInDeepFreeze

I'm aware, and I agree (besides the trivial quantifier-defined one), I was simply noting that modal logic is not a prerequisite to having existence predicates in any sense: most logics with existence predicates are not modal (to this, I think you agreed)

Are you referring to the E formula from FOL= (and similar systems), such that Exists(x) =df ∃y y=x?

While that's certainly an 'existence predicate', it is usually not what is really at stake in the debate of an existence predicate (i.e., it's sorta trivial). Usually, the controversial kind of existence predicate that we're interested in is the one that allows us to say ∃x ¬Exists(x), aka quantify over non-existent things, whatever those are.

That's all fine, but the more general point I mentioned is that we need to move to modal logic to have existence as a predicate. — TonesInDeepFreeze

Several families of wholly non-modal logics have existence as a predicate: free logics, inclusive logics, Meinongian logics et. cetera.

Traditional modal logics that extend classical logics, like FOL or FOL=, with modal axioms, also do not treat existence like a predicate. The modal logics that treat existence like a predicate are, at least all the ones that I'm aware of, just modal extensions of some of the non-modal systems above (i.e. Barba's free modal logic).

"Will this post have good outcomes, will it be productive, is it free of any breach of virtue that will harm my character?" and it would be nuts for me or anyone to expect they would. — TonesInDeepFreeze

I will say that the benefit of virtue ethics is that you'll no longer have to reconsider this in straightforward situations that are not ethical dilemmas. In acting virtuously, virtuous action becomes habit

Sadly, it seems math discussions on TPF are doomed to descend to the level of farce. — Real Gone Cat

Absolutely correct: the level of crank on this site is ridiculous

Thus, the diagonal of the square is now commensurable with its side, which we've known to be untrue since the time of the Pythagoreans! — Real Gone Cat

Interestingly enough, Avicenna's argument against (mereological) atomism was that applying the Pythagorean theorem is empirically successful, and it could not have been had our physical space been akin to a taxicab geometry, something like what the atomists suggested (in fact, in a taxicab geometry, using the Pythogrean theorem would not even approximate our values!). This later on came to be known as the distance function argument

What I'm proposing is that there is no "contest" involving infinitely many "contestants". For example, I'm proposing that to do calculus we don't need to assume that a continuum is built from the assembly of infinitely many points. Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points? — keystone

Your proposal is finitism. It's a cool proposal and an interesting philosophical topic on its own right, but entirely unrelated to the paradox you were trying to solve (i.e. irrelevant).

When you throw a dart at a dartboard, you don't hit a point, you hit an area. Any discretization of a dartboard into areas produces a finite number of areas each with a finite probability, all summing to a probability of 100%. What's wrong with this view? — keystone

Nothing is wrong with this view except that it misses the point of the paradox, which isn't related to a literal physical dart (we have no idea ifphysical space is discrete or not; this is a debated metaphysical topic I won't get into) rather the very fact that anytime you have probability with infinitely many "contestants", whether it's dense space or whatever, you will necessarily either give the "contestants" a probability of 0 or be faced with adding up over 100% (since reiteratively summing any non-zero quantity indefinitely will approach over 100% at some point).

Your "solution" isn't a solution in that it doesn't talk about what the problem talks about. The "problem" is referring to continuity in dense contexts: it's not at all a "problem" in nondense contexts, this is equivalent to solving the Liar paradox by just saying "what if the guy doesn't lie?"

In any case, there are two mathematically respectable solutions to this 'paradox'. One is the philosophical thesis of saying that zero-probability is not the same as modal impossibility, which is so far the most widely accepted solution. The other solution is to introduce hyperreal numbers, particularly nilpotent infinitesimals, such that each contestant has probability ε but reiterative summation does not eventually yield anything over 100%, primarily because while ε isn't 0, ε+ε can still equal 0. This approach is not as widespread.

It is true that in the US we are doing fine without monarchs, but these Britons have something that many liberal nations lack today, they have role models. — Eros1982

The royal family doesn't make for the best role models.

Are you not disquieted that a probability of 0 does not mean impossible? — keystone

All impossible propositions have probability 0, but not all probability 0 propositions are modally impossible. For instance, the probability of number being picked by randomly selecting a random
real number between 0-5 is zero, but a number will be selected. The fact that a number will be selected is not impossible, in fact, it will actually occur in this situation.

