For example, in set theory, we have a defined predicate 'is an natural number' (I'll use 'B'). So theorems such as:

Ax(Bx -> (x is even or x is odd))

And in the intended interpretation, 'B' maps to the set of natural numbers that is a subset of the universe for the model. But for another model, it might be a different universe, and 'B' might map to a different set from the set of natural numbers.

Or consider first order PA.

In the intended interpretation, the universe is the set of natural numbers, so the quantifier ranges over the natural numbers. But there are non-standard models (not just for the language but even of PA), so the quantifier ranges over a set very different from the set of natural numbers.

It is only by per a model that the domain of discourse is definite, and so it is only per a model that it is definite what the quantifier ranges over.

We're not talking about models here, though. We're talking about domains of discourse. — fishfry

model theory is not relevant to this conversation as I understand it. — fishfry

The question was:

wouldn’t the statement or conclusion «there is no set of all sets» be all-inclusive in one way or another if it really is true? — Sunner

"all inclusive in one way or another" is not definite. My choice was to give a mathematically definite framework for the the question. I surely do not presume to guarantee that I know what the poster in particular had in mind, but I responded correctly vis-a-vis the best way I know to make the question mathematically definite.

in general, there is an operation of unrestricted complement. — fishfry

Not in set theory (as I see you agree). And (just to be clear) in class theory, only with sets, not proper classes.

The complement of the set {1,2,3}, within set theory, is the collection of all sets that are not {1,2,3}. That complement is a well-defined collection, but it's not a set. — fishfry

It is a proper class (as you mention also) in class theory. In set theory, it doesn't exist. Clearly, the context of my posts was set theory.

I looked up Domain of Discourse on Wikipedia, and they did indeed say a domain of discourse is a set. I assumed they were mistaken, and were simply using "set" in its everyday, casual meaning, without regard for the issues of set-hood versus proper classes.

So I agree that if I looked it up, I'd find at least one source, namely Wiki, that claims a domain is a set. I just think they're wrong, and gave many examples to show why. — fishfry

Wikipedia is unreliable for mathematics (and other subjects). But it happens to be correct on this matter. And you don't need to wonder whether it's just a fluke of Wikipedia. Look in any textbook in mathematical logic or model theory. Or any PDF book or class notes on the Internet. The universe (aka 'the domain of discourse') for a model is a set.

any model of set theory has as a set, not a proper class, as its domain of discourse. For any model, the universal quantifier ranges over the members of the domain of discourse, which is a set.
— TonesInDeepFreeze

You are saying that when I make a statement such as, "Every set has a powerset," I am really saying:

1. I assume ZF is consistent.

2. By Gödel's completeness theorem, if ZF is consistent it has a model, which is a set.

3. The powerset axiom is implicitly quantifying over that set. — fishfry

No, I am definitely not saying that.

Phrases such as "quantifier ranges over" are not definite. To pin them down to a definite mathematical formulation, we turn to the mathematical logic. The method of models gives us definite formulations. A quantifier ranges over a universe. But what is that universe? Well, a universe is the carrier set for a model. So, per any given model, the quantifier ranges over the universe of that model.

In what I said, there is no need for an assumption that set theory has a model or is consistent. There is a crucial between (1) a model for the language of a theory and (2) a model of the theory:

(1) M is a model for a language L iff [fill in the definition here, in outline: M is a non-empty set together with a mapping of the non-logical symbols to (elements of the universe, n-ary functions on the universe, and n-rary relations on the universe)]

(2) M is a model of a theory T iff (M is a model for the language of T & every theorem of T is true in M)

Of course, if a theory is inconsistent, then there are no models of that theory. But whether or not a theory is consistent, there are models for the language of the theory.

To mention that a quantifier ranges over a universe per a model requires only (1) and no consideration whether any given model is or is not a model of the theory.

set theory can not prove its own consistency. — fishfry

Yes, if set theory is consistent, then set theory does not prove that set theory is consistent. Nothing I've said contradicts that.

The claim that every set has a powerset is true whether or not set theory has a model. All that is required is the axiom of powersets. — fishfry

No, a sentence is true or not depending on what model we're looking at. Truth is defined per models. The power set axiom is true in some models but false in other models (as you agree). It is true in any model of set theory (since set theory includes the power set axiom) but it is false in other models (ones that are not models of set theory).

