I am not skipping. You say:

So whether a set x is a member of itself or not a member of itself, if there is a set of all sets then x is a member of that set of all sets.
— TonesInDeepFreeze — Philosopher19

You're confused. What you skipped is my refutation (posted twice) of your claim that a set can't be both a member of itself and a member of another set.

So whether a set x is a member of itself or not a member of itself, if there is a set of all sets then x is a member of that set of all sets.
— TonesInDeepFreeze

This point implies that a set v can have more than one set that is a member of itself, as a member of itself — Philosopher19

"is a member of itself, as a member of itself" has no apparent meaning to me.

you perhaps fairly said "the z of all zs has no apparent meaning to me" — Philosopher19

Yes, so fix it.

L = the list of all lists
LL = the list of all lists that list themselves

Is L a member of itself in L? — Philosopher19

I gave you guidance before in how the notion of 'list' is couched in set theory. But you skip it. To avoid incoherence with a clash between 'lists itself' and 'member of itself', you could go back to what I wrote.

The v of all vs [...] The z of all zs — Philosopher19

You said that it is perhaps fair to say that such locutions have no apparent meaning, but then you proceed to post them again, as, for the second time, you've ignored my suggestion of the exact way you could reformulate so that you make sense.

you hold a boat load of mathematical knowledge — Vaskane

My knowledge in mathematics is quite meager compared with people more dedicated to the study.

the US education system does a massive disservice to the field of mathematics due to the fact that it divorces the philosophy of mathematics away from the applied version. — Vaskane

That might be true. Also, the fact that formal logic is not conveyed so that students could see how the mathematics is derived logically rather than simply decreed.

One can always learn vastly more about logic, mathematics, philosophy and the philosophy of mathematics, but my first interest in logic (which led to mathematics) came from my interest in philosophy, and as I learned logic and mathematics, I was learning about the philosophy of mathematics right alongside.

What's worse, a population of palm trees in a city, or a city in a population of palm trees?

Isn't the set of all sets equivalent to the set of all members? — Fire Ologist

Members of what?

Of course, every set is the set of all and only its members.

And in ordinary set theory, every set is a member of another set.

There aren't actually any sets within the set of all sets. There are only members. — Fire Ologist

If by "within" you mean 'member of', then the above nonsense. If x is the set of all sets, then, by that description alone, every member of x is a set and every set is a member of x

And in ordinary set theory (without urelements and without proper classes) every object is a set.

.

Or does it contain 2 more members, total of 38, being the prior sets called "numerical" and "alphabetical" plus their members? — Fire Ologist

The set {0 1 2 3 4 5 6 7 8 9 a b c d e f g h i j k l m n o p q r s t u v w x y z} has 36 members.

If we call it the set of two sets — Fire Ologist

We call it 'the union' of two sets.

{0 1 2 3 4 5 6 7 8 9 a b c d e f g h i j k l m n o p q r s t u v w x y z}
is the union of
{0 1 2 3 4 5 6 7 8 9} and {a b c d e f g h i j k l m n o p q r s t u v w x y z}

But let N = {0 1 2 3 4 5 6 7 8 9} and L = {a b c d e f g h i j k l m n o p q r s t u v w x y z}.

Then we also have the set:

{0 1 2 3 4 5 6 7 8 9 a b c d e f g h i j k l m n o p q r s t u v w x y z N L}

which has 38 members.

Aren't we just overcounting this new "set of all sets" if we count the sets within it, and not just the members of those sets — Fire Ologist

No.

The set of all sets encompasses all sets that are not members of themselves (precisely because they are members of it and not themselves) as well as itself (precisely because it is a set). — Philosopher19

That is confused. A set of all sets has as members all the sets. Let that sink in. ALL sets are members of the set of all sets. So whether a set x is a member of itself or not a member of itself, if there is a set of all sets then x is a member of that set of all sets.

You claim that a set cannot be a member of itself and also be a member of another set.

I've refuted that claim. You skip the refutation.

A subset of "all sets that are members of themselves" and "all sets that are not members of themselves" is contradictory — Philosopher19

Maybe you mean that no set x can be both a subset of a set of all sets that are members of themselves and a subset of a set of all sets that are not members themselves.

That is true of all sets except the empty set.

