Parsing counterfactuals in terms of possible world semantics makes explicit the relation involved in the counterfactual... — Banno

...with respect to our world.

I just needed to finish that sentence to avoid confusion.

If we stipulate that we're using phrases like "there is a possible world such that X" to mean "X is logically possible", then that's what we mean when we use those phrases- if you don't like it, too bad. — busycuttingcrap

It seems you misunderstand the situation. I have no problem with you using the modal language mentioned here. Sure, "there is a possible world..." means ... is logically possible. That's obvious, and not an issue.

The problem is when people like you and Banno, also Kripke, use ontological language to talk about how an individual "exists" and the "existence" of an individual within your modal logic. This is what creates confusion for people. Aristotle set this separation years ago, to combat sophism. The individual exists as primary substance, and is therefore separated out from the logical structures. That is the difference between the subject which serves for predication, and the object which has separate, independent existence.

I'm sorry that you object to people using ontological-sounding language to talk about modality and possibility rather than existence, — busycuttingcrap

The problem is that you use "existence" to talk about something other than existence. What's the sense in that? If you're talking about modality rather than existence, then obviously the appropriate thing to do is not to use "existence". Let me remind you again what you said.

So yes, an individual "exists" in numerous, maybe even uncountable, possible worlds, because there are numerous, maybe even uncountable, logically-possible propositions, predicates, etc that we can say of a given individual. — busycuttingcrap

It appears that you stand corrected. The same individual does not exist in numerous different possible worlds, because if individuals did exist in these possible worlds they would be distinctly different individuals, according to the possibilities proposed. Therefore they would not be the same individual. If they were considered to be the same individual, the law of non-contradiction would be violated.

This is what creates confusion for people. — Metaphysician Undercover

Sure, the first time they hear the phrase "there is a possible world such that blah-blah-blah". Then someone explains it to them, and they're all good. The only problem here is your stubborn insistence that people can't or shouldn't use terms in a way you don't like or agree with. But that's a problem on your end: possible-world semantics works, it is a useful tool, and so logicians and philosophers are going to continue to use it. If you don't like it, you're free to not participate.

There simply isn't any objects in logical possibilities (possible worlds), and nobody actually believes that there is, despite the fact that many people like busycutter, and Banno, argue that there is. — Metaphysician Undercover

In the expression "an individual exists in a possible world", the word "exist" is being used metaphorically, not literally, in the same way that it is being used metaphorically in the sentence "I existed on my desire for vengeance". The problem with a metaphorical language is that meaning depends on context and if the context is vague then the meaning is vague.

The problem is, that if we said "an individual exists in our actual world", are we still using "exists" metaphorically or literally ?

And then again, where does this "actual world" exist. I think it exists in the mind, though others would disagree. But even "the mind" is a metaphor.

IE, an individual exists in a possible world metaphorically, a possible world is a metaphor, exists in our actual world is being used either metaphorically or literally, and our actual world exists either metaphorically in our minds or literally as mind-independent.

“…. Now it may be taken as a safe and useful warning, that general logic, considered as an organon, must always be a logic of illusion, that is, be dialectical, for, as it teaches us nothing whatever respecting the content of our cognitions, but merely the formal conditions of their accordance with the understanding, which do not relate to and are quite indifferent in respect of objects, any attempt to employ it as an instrument (organon) in order to extend and enlarge the range of our knowledge must end in mere prating; any one being able to maintain or oppose, with some appearance of truth, any single assertion whatever.…”
(CPR, A61/B86)

Possible world semantics: amusing to play with, but don’t think for a minute there’s any knowledge to be gained by it. And if there’s no knowledge to be gained, whatever amusement there is, is time poorly lost.

Sure, the first time they hear the phrase "there is a possible world such that blah-blah-blah". Then someone explains it to them, and they're all good. The only problem here is your stubborn insistence that people can't or shouldn't use terms in a way you don't like or agree with. But that's a problem on your end: possible-world semantics works, it is a useful tool, and so logicians and philosophers are going to continue to use it. If you don't like it, you're free to not participate. — busycuttingcrap

Again, you're failing to grasp the issue. The separation between the logical subject and the physical object provides the force for Aristotle's refutation of Pythagorean idealism, commonly known as Platonism. So the fact that logical subjects are not individuals, or particulars, is the premise whereby the illusions of Platonist fantasies can be dispelled.

Take the proposition of the op for example. "We are math". If the "we" of this statement refers to a multitude of physical individuals (conventional usage), then the answer to the question of the op is no, because there is a separation of category between these physical objects referred to with "we", and the logical subjects of mathematics. To say "we are math", when "we" is understood in this conventional way would be a category mistake. But if "we" is understood as a logical subject instead of a collection of particulars, then there is no such category mistake, and Platonism is allowed to flourish. Then there is nothing to prevent "we are math" from being a true proposition.