Indeed, "Every set has a powerset" is NOT a semantic claim; it's a syntactic one. It follows from the axiom of powersets. There are models lacking the axiom of powersets where the claim is false. — fishfry

A sentence is an uninterpreted syntactical object. Aside from what the word 'claim' means, a sentence is interpreted per models.

And the phrase "models lacking the axiom of powersets" doesn't make sense. Models don't have axioms. Rather, axiomatizations of theories have axioms. What you might mean is "models in which the power set axiom is false".

And "every set has a powerset" is just an English way of saying the power set axiom.

Perhaps we're arguing about syntactic versus semantic domains. — fishfry

I have never read of a "syntactic domain". I don't know what you mean by it.

On the other hand, we can always define unary predicate symbols. For example, in set theory, we can have the predicate symbol 'G' defined by

Gx <-> [fill in the requirements for x being a group]

Then we have universally quantified conditionals:

Ax(Gx -> P)

Note the quantifier ranges over the universe, but it happens that the formula it applies to is a conditional in which x being a group is the antecedent.

So that is a relativization of P to groups.(That's a simplification. Relativizations are recursive so that P and its subformulas are themselves relativized.]

For example, we define a unary predicate symbol 'L' and 'V':

Lx <-> [fill in the requirements for x being constructible]

Vx <-> x=x

So the relativizations such as:

Ax(Lx -> P)
read as "P holds for constructible sets".

and

V = L
for
Ax Lx

And per a given model, 'L' will map to a subset of the universe, and 'V' will map to a subset of the universe (and if the model is a model of "Vx <-> x=x" then V maps to the subset of the universe that is the universe itself).

that impressively buzzword-compliant paragraph — fishfry

I didn't use the terminology to impress anyone with buzzwords. And they are not buzzwords. They are terminology of mathematics.

there is no mention of the domain of discourse. So again, none of this is relevant. — fishfry

It's about models. A model is a domain of discourse along with a function on the nonlogical symbols. So where I mentioned a model, there is a domain of discourse associated with that model. And the paragraph was not meant to address the original question, but rather to give an idea of how notions of proper classes as models are (as I hope I recall correctly) reducible to the syntactical approach of relativizations.

You're perfectly correct that ZFC is consistent if ZF is, but what has that got to do with the conversation? — fishfry

It was added merely as an illustration of an important theorem that comes from relativizations.

There also is the notion of proper classes as models, or more specifically, inner models. However, I think (I am rusty on this) that when we state this formally, it actually reduces to the syntactical method of relativization, so that when we say L is an inner model of set theory, we mean something different from the plain notion of a model. If I recall correctly, roughly speaking, relative to a theory T, saying 'sentence P is true in "class model" M' reduces to: In the language for T, we define a unary predicate symbol 'M', and P relativized to M is provable in T. So, for example, when considering the consistency of the axiom of choice relative to ZF, we find that the axiom of choice is true in the constructible universe L ("L is a model of AC"), which, in one way of doing this, reduces to: In the language of set theory, define a unary predicate symbol 'L', then we show that AC relativized to L is a theorem of ZF. So if we have a model D of ZF, then the submodel that is D intersected with L (the intersection of a set with a proper class is a set) is
a model of ZFC. That entails the consistency of ZFC relative to the consistency of ZF. But I am rusty here, so I may be corrected.

There are set theories in which classes are formalized — fishfry

Right. But even with those theories, the domain of discourse for a model for the language of the theory is a set.

Even a class theory such as NBG has only models that have a domain of discourse that is a set.

Moreover, if we tried to allow a proper class to be a domain of discourse, we'd get a contradiction:

For example, suppose we are doing model theory in a class theory in which there are proper classes. Okay, so far. Now suppose U is a proper class and, for simplicity, we have a language with just one nonlogical symbol. And let R be the relation on U that, per the model, is assigned to the nonlogical symbol.

Then we have the structure <U R>. But then, unpacking the ordered tuple by the definition of tuples (such as Kuratowski), we get that U is a member of a class, which contradicts that U is a proper class.

If you look at textbooks in mathematical logic, model theory, and set theory, you will see that without exception the definition of a model stipulates that its domain of discourse is a set.

Note: I put strikethrough there to accommodate the following post:

Well, we are quantifying over the collection of all sets. — fishfry

Not formally. Formally, any model of set theory has as a set, not a proper class, as its domain of discourse. For any model, the universal quantifier ranges over the members of the domain of discourse, which is a set.