The z of all zs i — Philosopher19

"The z of all zs" has no apparent meaning to me. But if you mean a set z such that every set is a member of z, then that is said as "the z such that for all x, x is a member of z". Let me know that you've fixed that, then maybe I'll go to your next sentence.

We need a meaningful distinction between "member of self" and "not member of self" — Philosopher19

We have it.

We need a set of all sets (math/logic would be incomplete without it, if not contradictory) — Philosopher19

Again, you skip my whole explanation to you about incompleteness.

Again, you skip my refutation of the claim that without a set of all sets we have a contradiction.

Again, discussion goes in a circle with you as you skip the counterpoints given you. And, again, you've been given a refutation that it is your interlocuters who are circular, so skipping that refutation and instead yet again claiming the fault is with your interlocuters would be yet more circularity from you.

I do not think I was ever subjected to new math. — Metaphysician Undercover

Virtually any student is subjected to certain instruction whether they like it or not. It would be fair to say that New Math is not good only if one at least knows what it is.

my education in mathematics — Metaphysician Undercover

You have virtually no education or self-education in the mathematics you so obdurately opinionate about.

The whole confusion resulted from the wrong premise that infinite numbers do exist. — Corvus

What is the "whole confusion"? Yes, there are people who don't know about set theory and are confused about it so that they make false and/or confused claims about it. But the axioms of set theory don't engender a confusion. They engender philosophical discussion and debate, but there is no confusion as to what is or is not proven in set theory. Whether any given axiom is wrong or not is a fair question, but it doesn't justify people who don't know anything about axiomatic set theory thereby spreading disinformation and their own confusions about it.

the expression "different sizes of infinite sets" is ungrammatical. — RussellA

I don't think so. And it's clear to me. There are infinite sets that have sizes different from one another. More formally:

There exist x and y such that x is infinite, and y is infinite, and the size of x is not the size of y.

we find lots of confusions in math and also the math students — Corvus

There are areas of great puzzlement and disagreement in the philosophy of mathematics. But I don't know what specific confusions you refer to, specifically in formalized classical mathematics.

It is not the bible, to which you have to take every words and sentences as the objectivity that everyone on the earth must follow. — Corvus

Of course. And I have many times explicitly said that no one is obligated to accept, like, or work with any given set of axioms and inference rules. But if the axioms and inference rules are recursive, no matter what else they are, then it is objective to check whether a given sequence purported to be a proof sequence is indeed a proof sequence per the cited axioms and rules. If you give me formal (recursive) axioms and rules of your own, and a proof sequence with them, then no matter whether I like your axioms or rules, I would confirm that your proof is indeed a proof from those axioms and rules.

But what you claim to be objectivity is from the textbooks. — Corvus

What I said was that it is objective to mechanically check that a purported formal proof is indeed a proof from the stated axioms and rules of inference. If there is anything more objective than verification of application of an algorithm, then I'd like to know what it is.

But as you say, this is problematic as it suggests that infinity is an object, such as a mountain or a table, which can be thought about. — RussellA

I don't say that.

I say that 'infinity', applied to set theory, is not advisable, because in set theory there is no object called 'infinity', especially one that has different cardinalities. It's not a matter of can be thought about, but rather that there are many infinite sets, not just one called 'infinity'.

within mathematics is the infinity symbol ∞ — RussellA

The lemniscate is usually used to indicate a point of infinity on a number line, which is very different from the context of the cardinalities of infinite sets. Such a point of infinity is some designated (or sometimes, less formally, unspecified) object along with a set, such as the set of real numbers, and an ordering is stipulated. If the treatment is fully set theoretical, then the object itself can be infinite or not.

So what does the word "infinity" refer to, if not a noun inferring an object? — RussellA

I am not saying that one should not use 'infinity' as a noun. It is a noun. And people can use it for many things. But it is an invitation to confusion to use 'infinity' regarding set theory or mathematics in a context such as discussing infinite cardinalities. Set theory does not define an object named 'infinity' in this context. Rather, it defines a property 'is infinite'. Keeping that distinction in mind goes a long way to avoiding confusions.

As the Wikipedia article on Infinity writes: Infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object. — RussellA

(1) There are better sources than Wikipedia.

(2) The quote does not say that mathematics refers to some set that is named 'infinity'.

(3) The quote then defers to 'infinite', the adjective, which is correct.