Of course, it ought to be obvious to you, that "we" is not a proper logical subject, it is vague, ambiguous, and not well defined. So the latter use of "we" ought not be allowed into any logical proceeding because of the ambiguity it brings with it. And this is exactly the problem with your and Banno's use of "individual". You insist on allowing an individual to be a logical subject (above mentioned category mistake), thereby introducing this ambiguous, ill-defined, form of logical subject into your logical proceedings, 'the individual'. I've argued against this practise in many threads on mathematics, where the ill-defined logical subject which is claimed to be a particular entity is called a mathematical object. But this is just a well-known category mistake, which was thoroughly exposed by Aristotle in his efforts to disclose the pervasiveness of sophistry in his time.

So I agree completely with you in your assessment of "your stubborn insistence that people can't or shouldn't use terms in a way you don't like or agree with". But I disagree with your characterization of this being a "problem", in this particular instance. My stubbornness and insistence is well justified and supported by the fact that what you and Banno propose constitutes a well-known category mistake. And this type of behavour, of insisting on allowing such ambiguity into your descriptions of logical possibilities, displayed by you and Banno, has been well documented as the basis for logical sophistry. So my insistence is warranted as well. Therefore my insistence that you use terms in a way consistent with good philosophical practise is not a "problem" at all, but has already been well demonstrated to be the solution to a problem, while your practise is the problem.

In the expression "an individual exists in a possible world", the word "exist" is being used metaphorically, not literally, in the same way that it is being used metaphorically in the sentence "I existed on my desire for vengeance". The problem with a metaphorical language is that meaning depends on context and if the context is vague then the meaning is vague. — RussellA

RusselA, we know that metaphor has its uses. But creating ambiguity in terms which already are well-defined in philosophy, for the purpose of sophistry, is clearly not a good use. That is very poor epistemology.

The problem is, that if we said "an individual exists in our actual world", are we still using "exists" metaphorically or literally ?

And then again, where does this "actual world" exist. I think it exists in the mind, though others would disagree. But even "the mind" is a metaphor.

IE, an individual exists in a possible world metaphorically, a possible world is a metaphor, exists in our actual world is being used either metaphorically or literally, and our actual world exists either metaphorically in our minds or literally as mind-independent. — RussellA

Metaphors do not provide good premises for logical proceedings. That is why we separate out ill-defined things like "particulars", "individuals", "objects", and speculate metaphysically about the existence of these things, rather than allowing them into our logical premises. When we allow ambiguity into the premises, soundness suffers.

If I were an 8 year old and someone told me we are math, I'd think we're either a number (arithmetic) or a shape (geometry). Of course these two fields turned out to be interchangeable (coordinate geometry, courtesy Descartes). I believe I'm a fly.

Metaphors do not provide good premises for logical proceedings. — Metaphysician Undercover

True. In the sentence "Mary exists in a possible world", "exists" means "could exist", so the sentence is incorrect. It should be "Mary could exist in a possible world".

However, if it was a deliberate intention to use "exists" as meaning "could exist", then this would have been a valid metaphorical use of language. Confusing, but valid.

Sorry, but it's entirely legitimate to ascribe the predicate of existence of Mary in a possible world. Why is there so much confusion about counterpart theory or possible world semantics?

Sorry, but it's entirely legitimate to ascribe the predicate of existence of Mary in a possible world. Why is there so much confusion about counterpart theory or possible world semantics? — Shawn

The point I was making, is that in this situation, the predicate "exists", is predicated as a possibility, therefore a possible predication, as is the case in "possible world" language use. "Mary", as the subject, on the other hand, must be given a place in relation to the possible predication. "Mary" does not signify a part of the possibility, the predication is the logical possibility. We can say that "Mary" represents an individual, but we still cannot assign "existence" to this individual without justification, as is the case with all such logical subjects. That it is logically possible that the individual represented by "Mary" does not exist demonstrates that we cannot assign existence to that proposed individual without justification. Only when "Mary" is shown to refer to a real physical individual (substance), can we say that Mary exists.

Otherwise "Mary" just signifies an individual in the general sense, in abstraction. And when "Mary" signifies an individual in the general sense, the logical possibility that Mary exists, takes on a completely different meaning. When "Mary" is not assigned to any specific individual, then the logical possibility of Mary's existence just means that it is possible that there is an existing person named Mary.

I believe I'm a fly. — Agent Smith

And you very well may be. My condolences.

However, you may exist as only a possibility in another philosophical realm where the word "you" can mean annihilation by fly-swatter. Or not. This is serious stuff.

:lol:

Sorry, but it's entirely legitimate to ascribe the predicate of existence of Mary in a possible world. Why is there so much confusion about counterpart theory or possible world semantics? — Shawn

The confusion is not about possible world semantics, the confusion is about the mixing up of metaphoric and literal meaning.

There is no confusion as to what "Mary exists in a possible world" means, as there is no confusion as to what "Mary has a heavy heart", "Mary is down in the dumps" or "Mary is as happy as Larry" mean.

There is nothing wrong with using poetic or metaphoric language, as such words are an integral part of language. The problem arises when poetic and metaphoric language becomes mixed up with language that is intended to be literal, after all, this is a philosophy forum where the meaning of words is important, not a poetry forum.