On the contrary, domains of discourse are often truly universal. They're just not always sets. — fishfry

In mathematical logic, a domain of discourse is a set. You may look it up anywhere.

The unrestricted complement of a set always exists. It just may not be a set. — fishfry

The context of my remark was set theory. In that context, there is no operation of absolute complement but only relative complement.

A domain of discourse is a set.
— TonesInDeepFreeze

I'm afraid I can't agree. — fishfry

The official, formal definition of a 'model' is that the domain of discourse is a set.

Your question is insightful. You're thinking along the right lines. But there is no set of all things not in the domain of discourse.

We take only relative complements of sets:

df. T\S = {y | yeT & ~yeS}

That's fine. The set of all things in T but not in S.

But we don't have:

{y | ~yeS}

That is, we don't have the set of all things not in S.

In set theory, given any set S, there does not exist the set of all sets that are not in S. Because if there were, then the binary union of S with the set of all sets that are not in S would be the set of all sets:

thm: ~ExAy(yex <-> ~yeS)
proof: Suppose ExAy(yex <-> ~yeS). Then it would be {y | ~yeS}. Then S u {y | ~yeS) = the set of all sets.

A domain of discourse is a set. And there is no set of all sets that are not in that domain of discourse.

So we might be tempted to say there are more things outside the domain of discourse than in it. Except, since there is not a set of all things outside the domain of discourse, we can't even give the totality of things outside a domain of discourse a cardinality even to use expressions such as "more".

df. x is countable iff (x is one-to-one with a natural number of x is one-to-one with the set of natural numbers).

As far as I can tell, that is different from the everyday sense, since the everyday sense would be that one can, at least in principle, finish counting all the items, but in the mathematical sense there is no requirement that such a finished count is made.
— TonesInDeepFreeze

No, not in my neck of the woods. The everyday sense would be that one could, in principle only, count an infinite series of elements/units/items for all of eternity yet to come and still never get to finish. — javra

Your neck of the woods is a fantasy place. People in everyday life don't take 'countable' to mean "one could, in principle only, count an infinite series of elements/units/items for all of eternity yet to come and still never get to finish."

I'd like to see just one person at your local supermarket in your neck of the woods who says anything like, "My understanding of what 'countable' means? Oh, it means that in principle you could count an infinite number of things for eternity but still not finish."!

I may not be a mathematician but I can take care of my own bank-account via numerations of various sorts just fine — javra

No one doubts that you are a grown up person who can do numerical reckoning just fine. The point is that you don't know anything about the mathematical notions that you have referred to in this thread. And I don't mean your philosophical notions, but rather the specific mathematics you have also referenced. Being able to balance a checkbook is nice, but it's not an informed understanding of the mathematics you've commented on.

It should not still be needed to say:

The notion that some infinite sets are countable is a special mathematical notion. It is technical and has a rigorous formal definition. There is no layman's version of it.

Oh come on! How captious can a person get?

'countably infinite' and 'countable infinity' are tantamount to each other.

'countable infinity' though is less apt, since there is no object that is infinity. Rather there are different things that are infinite.

And lately you wrote:

“the ‘non-mathematical’ countably of infinity” — javra

'countably of infinity' is not even English.

To move past your ridiculously captious objection that I said 'countably infinite' (a coherent notion, used as a favor to you) rather than 'countable infinity' (an unclear notion since there is no object that has the name 'infinity'), instead I'll couch using only 'countable infinity':

You said there is a layman's (or whatever synonym of 'layman' you used) notion of countable infinity. Then you asked what "“the ‘non-mathematical’ countably of infinity” would mean to the general audience (in other words, presumably, a layman audience).

And my point stands, there is no layman's notion of 'countably of infinity' (let alone that it's not English), nor layman's notion of 'countable infinity', nor (put better) layman's notion of 'countably infinite'.

/

Sorry, but I have better things to do that to spend more time in addressing such replies. — javra

So your excuse for not dealing with the point that there is no layman's notion of a countable infinity (better put, of countably infinite) is that I used such a slight variation in phrasing.

Meanwhile, there's a trail of other falsehoods, blatant misconceptions and nonsense you've posted, and I explained your errors. As well as your posts in this thread are an impenetrable morass of your ersatz undefined terminology, with various ersatz undefined qualifiers popping in and out and out and in again. Any fault for you not being understood is yours, not your readers.

You again blatantly misunderstand what I was saying. — javra

No, I clearly see what you actually posted. In earlier posts, you mentioned that there is a layman's notion of countably infinite. Then later you asked what that could mean. And I replied to the effect that that question is your problem alone, since indeed there is no layman's notion of countably infinite.