(4) The article is almost all about 'infinity' not applied to infinite sets and cardinalities. And the small part of the article that is concerned with infinite sets and cardinalites correctly, when talking about sets, uses 'is infinite', the adjective for the property of being infinite, not 'infinity' to name a set.

Infinity in math has been improvised to explain and describe continuous motion — Corvus

Also, more simply, there are infinite sets of numbers, such as the set of naturals, the set of rationals, and the set of reals. With ordinary classical logic, for even just simple first order PA to have a set over which the quantifier ranges requires an infinite set.

Sets can have different sizes etc. It is OK to keep on saying that in math forums, and it sounds correct because that is what the textbook says. — Corvus

Not just because it's what a book says. Rather, textbooks provide proofs of theorems from axioms (including definitional axioms) with inference rules. One doesn't have to accept those axioms and inference rules, but if one is criticizing set theory then it is irresponsible to not recognize that the axioms and inference rules do provide formal proofs of the theorems. Moreover, intellectual responsibility requires not misrepresenting the mathematics as if the mathematics says that the theorems claim simpliciter such things as that there are infinite sets of physical objects or even that there are infinite sets in certain other metaphysical senses of 'infinite'.

the whole picture was based on the fabricated concepts, which are not very useful or practical in the real world. — Corvus

Fabricated in the sense of being abstract. And it is patently false that classical infinitistic mathematics is not useful or practical. Reliance on even just ordinary calculus is vast in the science and technology we all depend on.

So he spent $85,000 for just a fancy noun. I told him it was not a wise purchase.

What is one sentence where "infinity" is used as an adjective? — Lionino

My cousin spent $85,000 on an infinity pool because he thinks that if he swims in it he will live forever.

What is required is trust in the teacher's ability to recognise and adapt their teaching to the student. But that's contrary to the very notion of a curriculum. — Banno

That seems to me to be a trenchant observation.

distinguish between "member of self" and "not member of self" — Philosopher19

That is one thing you say that makes sense and is correct.

For me, as a kid, New Math was wonderful. It opened my imagination to different ways of looking at mathematics, not just learning by rote how to do long division and stuff like that. For example, the idea of numbers in binary, modular arithmetic, intersections and unions, truth tables. I think maybe the idea behind it was to get children primed for the upcoming age of computers, such as binary numbers. It blew my mind, I savored it and it served me well.

Of course, but I'm saying that in context of sets in mathematics, 'infinity' as a noun invites misunderstanding, especially as it suggests there is an object named 'infinity' that has different sizes.

Is that why you felt the need to correct me when I said
To me it's just silly to argue the point of how big an infinity is when infinity is a concept considering continuity, not size.
— Vaskane
Because it was UNREASONABLE for me to not assume mathematics simply because numbers were involved? That's the real Dunning-Kruger here. — Vaskane

It’s the reverse.

In recent threads, the notion infinity has been raised with reference to mathematics - in the original posts and in replies. And I have not said that therefore the subject must be contained to mathematics. But the mathematical aspects should not be mangled, so I have commented to correct and articulate points about the mathematics that is being referenced. And still that does not even insist that one may not have an alternative mathematics; rather that if, for example, one claims to disprove that there are sets of higher infinite cardinalities, then the context in which there are higher infinite cardinalities should not be misconstrued or misrepresented. And ordinarily, the context of higher cardinalities would be classical set theory. Or for example, if the context begins with intervals on the real number line, then the ordinary context is classical real analysis and it should not be misconstrued or misrepresented. And, again, if one wants to discuss it in some other context, mathematical or otherwise, then that is fine, but that doesn’t preclude that we also discuss it in context of ordinary mathematics.

On the other hand, you post to say that you view it as “silly” to consider the notion of infinity regarding size when it is not regarding size but rather continuity. Then it is reasonable for one to say, “No, this discussion is not silly for talking about size, as indeed size is central to the ordinary context of sets in mathematics as indeed the definition leads right into size rather than continuity.”

Yes, continuity is an important topic related to the infinitude of certain sets. But the notion of infinitude applies even where the topic of continuity is not involved. So it is not silly to talk about the sizes of infinite sets.