A possible world may or may not exist. If the possible world doesn't exist, Mary cannot exist in it, so "Mary does not exist in a world that does not exist" is true. If the possible world exists, then it is not a possible world, it is an actual world, so "Mary exists in a world that exists" is true. As a possible world is a modal world, if Mary exists within it, then Mary's existence should also be a modal existence. Therefore it would be better say "Mary may exist in a possible world", "Mary might exist in a possible world", "Mary can exist in a possible world" or "Mary could exist in a possible world".

Your claim was:

according to set theory, all logically possible (consistent) collections exist — litewave

I refuted that claim:

https://thephilosophyforum.com/discussion/comment/766312

Your reply does not refute my refutation, as well as there are other errors in your reply:

First, here are the points in my refutation, most of which you skipped:

(1) It's not even clear how "all logically possible (consistent) collections exist" could be stated as a mathematical statement in set theory.

(2) It's not even clear how "all logically possible (consistent) collections exist" could be stated as a rigorous philosophical principle regarding set theory.

(3) Even if we did have a rigorous statement of such a philosophical principle, it's not a given that it is the consensus of set theorists and philosophers of mathematics that it is true.

(4) There are infinitely many statements formalizable in the language of set theory that state the existence of sets with given properties but such that it is consistent with set theory there exists such a set, but it is not a given that set theorists endorse that any given one of those sets exists.

You did reply to that point, but your reply fails, as I'll explain later in this post.

(5) Set theory does preclude certain kinds of sets that otherwise it would be consistent to say they exist. In particular, the axiom of regularity precludes certain kinds of sets that otherwise would be consistent to say they exist.

Since you did not reply to that, I'll add: I surmise that naively (informally, intuitively) most set theorists' notion of 'set' includes that sets are not members of themselves, and that, more generally, every set has a minimal member. That is especially witnessed as the axiom of regularity is a standard axiom, which is especially relevant since you say that naive set theory is "elaborated upon" by axiomatizations such as ZFC. This is a point blank refutation of your claim that "according to set theory, all logically possible (consistent) collections exist", as indeed both the naive notion of sets and the standard axiomatizations exactly preclude the existence of certain kinds of sets that would not be inconsistent to assert their existence otherwise. That point cannot be skipped and it alone decisively refutes your claim.

Added errors in your reply:

(6) When we say 'set theory' in the last 100 years, we mean one of the axiomatized set theories, not naive set theory. So saying 'according to set theory' would not be understood as 'according to naive set theory'.

(7) 'naive set theory' is ambiguous, as it means different things to different people and in different contexts. In any case, it is not usually understood as "it just says that a set is a collection of objects". And even if that were the meaning of 'naive set theory', it would not follow that according to naive set theory "all logically possible (consistent) collections exist" (whatever exact claim that might be).

The most salient sense of 'naive set theory' is the inclusion of the informal principle that to each property there is the set of all and only the objects that have that property. Or, formally, the axiom schema of comprehension: For every formula P in which 'y' is not free, we have E!yAx(xey <-> P). That schema is famously inconsistent. So taking 'naive set theory' in that sense is of no use to your claim.

And taking 'naive set theory', as you mention, as merely meaning an informal understanding that is nevertheless formalized in a theory such as ZFC also is of no use to your claim, since such theories exactly preclude the existence of certain kinds of sets that would not be inconsistent to assert their existence otherwise.

(8) It is not clear what you intend with "a set is a collection of objects". Is that intended as a definition of 'set'?

As far a quite informal notion, it's perhaps okay though it merely shifts from 'set' to 'collection'. But it is also widely viewed that 'set' is an informal notion that is not defined, especially as you say that naive set theory is explained by axiomatic set theory. Also, formally, in class theory (a variation of set theory), we make take 'is a set' as a primitive (undefined) predicate. But also, even in set theory, we may define 'is a set' as follows:

df. x is a set <-> ((x=0 or Ey yex) & Ez xez)

Or put informally: a set is not a urelement and not a proper class.

Also, the understanding of sets has been refined greatly since early definitions such as Cantor's. Especially we countenance the iterative concept. (For an excellent argument see Boolos's essay "The Iterative Conception of Set" in the great volume 'Logic, Logic, and Logic'.)

Edit: Also, ordinary set theory take sets to be hereditarily sets.

Returning to the main point: Such notions do not entail that all logically possible sets exist. We already saw that the axiom of regularity exactly disputes that all logically possible sets exist, but also it is just a non sequitur to jump from a definition of 'is a set' to asserting the existence of "all logically possible collections". From the definition of 'unicorn' we don't infer that unicorns exist (presumed counterfactual), let alone that all possible unicorns exist. From the definition of 'extraterrestrial creature' we don't infer that extraterrestrial creatures exist (unkown), let alone that all possible extraterrestrial creatures exist. From the definition of 'dog' (known fact but not inferred merely by definition), we don't thereby infer that dogs exist, let alone that all possible dogs exist.