Yes, this will seem very counterintuitive. The simplest way I can explain it in a non-technical fashion is that selecting any non-zero probability for each number will force us to add up way over 100%, because there are infinitely many other "participants" (numbers), which means the only probability we can assign to each participant is zero.

There's actually a way out of this being nonstandard analysis giving us infinitesimals and particularly nilpotent infinitesimals in several hyperreal systems (*R). These can be very elegant in several applications including this one, where we can include numbers so small that even their squares (n^2) are zero but the numbers themselves, n, may not be.

But I accept that you don't consider 'utilitarian' as a correct description of your productivity and outcomes argument. Indeed, my point doesn't rely on the particular rubric 'utilitarian' but rather that I reject your productivity and outcomes argument, whatever rubric it correctly falls under. — TonesInDeepFreeze

It correctly falls under 'contemporary analytic virtue ethics', and is not consequentialist in the substantive sense of the word but the trivial one (the trivial one where all ethical theories are 'consequential' in that Kantians would care about that consequence of violating the imperative, or Aristotelians about virtue, and so on). The substantial sense of 'consequentialism' and 'utilitarianism' (which falls under the former) is not as broad, and the argument translates to the fact that your actions fall short of moral excellence (in that indignancy fail prudence and justice): indeed, no virtue ethicist would ever grant you that your personal frustrations offset this.

But I don't agree with the utilitarian framework you apply here*. First, I don't think utilitarian result is the only consideration. Second, for utility it doesn't matter anyway: The crank will continue to spew disinformation no matter whether left unresponded to or responded to with correction.

* I don't claim you adhere to utilitarianism. I am just saying that in this particular context your framework is utilitarian. — TonesInDeepFreeze

If you know a thing or two about that quote you agree with, you'd know that I'm not employing a utilitarian framework here* (though I'm delighted that utilitarians and Kantian deontologists alike would also agree with me): this is simply the framework of Hellenic virtue ethic theories, and I'm saying that, in a less blunt way, that in not acting rightly one retracts from the virtue of their character, which is less so a property of any particular fault or flaw in action rather than those habits which become second nature to them and come to form their decisions & methodology (which is what the discussion came about, the generalized implications versus the particular one, hence the relevance of the quote).

*I'm not saying this to imply that what I said disagrees with what utilitarians or deontologists have to say, just that I'm not myself using their theories.

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 :)

Rather than stamping out crackpottery, you are fanning its flames. — apokrisis

This is similar to my diagnosis.

I find myself as frequently frustrated as Mr. Tones with respect to the mathematical, logical or other formal/technical errors that are somewhat frequent on this forum. However, a rude attitude seldom yields anything productive: there is the option of politely leaving a discussion, perhaps at no fault of your own but the inadequacy of your interlocutor, or explaining their mistake at a reasonable level.

"Fanning the flame", so to speak, is unnecessary in whole. As a wise Greek philosopher said:

We do not act rightly because we have virtue or excellence, but we rather have those because we have acted rightly. We are what we repeatedly do. Excellence, then, is not an act but a habit.

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.

Merci, but where's the argument ... that proves/suggests reason is our go-to-person if our objective is to find the truth? — Agent Smith

The fact that you're asking for an argument to prove this kind of presupposes the thing being proven, but nevertheless, I'd cite Conee, who argues for reflective equilibriums which are epistemically but not logically circular (it's basically a type of a coherentist foundationalism which also doubles as a response to the Criterion problem).

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.

Suppose a lottery out of ten billion atoms in some universe, where there will only be one winner selected at random.

The probability of any atom being selected is astronomically low.

But should we be surprised that an atom would be selected? Does the selection itself have an astronomically low probability?

No. The fact of any particular atom, a, b, c, being selected (∃x Selected(x) & x=a) is very low, but just that very fact of an atom being selected, is in fact not unlikely at all. If anything, it's a certain fact: some atom must be selected, even if we don't know which one! (∃x Selected(x)). The earlier proposition specifying the atom being selected is just about 0.000(...)1% or some other low probability, and we should not expect it to be true nor would we be rational in supposing it'd be true. But the latter proposition of an atom being selected is a distinct one, and a certainty, one that we're rationally obliged to believe.

Similarly, the fact that people exist at the present, out of infinitely many arbitrary "potential people", is no surprising fact. It's an almost-certainty given that people in the past existed, and reproduced.