Ask any person at a busy street corner what their notion of counting the infinite is. Here are the three possible answers you will get:

"Huh?"

"You can't count infinity. Everybody knows that."

"I'm just trying to catch a cab here. Do you know any good Thai restaurants uptown?"

/

There's a bunch more written by javra that I'd like to address, but I'm out of time now.

But that it makes sense to conceive of any infinity composed of discrete items as "actual" rather than as "potential" (this in Aristotle's usage of these terms within this context - rather than what we understand by these term today) is not something that, for example, is amicable to mathematical proofs. — javra

It doesn't invite mathematical proof because 'actual' and 'potential' are not mathematically defined terms.

Meanwhile, formal set theory does not use the term 'is actually infinite' but instead plain 'is infinite', which is rigorously defined. And set theory is made of rigorous, formal, objectively verifiable, indeed machine-algorithmically checkable mathematical proofs.

On the other hand, the notion of 'is potentially infinite' has not, as far as I've ever found, been given a formal definition, let alone a system in which it used. Instead, it is an informal notion that is thought to be captured by (but not defined in) certain systems. Though, countenancing adoption of such systems raises questions about how much they can prove of mathematics, their own intuitive strengths and weaknesses, and their complexity in formulation and ease or difficulty in using.

Might be your added in snide insult. — javra

[said to Real Gone Cat]

Because javra is never snide, you see.

So, when the conceptual grouping (to not irk mathematicians by saying "set") of all natural numbers is taken to be a potential infinity it is still taken to be an infinity - else an infinite grouping - just not one that claims to be complete or else whole. Here, one can contrast the conceptual grouping of all natural numbers with - to keep thing as simple as possible - with the conceptual grouping all natural numbers that are even. There will be a one-to-two correspondence between them: for every one even natural number in the grouping of even natural numbers there will be two natural numbers in the grouping of all natural numbers. When both groupings are taken to be compete wholes, then the grouping of even natural numbers will contain a lesser cardinality than (more precisely, half the cardinality of) the grouping of all natural numbers contains - with both groupings yet being infinite. But when both groupings are taken to be never-complete, then for ever one item added to one grouping there will likewise be one item added to the other, and this without end. Such that one cannot compare the cardinality of infinities in each grouping, other than by affirming that they are both infinite in the same way. — javra

That is such an inpenetrable mess that it would be a task to unsort it all. But a couple of points:

If we said that there are half as many even numbers as natural numbers, then we can also say there two-thirds as many even numbers as natural numbers, and three-fourths as many even numbers as natural numbers, ad infinitum. For example:

0 2 1 4 6 3 8 10 5 12 14 7 ...

Between every odd number, there are two even numbers.

Moreover, the notion that there are half as many even numbers as there are natural numbers induces that there are infinite subsets ("groupings" or whatever you call them) of the the natural numbers in smaller and smaller size ad infinitum. We would have "half as many even numbers as natural numbers", "one-third as many multiples of three as natural numbers", "one-fourth as many multiples of four as natural numbers", ad infinitum.

Detractors of set theory are put off by the fact that an infinite set has the same cardinality of certain of its subsets. Okay. But then we would ask, "So what is your axiomatic alternative?" But the notion of infinite subsets of the naturals in infinitely descending chains of smaller and smaller cardinality is itself utterly unintuitive. You see, at least set theory does preserve the most basic and most intuitive notion of even everyday mathematical thought: Sets are the same size if there is a one-to-one correspondence: sheep and counting stones.

/

As to your "quantified mathematical infinites", "metaphysical infinities", etc., I would suggest that instead of getting vocabulary all mixed up with mathematics, you could stipulate terminology such as:

q-infinite for "quantified mathematically infinite"

m-infinite for "javra's personal metaphysical notion of infinite"

etc.

And perhaps you'd be so gracious as to provide crisp definitions of each.

That would at least show some respect to the people reading your posts by allowing it to be clear in which of the different contexts you are claiming.

I shouldn't point at anyone but you can raise your hand if it's you. — Mark Nyquist

I don't have "a magical view of how mathematics is physically done", so you can leave me out.

But I thought you might have some particular mathematicians or philosophers in mind. Or do you have a magical view of people existing that don't actually exist?