So I am not insisting that any discussion be confined to mathematics. But I do say that such a discussion may include mathematics, especially when it starts out with reference to notions that are usually regarded as pertaining to mathematics, especially as the notion of greater infinitudes is ordinarily in context of classical mathematics and especially where the original poster mentions it in connection with Russell’s paradox. But, in stark contrast, meanwhile you are the one who is telling other people that it is silly to talk about the subject in terms of a certain aspect when you incorrectly claim it not concerned with that aspect.

Language is bendable, and often done so for artistic effect. — Vaskane

Hear hear.

One can do whatever one wants with numbers. That doesn't vitiate that it is reasonable that I and others have commented on the mathematics.

doesn't equate to math — Vaskane

I didn't say anything about 'equating to math'.

Rather, the context includes mathematics, as also other posters have taken it.

Which is the exact same boneheaded mistake you made on the other post about infinity. — Vaskane

Actually, in that instance I responded to the poster who has written:

As I wrote before: ""infinity" as an adjective means something along the lines "any known set of real numbers can be added to"". — RussellA

That is a mathematical context.

if there are infinite whole numbers, and there are infinite decimals between 0 and 1, and there are infinite decimals between 0.1 and 0.12, and there are infinite decimals between 0.1111111 and 0.1111112 [...] — an-salad

Ordinarily, one would take that to be referencing mathematics, as have posters in this thread, not just me.

except it can be used as an adjective, so stop being a dumb cunt who only seems to know "maf." — Vaskane

'infinity pool' for example. But I'm talking about the context here.

it seems barmy to talk about different size of the infinite sets — Corvus

No set has different sizes. But there are infinite sets that have sizes different from one another. That follows from the axioms.

One is free to reject those axioms, but then we may ask, "Then what axioms do you propose instead?"

One is free to reject the axiomatic method itself, but then we may ask, "Then by what means do you propose by which anyone can check with utter objectivity whether a purported mathematical proof is correct?"

One is free to respond that we check by comparing to reality or facts or something like that. But then we may point out, "People may reasonably disagree about such things as what is or is not the case in whatever exactly is meant by 'reality' or in what the facts are, so we cannot be assured utter objectivity that way."

One is free to say that we don't need utter objectivity, but then we may say, "Fair enough. So your desideratum is different from those using the axiomatic method."

"infinity" as an adjective — RussellA

'infinity' is not an adjective.

An item in a subset cannot be both a member of the subset and the set. — Philosopher19

Do you mean: If S is a subset of some set T and x is member of S, then x cannot be a member of T ?

That's incorrect. By the definition of 'subset', if S is a subset of T, and x is a member of S, then x is a member T.

If it's a member of other than itself, this means that it's not a member of itself. — Philosopher19

Every set is a member of certain other sets. For example, every x is a member of {x}.

With the axiom of regularity, no set is a member of itself. So with the conjunction:

x is a member of x and x is a member of other sets

the first conjunct is false, therefore the conjunction is false even though the second conjunct is true.

Without the axiom of regularity, it is not precluded that there are sets that are members of themselves. Therefore, without the axiom regularity, we cannot infer that there is no set that is both a member of itself and a member of other sets too.

/

Or maybe your phrasing is not what you mean. When I take your phrasing literally, as best I can, I take you to be saying that a set cannot be a member of another set and also a member of itself.

But if all you mean is that a set cannot be both a member of itself and not a member of itself, then, of course, we have no disagreement.

Then you're a dumbass for arguing with me when I was correct that Infinities do indeed have different sizes if that's your stance too. Either way, you're an "egregious" dipshit. "Oh, this guy is arguing the same thing as I am, and he's being comical, I should "correct," him about infinities not having sizes even though that's MY STANCE! Oh, wait, it's my stance, noone else can have it!" Eitherway, the fact is, you're a dumbass when it comes to communication. — Vaskane

I have never argued against the point that there are different sizes (cardinalities). And I have never said that other people may not also point out that there are different sizes (cardinalities).

That's a specific, restricted definition of "incompleteness". The term is slightly different in physics for example. So this is an example of what I am talking about. — Metaphysician Undercover

That is silly. Mathematicians don't claim that the mathematical sense of incompleteness trumps all other senses of that rubric in other fields. Just as a biologist talks about cells in an organism does not begrudge a penologist talking about cells in a prison and vice versa.

Mathematics also uses a specific, restricted definition of "infinite" — Metaphysician Undercover

Yes, and no fair minded mathematician would thereby begrudge people in other areas of thought from using other senses of the words.