(9) You mention Hamkins's multiverse view. But a multiverse view decidedly contradicts naive set theory (in the sense of the schema of comprehension). As to naive set theory in your sense of informal understanding anticipating formal axiomatization, the multiverse view and your remarks about it actually hurt your claim that "all logically possible collections exist". Indeed, the multiverse notion suits my argument: It depends on what particular theory is considered. For example, if we adopt CH as an axiom, then there does not exist a set whose cardinality is strictly between the cardinality of the set of naturals and the cardinality of the set of reals. But if we adopt the negation of CH as an axiom, then there do exist sets whose cardinality is strictly between the cardinality of the set of naturals and the set of reals. There is not in set theory itself a universal principle that "all logically possible collections exist" (even setting aside, as I've mentioned, that it's not clear how we would rigorously articulate such a principle).

And you say, "every axiomatized set theory selects a limited collection of possible (consistently defined) sets." But that contradicts your own claim that "according to set theory, all logically possible (consistent) collections exist":

A set theory (a) proves the existence of certain sets, and certain kinds of sets, having certain properties, and (b) disproves the existence of certain sets, and certain kinds of sets, having certain properties, and (c) for certain kinds of sets, leaves neither proven or disproven that they exist. So, even the most common set theories preclude the existence of certain sets and leave unanswered whether other certain kinds of sets exist. So, again, it is not the case that "according to set theory, all logically possible (consistent) collections exist" (let alone, as mentioned, it is not clear how "all logically possible (consistent) collections exist" could even be exactly stated in the language of set theory or even as a rigorous philosophical claim).

(10) You say, "For me, as long as such a set is consistently defined, it exists."

First, using the method of formal definition, there is no such thing as an inconsistent definition. (See many a book in mathematical logic for explanation of the method of formal definition, while I think Suppes's 'Introduction To Logic' is the best one on the subject.)

Second, and most telling, that for you something is the case about sets doesn't imply that "according to set theory" it is the case. You overstated. You jumped from your own glib view to a sweeping claim about set theory itself.

/

It is true that set theorists have different perspectives: Some favor a "wider" view of sets; that our theory should allow a more "liberal" acceptance of kinds of sets. And other set theorists favor a "narrower" view of sets. But, again, to understand those perspectives as rigorous requires a lot more work. And, again, since there are such disagreements, it is not the case that "according to set theory, all logically possible (consistent) collections exist".

I believe I'm a fly. — Agent Smith

Look what happened to this guy!

I think I can clarify a lot by addressing this part of your post:

(5) Set theory does preclude certain kinds of sets that otherwise it would be consistent to say they exist. In particular, the axiom of regularity precludes certain kinds of sets that otherwise would be consistent to say they exist.

Since you did not reply to that, I'll add: I surmise that naively (informally, intuitively) most set theorists' notion of 'set' includes that sets are not members of themselves, and that, more generally, every set has a minimal member. That is especially witnessed as the axiom of regularity is a standard axiom, which is especially relevant since you say that naive set theory is "elaborated upon" by axiomatizations such as ZFC. This is a point blank refutation of your claim that "according to set theory, all logically possible (consistent) collections exist", as indeed both the naive notion of sets and the standard axiomatizations exactly preclude the existence of certain kinds of sets that would not be inconsistent to assert their existence otherwise. That point cannot be skipped and it alone decisively refutes your claim. — TonesInDeepFreeze

As I said, by "set theory" I mean set theory in the most general sense. I supposed that this is what is commonly understood as naive set theory, but to clarify, I mean the concept of a set or collection of objects that is elaborated in all consistent axiomatized set theories together. In other words: if a set is included at least in one consistent axiomatized set theory, then such a set exists. As I said, there are uncountably many axiomatized set theories. ZFC set theory is just one of them. So I include also sets that are members of themselves and sets that don't have a minimal member, as long as these sets are consistently defined. Elsewhere you stated that there is no such thing as an inconsistent definition, so let me give you an example of an inconsistently defined set: an empty set that has one member.

Also, the example with CH may be clarifying. If there is a consistent axiomatized set theory that includes a set with cardinality between the cardinalities of the set of naturals and the set of reals, then such a set exists, simply because it is included in a consistent axiomatized set theory (that has an axiom that is the negation of CH). It's no problem that in a different axiomatized set theory, which has CH among its axioms, such a set doesn't exist. A set theory with CH as an axiom simply selects only certain sets among which a set with a cardinality between naturals and reals is not included. Every consistent axiomatized set theory selects certain sets, and by "all logically possible (consistent) collections" I mean sets or collections selected by all consistent axiomatized set theories together - that's the multiverse view in set theory. If a set is included at least in one consistent axiomatized set theory, then it exists in the set-theoretic multiverse.

The 1986 remake as the classic?

I grieve for my people.

No one could predict that by "set theory" you meant your own personal concept (a concept that is not at all what people ordinarily mean by "set theory"). So my original point stands, per an ordinary understanding of "set theory", it is not the case that "according to set theory, all logically possible (consistent) collections exist", as well as we don't have a suggestion as to how that would be rigorous mathematical or even philosophical statement.