A point goes from itself into segments — Gregory

No it doesn't. There are points and there are segments between points. A point doesn't "go from itself" to something else.

foundations/set theory Tones is the resident expert — jgill

I'm not an expert.

It would be helpful if the philosophy of mathematics was ungraded to address the problem of abstract concepts. — Mark Nyquist

I don't know what 'ungraded' means there, but there many many library shelves worth of articles and books in the philosophy of mathematics on the subject of abstractions and concepts vis-a-vis questions of existence and truth.

It seems some hold a magical view of how mathematics is physically done. — Mark Nyquist

Who?

does the mathematician’s specialized definition of countability thereby take precedence over what layman understandings — javra

No. It just needs to be clear what the context is.

the two Wiktionary definitions of infinity — javra

As I mentioned, one of those is claimed as a mathematical definition. But it is not.

“mathematical” notion of “countability” — javra

Whatever your questions about it, it would be best to start with knowing exactly what it is.

df. x is countable iff (x is one-to-one with a natural number of x is one-to-one with the set of natural numbers).

As far as I can tell, that is different from the everyday sense, since the everyday sense would be that one can, at least in principle, finish counting all the items, but in the mathematical sense there is no requirement that such a finished count is made.

Must I express “non-mathematical” for every term I use in attempts to define a certain species of concepts regarding infinity? — javra

Best would be to state that you are using your own vocabulary as adapted from various everyday senses, then to state your definitions, and not just in an ostensive manner, or listing of cognates, or blurry impressionistic mentions using more terminology that is itself undefined.

But then, what on earth would “the ‘non-mathematical’ countably of infinity” signify to a general audience?! — javra

Indeed! You are the one who claims to represent a layman's non-mathematical notion. It's a safe bet that no one unfamiliar with set theory or upper division mathematics has any notion of all of a countably infinite set. There is no layman's notion of this. So it's silly trying to represent it.

Can we at least mutually understand a differentiation between “quantifiable infinity” which can thereby be addressed in the plural and “nonquantifiable infinity” which cannot thereby be properly addressed in the plural? — javra

Define 'quantifiable infinity' and 'unquantifiable infinity'. The comments you then added are not definitions.

In sum: You seem to want to investigate notions of infinity in non-mathematical senses or contexts. Fine. But then you'd do well to leave mathematics out of it if you don't know anything about the mathematics.

I have instead used “mathematical infinities” in layman’s terms from the get-go — javra

You said you were distinguishing the mathematical notion from a metaphysical notion. You didn't say anything about the mathematical notion being a layman's notion. Anyway, what layman's notion would that be? Which laymen? There is not a distinct layman's notion about infinite sets as they occur in mathematics.

And your claim is belied by a passage such as this:

As in, the infinity of real numbers is infinity #1, the infinity of natural numbers is infinity #2, and the infinity of transfinite numbers is infinity #3. Each of these three infinities is in turn other than the infinity of surreal numbers, for example, which on this list would be infinity #4, making a total sum of four infinities that have been so far addressed. — javra

The differences in cardinalities among infinite sets is not a notion a layman has even ever heard of. (Let alone surreal numbers.)

Yes, my bad for not understanding beforehand how use of these terms would be strictly understood by those with a mindset such as your own. — javra

Gotta admire the knack that goes into turning a retraction regarding terminology back around as a sarcastic dig such as "mindset such as your own". What mindset would that be? The mindset of someone who happens to know how the terminology is actually used in the subject under discussion.

insult — javra

It's not much of an insult to point out that someone is throwing around mathematical terminology without knowing what it means and that they know virtually nothing about the subject while spouting opinions about it nonetheless.

Or as a great filmmaker said:

https://www.youtube.com/watch?v=9wWUc8BZgWE

'P' for 'is an appearance'

'F' for 'is known in a filtered way'

'U' for 'is known in an unfiltered way'

'r' for 'our action'

(1) Ax(Fx -> ~Ux) premise

(2) Ax(Px -> Fx) premise

(3) Ur premise

(4) Therefore, ~Pr

That's valid.

You left out the premise Ax(Fx -> ~Ux). In other words, you did not make explicit that if something is known in a filtered way, then it is not also known in an unfiltered way.

The theorem is:

~ExAy yex

"It is not the case that there exists an x such that every y is a member of x."

or, in context, "There is no set of which every set is a member."

That is not a definition. That is a statement that there does not exist a set having a certain property (the property of having every set as a member).