I've said it over and over: Philosophers, scientists, theologists, et. al should be permitted to use terminology as suits them, and to express concepts as suits them. But when someone says the mathematics is wrong for using terminology in its way too, then that is quite unreasonable, and even worse when cranks claim that the mathematics is thereby wrong and worse, premises that claim on horribly misconstruing the mathematics and outright fabricating that makes it says things it actually does not say.

Words have different meanings in different fields of study. The point is that reasonable people allow that. What is unreasonable is when the crank has his own ways of using words and their related notions and then dictates that mathematicians are wrong for their specialized sense not conforming to the crank's own sense.

The "math boys" here at the forum tend to respond with 'go read some math texts' to anyone who disagrees with them on fundamental principles. — Metaphysician Undercover

Again, the intellectual dishonesty of the crank in action. In this case, blatant strawman by misrepresentation of what his interlocutors have said. And even more egregiously by dint of the fact that this strawman has been pointed out to him many many times.

It's not a matter of disagreement on principles, but rather ignorant and confused misrepresentation of the mathematics that is supposedly being discredited. It is fine and even essential that there be different points of view about foundations including critiques of classical mathematics. But it is pernicious against knowledge and understanding when the attacks on mathematics claim things about the mathematics that are crucially false, when the attacks are premised in an ignorant prejudice that the mathematics works in certain ways that it definitely does not. After the crank's error about this have been explained to him over and over and over and he still persists to spread the disinformation, then the best thing is to recommend that he get a basic textbook to inform himself in the subject that he has spent so much time already cultivating his self-imposed terrible misunderstandings.

That's like telling an atheist to go read some theology, as if this is the way to turn the person around. — Metaphysician Undercover

It's nothing like that. It's the reverse. It's like telling the zealot denouncing scientific theories to get a textbook in biology.

I don't know anything about microbiology, so I don't spout a bunch of nonsense about. If I did, I should expect someone to kindly tell me to shut up about it and get an introductory text.

concepts like infinity, incompleteness, and even computability, extend beyond mathematics. So, the mathematical approach is only one approach to such concepts. — Metaphysician Undercover

Of course.

Except incompleteness (in the sense of the incompleteness theorem).

To say that such concepts are the domain of mathematics, therefore mathematicians ought to define them — Metaphysician Undercover

I never said such a thing. Maybe other people have.

It could not be more clear.

You wrote:

"if we are considering the set of all natural numbers, then we thereby know that this set is infinite because there is an infinite amount of them."

But:

"there is an infinite [number of] natural numbers" is just another way of saying "[the] set [of natural numbers] is infinite".

So your argument is just that we prove the set of natural numbers is infinite because it is infinite (has infinitely many members).

Proving that a set is infinite is the same as proving that it has infinitely many members.

So it is question begging to assume the set of natural numbers has infinitely many members when that assumption is just another way of saying what you want to prove.

But you can look up actual proofs that the set of natural numbers is infinite.

Yes, he is prominent, therefore natural to refer to him.

Works both ways. It would be better if I had said that. The philosophy of mathematics needs for there to be mathematics to philosophize about and developments in mathematics do inform philosophy; and mathematics is liable to being philosophized about by philosophers.

No clue, I could not find it, I only know that he works on it — Lionino

I'll believe that he has anything when I see it. Especially, does he purport to offer an axiomatic system? I don't recall, but perhaps he rejects the axiomatic method. That would be fine. But there is no comparing, on one hand, an ostensive treatment of mathematics in which one can leave a lot unexplained, unsupported and without the ultimate objectivity of access to mechanical means of checking proofs with, one the other hand, an axiomatization that submits itself to the constraints and discipline required for that ultimate objectivity.

I’m wondering what a thread on mathematics is doing on a philosophy forum. — Joshs

The philosophy of mathematics is a rich area.

(1) Unfortunately, cranks, who are ignorant and confused about the mathematics post incorrect criticisms of the mathematics, from either a crudely conceived philosophical or a crudely imagined mathematical perspective. That calls for correcting their misinformation about the mathematics itself.

It is great to challenge classical mathematics, but a meanginful challenge needs to not misrepresent that mathematics. Otherwise the effect is inimical to knowledge and understanding of the subject.