But now we have a revision: According to littwave's ersatz concept of set theory (not one ordinarily understood as "set theory"), call it 'L-theory', all logically possible (consistent) collections exist (setting aside how that would be a rigorous or mathematical or even philosophical statement).

I mean set theory in the most general sense. I supposed that this is what is commonly understood as naive set theory, but to clarify, I mean the concept of a set or collection of objects that is elaborated in all consistent axiomatized set theories together. — litewave

And that is not what anyone means by 'naive set theory'. So your notion is not set theory and it's not naive set theory. I'll call it 'L-theory'. Moreover, you skipped my point that you enlist a multiverse view, yet a multiverse view clearly contradicts L-theory, as indeed a multiverse view is the very opposite of L-theory.

In other words: if a set is included at least in one consistent axiomatized set theory, then such a set exists. — litewave

What are all the axiomatized set theories? There is no definitive list, and there is no conceptual limit. For that reason alone your notion is fatally vague. And what does "included" mean? Does it mean that a theory proves a theorem of the form E!xAy(yex <-> P) so that we can name that unique set? That would entail that there are only countably many sets (since there are only countably many names). Otherwise, in what sense does a set "exist" in your framework? Exist as a member of some model of some set theory (as, again, it is not known what theories are allowed to be considered for this purpose)? Or does it mean that there is a proof that there are certain classes of of sets having a certain properties? All and only properties expressible as a formula of set theory? Or sets that are subsets of model theoretic universes? Or what? 'exist' has at least two set theoretic meanings: To be named by a constant symbol in a definition or to be a member of a universe of a model of set theory.

Without providing something a lot more mathematically or philosophically substantive than you have, your notion is arm waving in a "dorm room".

So I include also sets that are members of themselves and sets that don't have a minimal member — litewave

And that is not set theory. And, for consistency then, you have to exclude the ordinary set theories, including ZFC, since they prove that there are no such sets. Your notion is not set theory; it is, at best, L-theory.

Also, how would you know that such a stew that provides that "all sets exist if they exist in at least one theory" is itself consistent?

you stated that there is no such thing as an inconsistent definition, so let me give you an example of an inconsistently defined set: an empty set that has one member. — litewave

That's not a definition of a set. That's a definition of a property, viz. the property of having no members and having a member.

A definition of a set is of the form (where 'c' here is a constant, so it's actually a definition of a constant):

x = c <-> P

where 'P' is a formula in which 'c' does not occur, and at most 'x' occurs free, and we have a previous proof of E!xP.

A definition of a property (actually a predicate symbol) is of the form:

Fx <-> P

where 'F' is a predicate symbol, and 'P' is formula in which 'F' does not occur, and at most 'x' occurs free.

Moreover, any theory that has the axiom of regularity makes inconsistent this formula.

Ex xex

since the axiom of regularity implies ~Ex xex.

And, since you place no limit on how we may extend any of the set theories, we can have many theories that preclude that there exist sets of certain kinds, including sets with cardinality strictly between the cardinality of the naturals and the cardinalities of the reals, or inaccessible cardinals, etc. Indeed, we can have consistent theories that, by dropping some axioms and adopting others, consistently preclude any kind of set we want.

Your thinking about all of this is thoroughly half-baked.

A set theory with CH as an axiom simply selects only certain sets among which a set with a cardinality between naturals and reals is not included. — litewave

No, you skipped what I pointed out:

A set theory (a) proves the existence of certain sets, and certain kinds of sets, having certain properties, and (b) disproves the existence of certain sets, and certain kinds of sets, having certain properties, and (c) for certain kinds of sets, leaves neither proven or disproven that they exist. So, even the most common set theories preclude the existence of certain sets and leave unanswered whether other certain kinds of sets exist. So, again, it is not the case that "according to set theory, all logically possible (consistent) collections exist" (let alone, as mentioned, it is not clear how "all logically possible (consistent) collections exist" could even be exactly stated in the language of set theory or even as a rigorous philosophical claim). — TonesInDeepFreeze

A theory does not "select only certain sets". There are properties such that a consistent set theory leaves unanswered whether or not there are sets having that property. So it is utterly vague to say what sets a given theory "selects" beyond those we know that the theory proves to exist. And even worse to hand wave that there is some universal criterion of existence based on a union of an undetermined set of theories. And again, since ZFC+CH is now ruled out by your requirement that there exists a set of cardinality strictly between the naturals and the reals, your vague L-theory cannot speak for set theory itself, since set theory itself is not settled as to CH.

I mean sets or collections selected by all consistent axiomatized set theories together - that's the multiverse view in set theory. — litewave

I gather that you mean "in at least one" and not "selected by all" [emphasis original]. Using 'all' as you do confuses your own point.

But more importantly: Please cite where Hamkins says anything tantamount to "[all sets exist that are] selected by [at least one] consistent axiomatized set theory."
/

You've been corrected on a number of points: What set theory is. What naive set theory is. That L-theory is a sufficiently definite notion - mathematically or philosophically. That L-theory doesn't work out the way you think it does. What a definition of a set is as opposed to a definition of a property of sets. And I wonder from what exact specific passage you infer that Hamkins holds that all sets exist that are "selected" by at least one consistent axiomatic set theory.