But that theorem is also equivalently:

Ax~Ay yex

"For every x, it is not the case that for every y, y is a member of x."

or, in context, "For every set, it is not the case that every set is a member of it."

That also is not a definition, but it is a universal quantification. For any domain of discourse for a model of the languge, the quantifier ranges over all members of that domain of discourse.

And, yes, that domain of discourse is a set. So, naturally, and fair enough, one might ask, "But then isn't that domain of discourse the set of all sets?"

The answer is 'no'. Given any domain of discourse, it is a set, but we don't stipulate that every set is a member of it. Indeed, we can't do that, since there is no set that has every set as a member.

Moreover, using a given set theory (for example, ZFC), we cannot, using only that particular set theory, specify a particular model (nor domain of discourse for the model) of the theory (per the second incompleteness theorem). But, for example, using a theory such as ZFC+"exists an inaccessible cardinal", we can specify a model (with a domain of discourse) of ZFC (but not of ZFC+"exists an inaccessible cardinal"); and that model would be a set, but still not a set that has every set as a member.

In sum: The universal quantifier ranges over some domain of discourse that is a set; but that set itself does not have every set as a member. So when informally we couch the universal quantifier as "For all sets", to be more accurate, we actually must mean, "For all sets in the domain of discourse" (and, as mentioned, that domain of discourse does not have every set as a member).

I made a mistake in this thread. I didn't know it then, but I know now, that I used the word 'onto' in a way that is not standard English, which got tangled into a larger disagreement.

I said that the statement "There is a universal set" is not, onto itself. a contradiction.

I should have said that the statement "There is a universal set" is not, in and of itself, a contradiction.

The correct English there is not 'onto itself' but 'in and of itself'.

It seems that the other poster might have thought I meant 'onto' in the sense of a surjection, which I did not mean and would not even make sense in that context. But I don't blame the other poster on that particular point, since it was my mistake in English.

However, my substantive point, as I explained it clearly in other passages not with the word 'onto', stands:

ExAy yex

(read in context as, "There is a set of which every set is a member").

is not a contradiction, but it is inconsistent with the axiom schema of separation.

What is your definition of 'all-inclusive' in this context?

The theorem is:

~ExAy yex

"It is not the case that there exists an x such that every y is a member of x."

or, in context, "There is no set of which every set is a member."

So the cardinality of the integers is "less than" that of the reals.
— jgill

This will be true only when one assumes the occurrence of actual infinities, in contrast to potential infinities. As an easy to read reference: https://en.wikipedia.org/wiki/Actual_infinity From my readings the issue is not as of yet definitively settled - or at least is relative to the mathematical school of thought.

At any rate, the issue of whether infinities (in the plural) are determinate, indeterminate, or neither has dissipated from this thread some time ago. I'm looking to follow suit. Best. — javra

If there are no infinite sets, then there is no set of all the integers nor set of all the reals.

But the observation about them could still hold in the sense of recouching, "If there is a set of all the integers and a set of all the reals, then the cardinality of the former is less than the cardinality of the latter.'

Moreover, in any case, even without having those sets, we can show that there is an algorithm such that for every natural number, that natural number will be listed; but there is no such algorithm for real numbers.

/

Your readings about infinity are grossly inadequate. Among the vast vast number of mathematicians, it is settled that there exist the set of natural number and the set of real numbers; that infinite sets exist. Meanwhile, there are some, but relatively very few, mathematicians who insist on countenancing only "potentially infinite" sets (even though 'potentially infinite' has only a heuristic but not formal mathematical definition). But to say it is not "settled" is as good as a truism in the sense that any question will always have dissenters, but not a substantively correct claim since infinite sets are basic in mathematics, including ordinary calculus.

As in, the infinity of real numbers is infinity #1, the infinity of natural numbers is infinity #2, and the infinity of transfinite numbers is infinity #3. — javra

No, that is plainly wrong.

Both the set of natural numbers and the set of real numbers are transfinite sets.

Why are you making pronouncements on a subject that you know virtually nothing about?

Countable in the sense of: one infinite line and another infinite line make up two infinite lines. — javra

Yes, a set of two lines is a countably (and finite) set.

The set of points in a line is uncountable. And the union of the sets of points in any number of lines is uncountable.

Or: the infinity of real numbers and the infinity of natural numbers and the infinity of transfinite numbers make up three numerically distinct infinities. More technically, make up three numerically distinct infinite sets. — javra

'transfinite' is just another word for 'infinite'. There are many infinite sets, such as the set of natural numbers and the set of real numbers.