However, when it turns out that an atom gets selected, or a person is born, while this satisfies the truth of the general facts, any particular person must be a particular someone, in the same way any particular atom must be a particular atom. That fact should carry with itself no surprise. 'Tis but an illusion.

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

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.

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)

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.

I reiterated the point that I was correct to support the additional point, which you did not mention, that I was also not arrogant about it. — TonesInDeepFreeze

Sure. You were not arrogant about it. Extra brownie points.

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

You were wrong to claim I was not correct — TonesInDeepFreeze

No need to hammer something down when my previous post already agreed that, per set-theoretic context, you were correct. You can relax. I simply made the point that my misinterpretation was not unreasonable.

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.

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

I’m 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. — Michael

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.

You are either confused about the context of the posts or you are willfully fabricating about it. — TonesInDeepFreeze

I responded based on the quotation of him where he says "part is equal to whole" - this is what you've included in your post in whole with no further passage/text in the quote. Perhaps you could've made a larger quotation for context, but otherwise I do not think my presumption was particularly irrational (since, reading "part is equal to whole" at face-value, just is a mereological truism.)

You are entirely correct in that if he refers to proper subset and set by "part and whole", which, per context seems to be the case, then it is completely inaccurate.

The set of all that physically exists isn’t contradictory, which is what that argument shows. — Michael

Nope. This has already been addressed:

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!) — 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

Where’s the contradiction here? — Michael

The set of all that exists is contradictory.

Your response before was just to say that if physicalism is true then there is no set of all that exists. — Michael

Indeed. No sets exist if physicalism is true.

No, set theory does not say that there is a proper subset of a set such that the proper subset is the set. Set theory does say that there are sets such that there is a 1-1 correspondence between a proper subset of the set and the set.

This is another example of you running your mouth off on this technical subject of which you know nothing because you would rather just make stuff up about it rather than reading a textbook to properly understand it. — TonesInDeepFreeze

It's kind of funny in retrospect how arrogant you came off in accusing Agent Smith, while, unknowingly, you are completely wrong! Part and whole have nothing to do with set and subset: one is mereological, the other is set-theoretic, yes, they can overlap, but no, they're not the same thing.

To interpret his statement set-theoretically, when "part and whole" are specifically the technical terms used in formal mereology, is either (1) delibarately uncharitable or (2) you not being aware of what he's referring to. But whatever the case is, (1) or (2), it does not excuse your hostility.

In any case, (non-proper) parthood is a transitive, antisymmetric, and most importantly reflexive relation such that ∀x P(x, x): every whole is a (non-proper) part of itself. This axiom is true in virtually all of contemporary formal mereologies (X, M, AX, GEM, AEM, AMM, EM, etc.) and is perhaps the least controversial mereological axiom: even the transitivity of parthood is sometimes disputed!

So AgentSmith was correct, and your "correction" of him is a result of conflation of mereology with set theory on your part, so before telling him to do his homework, do your own.

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? — Michael

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:

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. — 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)

Mathematical anti-realists and physicalists are quite capable of doing maths with sets. — Michael

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.

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. — Michael

The set itself asserted by that premise doesn't exist. Not its powerset: just the set of all that physically exists.

P1: if God exists, nonresistant nonbelievers would not exist
P1: nonresistant nonbelievers do exist
C: God does not exist — aminima

Deism and other forms of theism will not necessarily want people to believe in them

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 — Michael

(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:

(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) — 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 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. — Michael

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.

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. — 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:

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). — Kuro

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. — 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.

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). — 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:

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 — 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.

I follow Russell, Quine, et. al in seeing there just being one mode of inquiry, which can be arbitrarily subdivided into that one inquiry across several domains (philosophy, science, and mathematics, then, form a spectrum instead of being fundamentally distinct from each other)

In this sense, to me it makes sense only to evaluate philosophy on the standard of truth. I understand this notion of danger you speak of, but these can also be habituated in this notion of truth, in propositions along the form "x ideas influence y people to do z things", which can still be evaluated as fact or false. Based on this, I think the standard of truth is the most general and appropriate standard for philosophical evaluation.

(Also, I may even argue other standards have to collapse to this standard: otherwise, are these other standards claiming they're the right standard? Surely my hypothetical opponents will have it that I'm wrong and that they're right)