/

Every consistent first order theory is extended by a consistent maximal first order theory, in the sense that every sentence in the language is a theorem of the maximal theory or its negation is a theorem of the maximal theory. But it's doubtful whether that would help L-theory: (1) The maximal theory is taken as an infinite union of theories with a sequence, as at each step the consistent alternative is included, but if both are consistent, then either (but not both, of course) may be included. This is nothing like the handwaving of L-theory. (2) From (1), there is not just one maximal theory. Not a definitive union of theories by which it is determined what sets exist or not. (3) The maximal theory is not necessarily recursively axiomatizable, so, to the extent that L-theory might be hoped to provide a formal theory, the theorem of maximal theories does not in and of itself provide a recursively axiomatizable theory.

↪universeness The 1986 remake as the classic?

I grieve for my people. — Banno

The fly-people?

And that is not what anyone means by 'naive set theory'. So your notion is not set theory and it's not naive set theory. — TonesInDeepFreeze

Well, Wikipedia article on naive set theory just mentions the general concept of a set as a collection of objects, and related general concepts like set membership relation, equality, subset, union, intersection etc. There is no requirement that a set cannot be a member of itself or that a set must have a minimal member, which you tried to impose on naive set theory.

What are all the axiomatized set theories? There is no definitive list, and there is no conceptual limit. For that reason alone your notion is fatally vague. — TonesInDeepFreeze

Sure, the number of axiomatized set theories is uncountable. I see no reason to set any arbitrary limit to them.

And what does "included" mean? — TonesInDeepFreeze

Look at Hamkins' paper on multiverse in set theory. As an example, he talks about a world defined by axioms of ZFC + CH and a world defined by axioms of ZFC + negation of CH. He claims that both worlds exist in the set-theoretic multiverse. That means that a set with cardinality between naturals and reals doesn't exist in the (ZFC + CH) world but it exists in the (ZFC + negation of CH) world and thus exists in the multiverse. I cannot speak about set theory with the same rigor as Hamkins or you but this seems to be what I am trying to say.

since ZFC+CH is now ruled out by your requirement that there exists a set of cardinality strictly between the naturals and the reals — TonesInDeepFreeze

ZFC+CH is not "ruled out", it just defines a part of the multiverse, a part in which there is no set with cardinality between naturals and reals.

First, you falsely put words in my mouth:

There is no requirement that a set cannot be a member of itself or that a set must have a minimal member, which you tried to impose on naive set theory. — litewave

Contrary to your mischaracterization of my remarks, I didn't opine as to naive set theory regarding the axiom of regularity, but rather as to set theory:

set theory does preclude certain kinds of sets that otherwise it would be consistent to say they exist. In particular, the axiom of regularity precludes certain kinds of sets that otherwise would be consistent to say they exist. — TonesInDeepFreeze

And that was before you ever mentioned 'naive set theory'.

In a following post, I said:

I'll add: I surmise that naively (informally, intuitively) most set theorists' notion of 'set' includes that sets are not members of themselves, and that, more generally, every set has a minimal member. — TonesInDeepFreeze

And that is true. I don't impose any particular axioms or principles onto naive set theory, since the rubric 'naive set theory' is not definite enough to say what its axioms (if any) are. Whether described as 'naively', 'informally' or 'intuitvely' (let alone formally), the vast number of set theorists regard sets as not being members of themselves.

I said what obtains in set theory and naively, informally, or intuitively with most mathematicians; I didn't say what does or does not obtain in an undefined 'naive set theory'. For that matter, I don't have any need to even mention 'naive set theory' other than that you brought it up.

So, please do not put words in my mouth again.

Wikipedia article on naive set theory just mentions the general concept of a set as a collection of objects, and related general concepts like set membership relation, equality, subset, union, intersection etc. — litewave

First, never take Wikipedia as authoritative on mathematics.

Second, please read the very article you cite. Wikipedia mentions three notions of 'naive set theory':

"Naive set theory may refer to several very distinct notions. It may refer to

Informal presentation of an axiomatic set theory, e.g. as in Naive Set Theory by Paul Halmos.

Early or later versions of Georg Cantor's theory and other informal systems.

Decidedly inconsistent theories (whether axiomatic or not), such as a theory of Gottlob Frege[6] that yielded Russell's paradox, and theories of Giuseppe Peano[7] and Richard Dedekind." [Wikipedia]

So, even according to Wikipedia, it is not a given that we take 'naive set theory' as you stipulate.

Look at Hamkins' paper — litewave

That is a dense article. (And I certainly don't trust that you have sufficient technical background to accurately represent anything it says.) So, please quote the specific passages you contend claim that all sets exist that are "selected" by at least one consistent axiomatic set theory. Tell me the exact formulations you have in mind that Hamkins mentions in his own words. And please cite where Hamkins says there is a set with cardinality between the naturals and the reals and that that is decided by the fact that ZFC+~CH is consistent.