As to your notion of "three", there are only two sets you mentioned: the set of natural numbers and the set of real numbers. And there are two kinds of infinite sets, countable ones (such as the set of natural numbers) and uncountable ones (such as the set of real numbers), and every set is either countable or uncountable. But we are not limited to only that bifurcation. Among uncountable sets, we can specify even more adjectives, such as 'inaccessible', etc.

But one trifurcation (among many we could define):

finite

denumerable (countably infinite)

uncountable

Both finite and denumerable sets are countable.

I too wonder how a continuum makes up something discrete
— Gregory

Yea. That appears to roughly sum up the issue. — javra

If it is the sum of the issue, then the sum of the issue is meaningless until 'discrete' is given a definition.

Meanwhile, the continuum is the pair <R L> where R is the set of real numbers and L is the standard ordering on R.

If there is a problem with that, then it awaits a clear statement of the problem.

Are the infinities of natural numbers and of real numbers two different infinities? Or are they the same nonquantifiable infinity? — javra

The cardinalities of the set of natural numbers and the set of real numbers are both infinite cardinalities, but not the same infinite cardinalities.

'nonquantifiable'. What is your mathematical definition?

It's fine to philosophize about mathematics. But it's silly to philosophize about it when you know virtually nothing about it.

It's fine to work out one's own metaphysical notions and try to make them not have conflicts. But then those are your notions, not notions of mathematics, so of course when you mix them together then you can get conflicts.

If your point boils down to the observation that mathematics handles notions in ways that conflict with the way you handle the notions, then, yes, of course, that is fully granted.

Yet the same issue results: if a unique line is so determined by any two points on a plane (in the sense just provided above) how does one then commingle this same stipulation with the fact that that which is being so determined is - at the same time and in some respect - infinite and, hence, does not have some limits or boundaries in any way set by its determinants. Here, the two point on a plane do not set the limits or boundaries of the line's length - despite setting the limits or boundaries of the unique line's figure and orientation on the plane. Again, once the line is so determined by the two points on a plane, it is fully fixed or fully set; hence, fully determined in this sense. But going back to the offered definition above, this would imply that "the line, aka that determined, has it limits or boundaries fully set by one or more determinants" - which it does not on account of being of infinite length. — javra

That is yet another variation on conflating the mathematics with personal undefined terminology.

If only you would carefully read the mathematical treatment of 'line' (either as an undefined primitive of axiomatic geometry, or a defined terminology of geometry developed set theoretically), 'length' and 'bounded'.

Then not mix up those definitions with your own personal undefined notions.

1. (uncountable) endlessness, unlimitedness, absence of a beginning, end or limits to size.
2. (countable, mathematics) A number that has an infinite numerical value that cannot be counted. — https://en.wiktionary.org/wiki/infinity

1. is not the mathematical definition.

2. is not the mathematical definition, and it is in error by claiming to be so.

/

It's fine to make whatever arguments you want about a non-mathematical use of 'infinite' in metaphysics, but it is a disaster to mix that up with the mathematical definitions.

Mathematics is a special subject matter that defines terminology in a special way. It is not at all to be taken that mathematical usage is the same as either everyday usage or usage in non-mathematical areas such as metaphysics.

The mathematical context is not the same as your personal metaphysical context. Be clear what context you are in at any given point in a discussion. Otherwise, we get yet more incoherent discussions that devolve into even greater incoherence.

countable as a unit on account of having some limits or boundaries — javra

You're using the word 'countable' differently from the definition of the word in mathematics. So, of course, confusion will ensue.

A mathematical infinity (in contrast to metaphysical infinity) is not limited or bounded in only certain respects and is thereby countable (metaphysical infinity is not limited or bounded in all possible respects and is thereby uncountable - and I won’t be addressing this concept). — javra

If the context is mathematics, then usually the notion of 'infinite' is referenced per set theory.

Of course, we are free to philosophize and use terminology in a non-mathematical way, but throughout your post, you mix mathematical terminology with your own personal meanings, whatever they may be. That is an invitation to confusion at the very onset.

To keep things straight, at least as to the mathematics itself, here is what mathematics provides:

In set theory, there is the defined adjective 'is infinite'.

Definitions:

A set is finite if and only if there is a one-to-one correspondence between the set and a natural number.

A set is infinite if and only if the set is not finite.

A set is countable if and only if (there is a one-to-one correspondence between the set and a natural number, or there is a one-to-one correspondence between the set and the set of natural numbers).