(2) And sometimes people post questions about mathematical subjects that have bearing on philosophy, such as about infinities, incompleteness and computability. The debate on realism v nominalism has as one of its major battlegrounds the ontological status of mathematical objects, especially infinitistic ones. And some may think that questions in epistemology are informed by such things as the incompleteness theorem and the unsolvability of the halting problem.

Brouwer v Hilbert itself is one of the very great debates in the history of the philosophy of mathematics, carried on by two mathematicians.

/

Meanwhile, one could also ask what are threads on such things as the U.S. presidential election, Gaza, and candy bars doing in a philosophy website. (Don't get me wrong, I am in no way saying those should not be in this website. Very much I say live and let live.)

Says the guy who tried arguing Cardinalities don't have size yet they do, as per the theorem I produced to prove you wrong. Since some Cardinalities are greater than others, we can say that some infinities are larger or even smaller than others. That you got your ass handed to you by someone suffering from "dunning-kruger" — Vaskane

You are egregiously and flagrantly putting words in my mouth.

I never said cardinalities don't have size.

But I'll say now that cardinalities are sizes.

Two sets are equinumerous iff there is a bijection between them.

The cardinality of a set is the cardinal number with which the set is equinumerous.

'the size of the set' and 'the cardinality of the set' are synonymous.

And we say that two sets have the same cardinality iff they are equinumerous.

And you have it backwards:

The original poster claims that it is contradictory to say that there are different infinite sizes. I have been saying that it is not contradictory to say that there are different infinite sizes. And I have been saying that in set theory it is easy to prove that there are different infinite sizes and indeed that some infinite sets are larger than other infinite sets.

It is amazing that you reversed it completely to characterize me as saying the opposite of what I have been saying.

/

There was no "ass handing" though you like the tough talk sound of that.

/

just goes to show you've got a lot to learn, but I'm happy to correct you any time pal. — Vaskane

I am continually overwhelmed by how much I don't know and could learn. But with you what I have learned is not about mathematics or philosophy.

I'm happy to correct you any time pal. — Vaskane

I'm happy to be corrected any time I am incorrect.

Or does it prove that every T has a "natural" example of a true and unprovable sentence, like the strengthened finite Ramsey theorem in Peano arithmetic? — Michael

That is a good question. We know that, for example, PA and set theory are such Ts. But, putting aside the ambiguity of 'natural' and assuming a general informal sense of it, I don't know whether it holds for every qualifying T.

If all it proves is that every T has the true and unprovable sentence "this sentence is true and unprovable" then it seems vacuous. — Michael

"This sentence is true and unprovable" is not the sentence we prove is not provable in T.

Rather, "This sentence is not provable" is the sentence we prove is not provable in T and we prove that it is a true sentence.

Don't forget that the predicate 'provable' can be emulated in T, but the predicate 'true' cannot be emulated in T.

I don't know what you mean by "vacuous" here. G is a sentence of arithmetic. It makes a certain true claim about natural numbers. Granted, the particular claim it makes about natural numbers is probably not of interest to anyone. But that's not the point. Rather, the point is that there is no recursively axiomatized and consistent system for basic arithmetic that is complete and thus, for any given such system, there are true sentences about the natural numbers that are not provable in the system. Moreover, we can then see that there are infinitely many such true and unprovable sentences. Moreover, we then see that it is possible that some of the sentences about arithmetic that are of interest to us might be undecidable (neither provable nor disprovable) in the system. Moreover, hastened by the previous point, we do go on to show specific sentences that are of interest that are undecidable. That leads to the work showing that there is no algorithmic method for solving Diophantine equations, which is not just of interest but is basic to mathematics, even basic to high school algebra, especially for any lazy teenager like me who ever wondered, "Isn't there a step by step procedure I could use to solve any possible equation that I might be asked to solve, so I wouldn't have to think over all these problems but instead could just apply the procedure?" Moreover, we are then led to showing the undecidability of profound and fundamental questions such as the axiom of choice in ZF and the continuum hypothesis in ZFC. Moreover the techniques used in the incompleteness prove lead to the profound find that there is no solution to the halting problem, etc. And to top all of that, the P v NP problem may be the most economically valued in mathematics, as solution to it would have vast ramifications for computing and business; and incompleteness informs us that it is possible that P v NP does not have a solution (though, granted, there are a lot of people who do think it does have a not yet discovered solution).