ZFC+CH is not "ruled out", it just defines a part of the multiverse, a part in which there is no set with cardinality between naturals and reals. — litewave

You switched from my point:

since ZFC+CH is now ruled out by your requirement that there exists a set of cardinality strictly between the naturals and the reals, your vague L-theory cannot speak for set theory itself, since set theory itself is not settled as to CH. — TonesInDeepFreeze

The point there, as I mentioned previously, yet you still don't get it, is that you can't speak for "set theory" when set theory in and of itself does not determine CH.

Rather than address refutations head on, and then gracefully admit that there are the errors in your posts I've mentioned, you instead put words in my mouth, cite a complicated and highly technical article (of which I see no reason to think you have even the basics for a background to understand the article), and then also misconstrue my point about CH.

Keeping a running tab (otherwise, the discussion does not progress healthfully as instead the items get buried by your continuing to ignore them or evade them:

You've been corrected on a number of points: What set theory is. What naive set theory is (as you didn't even read the very article about it that you cited). That your own personal glib hand waving is a sufficiently definite notion - mathematically or philosophically. That you conflate your own personal glib hand waving with what set theory and naive set theory actually are. That your own personal glib hand waving doesn't work out the way you think it does. What a definition of a set is as opposed to a definition of a property of sets. And I wonder from what exact specific passage you infer that Hamkins holds that all sets exist that are "selected" by at least one consistent axiomatic set theory. And now, falsely putting words in my mouth about regularity, and evading my point about CH by misconstruing it. And you present no even remotely definite sense of what you mean by 'exist' - whether by syntactical definition, or by membership in a universe for a model, or otherwise. And you show no recognition of the distinction between defining a particular set versus a claim that there exist sets having a certain property - which is a distinction crucial to allowing this subject to be discussed intelligibly.

So, please quote the specific passages you contend claim that all sets exist that are "selected" by at least one consistent axiomatic set theory. Tell me the exact formulations you have in mind that Hamkins mentions in his own words. — TonesInDeepFreeze

on page 2: "In this article, I shall argue for a contrary position, the multiverse view, which holds that there are diverse distinct concepts of set, each instantiated in a corresponding set-theoretic universe, which exhibit diverse set-theoretic truths."

And please cite where Hamkins says there is a set with cardinality between the naturals and the reals and that that is decided by the fact that ZFC+~CH is consistent. — TonesInDeepFreeze

The example with CH is in part 7: "Case study: multiverse view on the continuum hypothesis". See it for yourself. I tried to express the gist of it in my previous post.

And that quote is not at all tantamount to saying that we take as existing all the sets that are proven to exist according to different set theories. As it stands in and of itself, it could be mean the exact opposite - that there is no single universe that determines the totality of the sets. That what exists depends on each individual theory. He says there is are distinct concepts; yet your notion is that there is a unified concept that is arrived upon by collecting from all the distinct concepts, or from the union of what is proven among an uncountable number of theories. It remains to study Hamkins further to see exactly how his notion works, but at least from that passage, we cannot infer that it works as you claim it does.

But the more basic point is that, no matter your own views (or even Hamkins's, for that matter), it is not the case that "according to set theory, all logically possible (consistent) collections exist".

Set theory does not say that. There is no theorem of set theory that is anything like that. It's not even seen how it could be formulated in set theory.

Again, you conflate your own personal preference about how you wish to (mis)understand set theory with set theory itself. For that matter, I doubt you even know what set theory is.

And please don't dodge. Please say exactly what passages in part 7 you regard as saying that there is a set with cardinality between the naturals that is decided ('settled' in context) in general (not just per particular theories) by the consistency of ZFC+~CH.

And no retraction from you that you falsely put words in my mouth by claiming that I said naive set theory must obey the axiom of regularity. That you falsely put words in my mouth and then elected not to retract when it was pointed out witnesses intellectual dishonesty and poor faith.

And no admission by you that you omitted the key passages in the Wikipedia article that you cited yourself, as those passages directly support my point that naive set theory is commonly understood to be an informal framework that is informally inconsistent (and formally inconsistent if we formally spelled out the comprehension principle). Again, that's witness of your lack of sincerity to understand the subject matter on which you so freely opine and claim.

And no recognition by you that you missed my point, said more than once, about ZFC+~CH.

Back to the very start: It is not correct that "according to set theory, all logically possible (consistent) collections exist". I've demonstrated that in several ways. Apparently you are not willing to admit to even such a basic mistake.

And by now you've been corrected on these points:

* What set theory is. (I doubt you even know what the term 'set theory' actually refers to.)

* What naive set theory is (as you didn't even read the very article about it that you cited).

* That your own personal glib hand waving is not a sufficiently definite notion - mathematically or philosophically.

* That you conflate your own personal glib hand waving with what set theory and naive set theory actually are.

* That your own personal glib hand waving doesn't work out the way you think it does.

* What a definition of a set is as opposed to a definition of a property of sets.