A set is uncountable if and if the set is not countable.

A set is denumerable [aka 'countably infinite'] if and only if (the set is countable, and the set is infinite).

Theorems:

There exist finite sets.

There exist infinite sets.

There exist denumerable sets.

There exist uncountable sets.

So, contrary to your assertion, it is not the case that every infinite set is countable.

A geometric line, for example, is limitless in length but not in width (technically, it has 0 width, which is a set limit or boundary). — javra

No, it doesn't have a 0 width. It just doesn't have a width at all.

An infinite set, as another example, is limitless in terms of how many items of a certain type it contains but is limited or bounded in being a conceptual container of these definite items. — javra

If that is to have any mathematical import, then it requires mathematical definitions of 'limitless', 'in terms of how many items of a certain kind it contains', 'limited', and 'conceptual container of these definite items'.

However 'bounded' does have a mathematical definition. Per certain orderings upon which we evaluate boundedness, some infinite sets are bounded and other infinite sets are not bounded.

Mathematical infinity specifies a state of being. This state of being is defined by the lack of limits or boundaries. — javra

You say 'mathematical infinity', so I take it you're talking about mathematics. And about mathematics your are incorrect. 'is infinite' is not defined in terms of 'limits' or 'boundaries'.

imply that mathematical infinites are unchanging, unvarying, fully set, and fully fixed — javra

I take it that by 'mathematical infinites' you mean sets that are infinite.

"unchanging, unvarying, fully set, and fully fixed" as you use those words, are not ordinarily mathematical terminology (at least not in this introductory stage of discussion), but looking at mathematics from outside mathematics, and to indicate how mathematics is informally regarded, yes, we ordinarily think of the subjects of mathematics to be definite mathematical objects.

but, then, further entail that they have all their possible limits or boundaries determined? But then this sounds like a blatant contradiction: a mathematical infinity has all possible limits or boundaries set and, at the same time and in the same respect, does not have all its possible limits or boundaries set (for at least some of its possible limits/boundaries will be unset in so being in some way infinite). — javra

A contradiction is a statement and its negation. There is no known theorem of set theory that is a contradiction. Also, your argument fails because you have a false premise, which is that 'is infinite' is defined in terms of 'limit' or 'boundary'.

Anyone have any idea of where the aforementioned goes wrong? — javra

It goes wrong in these ways:

(1) Mixing formally defined terminology of mathematics with your own personal undefined informal terminology. (2) Adopting the premise that 'is infinite' is defined in terms of 'limit', 'boundary' or 'bounded'. (3) Thinking there is some kind of contradiction when only there is a pseudo-puzzle that results from the kind of strawman you set up by mixing terminologies and applying a false premise.

The word 'opposite' has caused problems in this thread because it is not been defined. An undefined word that causes confusion is less than needed. We have defined terminolgy (adding 'compatible' here as analogous to consistent):

* negation of necessary: "not necessary". the negation of Nq is ~Nq.

* negation of possible ("impossible"): "not possible". the negation of Pq is ~Pq.

* inconsistent (contradictories): "imply a statement and its negation". Q and R are contradictories iff together they imply a statement S and also ~S. Most starkly: Q and ~Q are contradictories, and R and ~R are contradictories. Also these are some contradictories:

Nq and P~q

Pq and N~q

* consistent ("compatible"): "not contradictory". In particular, saliently, in this thread:

Nq and Pq are consistent!

The crank is just hard cold plain wrong about it.

Why do I harp on that? Because:

Don't Normalize The Cranks!

The notion that necessary implies impossible is, as reiterated, daft.

So, we would not expect to find this as an axiom:

Nq -> ~Pq

But what about?:

Nq -> Pq

It is an axiom or theorem of just about all of the working systems of modal logic. (I think perhaps it is left off the initial "starter kit" axioms because whether it is needed as an axiom depends on whether it is derivable anyway from the other axioms, which may be formulated in various ways.)

As to 'impossible':

Yes, 'impossible' is the negation of 'possible'.

df. Pq <-> ~N~q

df. q is impossible <-> ~Pq

/

q is is necessary if and only if q is true in all worlds

q is possible if and only if q is true in at least one world

q is impossible if and only if q is is true in no worlds

q is contingent if and only if q is true in at least one world and false in at least one world

No one who knows anything writes:

p is true iff p.

The formulation is:

'p' is true iff p.