* That you have not shown a determining passage from Hamkins regarding your own notions vis-a-vis his.

* That you have not supported that Hamikins says what you claim he does about CH.

* That you falsely put words in my mouth about regularity, and without subsequent retraction, and evaded my point about CH by misconstruing it. And you present no even remotely definite sense of what you mean by 'exist' - whether by syntactical definition, or by membership in a universe for a model, or otherwise.

* You show no recognition of the distinction between defining a particular set versus a claim that there exist sets having a certain property - which is a distinction crucial to allowing this subject to be discussed intelligibly.

self-deleted. not needed.

But the more basic point is that, no matter your own views (or even Hamkins's, for that matter), it is not the case that "according to set theory, all logically possible (consistent) collections exist".

Set theory does not say that. — TonesInDeepFreeze

The union of all consistent axiomatized set theories does.

. Please say exactly what passages in part 7 you regard as saying that there is a set with cardinality between the naturals that is decided ('settled' in context) by the consistency of ZFC+~CH. — TonesInDeepFreeze

Hamkins regards the world defined by ZFC+~CH as equally real as the world defined by ZFC+CH and that both worlds exist in the multiverse. So that's how the continuum hypothesis is settled.

"Since we have an informed, deep understanding of how it could be that CH holds or fails, even in worlds very close to any given world, it will be difficult to regard these worlds as imaginary."

And no retraction from you that you falsely put words in my mouth by claiming that I said naive set theory must obey the axiom of regularity. — TonesInDeepFreeze

I didn't put words in your mouth. I thought that when you used the word "naively" you referred to naive set theory.

But if ZFC is the context of your notions, at least a kind of "base" theory for your uncountably many theories — TonesInDeepFreeze

I mentioned ZFC just as an example of axiomatized set theory.

Yes, I deleted that post, as I realized it failed to track with what you did say.

The union of all axiomatized set theories does. — litewave

That is an inconsistent theory. And even then it doesn't say what you say it does. You keep skipping the point that there is no apparent way to put your claim in the language of set theory.

And earlier you said ZFC is just an example. But now you premise on the union of all theories, of which ZFC is obviously one. But ZFC says no set is a member of itself, while you say that there are sets that are members of themselves, while you take existence from the union of all the theories. Now, granted, you might say that you are taking only from the individual theories the theorems that assert existences, not those that deny existences. It would take at least a bit of thinking to figure out how that would actually work, but at least it is overwhelmingly clear that it is nothing close to what set theory says.

Hamkins regards the world defined by ZFC+~CH as equally real as the world defined by ZFC+CH and that both worlds exist in the multiverse. So that's how the continuum hypothesis is settled. — litewave

That is the bare gist of it. And it doesn't say what you say it does. So I just repeat what you don't address:

And that quote is not at all tantamount to saying that we take as existing all the sets that are proven to exist according to different set theories. As it stands in and of itself, it could be mean the exact opposite - that there is no single universe that determines the totality of the sets. That what exists depends on each individual theory. He says there is are distinct concepts; yet your notion is that there is a unified concept that is arrived upon by collecting from all the distinct concepts, or from the union of what is proven among an uncountable number of theories. It remains to study Hamkins further to see exactly how his notion works, but at least from that passage, we cannot infer that it works as you claim it does. — TonesInDeepFreeze

I didn't put words in your mouth. I thought that when you used the word "naively" you referred to naive set theory. — litewave

Yet, I did not refer to naive set theory there. And I made clear previously that I was talking about set theory. And, if I now recall correctly, I had a least alluded to the fact that naive set theory is not definitely axiomatized (except usually it is understood to include the principle of comprehension). And in the second instance I made clear that I was talking about a naive, intuitive, informal way of thinking; and I did not in that regard say I was talking about naive set theory; moreover, since set theorists now do not work in naive set theory, it would make no sense to interpret me in the least charitable way possible - that I was talking about set theorists working in naive set theory.

So, please do not read my posts as carelessly as you read about set theory (I have no idea where you got your ersatz notions that you claim to represent "set theory", and you didn't even bother to read the Wikipedia article that you purported refuted one of my points, when actually it supported that point) and then purport that they say things that in fact they don't.

Back to the original point: You are incorrect that "according to set theory, all logically possible (consistent) collections exist". And you make it even worse with your handwaving about unions of theories and appropriating Hamkins (while you know not even the bare basics of the technical exposition). You are incapable of even conceding a single mistake, including the initial one.

The union of all axiomatized set theories does. — litewave


That is an inconsistent theory. — TonesInDeepFreeze

I suppose that you also think that a union of ZFC+CH and ZFC+~CH theories is an inconsistent theory. Yet according to Hamkins the worlds defined by these two theories are parts of a consistent multiverse.

It is as if you took these two statements:

(1) "This ball is red."

and

(2) "This ball is not red."

and concluded that these statements are contradictory. But you didn't notice that these statements are not about the same ball but about two different balls and thus there is no contradiction between them. Same with axiomatized set theories: they define different worlds and thus are not contradictory.