My point was and remains:

Are the infinities of natural numbers and of real numbers two different infinities? — javra

Yes. — Srap Tasmaner

By being 2 different infinities, they are thereby quantifiable as infinities wherein each individual infinity is demarcated from the other by some limits or boundaries. These infinities are thereby countable (in a non-mathematical sense): 2 infinities.

This cannot be so of metaphysical infinity (for reasons I've become tired of repeating).

I mean, they're different in quantity, not quality. They're both cardinal numbers, just of different sizes.

Now there are transfinite ordinals, but you'd have to ask someone else about those.

Why can't you count metaphysical infinity? I assume you only recognize one metaphysical infinity, so haven't you counted it? One.

Whereas metaphysical infinity would be infinite in length, in width, and in all other possible manners. — javra

But such a metaphysical infinity would still have a boundary of its identity because it would be differentiated from what it is not, for example from finiteness or from infinite lines.

I assume you only recognize one metaphysical infinity, so haven't you counted it? One. — Real Gone Cat

Not to be rude but, in this thread, you've made it a habit to assume things I haven't expressed and most often don't believe.

Whereas metaphysical infinity would be infinite in length, in width, and in all other possible manners. — javra

But such a metaphysical infinity would still have a boundary of its identity because it would be differentiated from what it is not, for example from finiteness or from infinite lines. — litewave

This will be true when it comes to it being a concept (a map of the territory). But it cannot be true of it as an ontic occurrence (as the territory itself). The boundaries you specify would nullify the possibility of its occurrence.

Mind, I've made no claim as to whether or not metaphysical infinity can ontically occur - and am intent on leaving this issue open ended. But - as with a) the infinity of nothingness or b) the infinity of at least certain understandings of God (each being a different qualitative version of what would yet be definable as metaphysical infinity) - it is possible for certain humans to conceptualize its occurrence.

Friend, as litewave pointed out, by your own argument, when you name a thing, you are placing a boundary on it. If you can, name two metaphysical identities. Now count them. Two.

Anyway, too many folks are trying to explain to you why you're wrong, but you keep doubling down. It's starting to feel like piling on. I'll keep reading comments, but I think I'm out.

But - as with a) the infinity of nothingness or b) the infinity of at least certain understandings of God (each being a different qualitative version of what would yet be definable as metaphysical infinity) - it is possible for certain humans to conceptualize its occurrence. — javra

Nothingness cannot have an ontic occurrence since it has nothing to occur, and if there were an infinite God he would be different from other objects, for example from us humans, so he would have a boundary of his identity too.

Nothingness cannot have an ontic occurrence since it has nothing to occur, and if there were an infinite God he would be different from other objects, for example from us humans, so he would have a boundary of his identity too. — litewave

OK. So you uphold that the concept (which has been around in the history of mankind for some time) is vacuous. While neither agreeing nor disagreeing with you, I see nothing wrong with that.

Friend, as litewave pointed out, by your own argument, when you name a thing, you are placing a boundary on it. If you can, name two metaphysical identities. Now count them. Two. — Real Gone Cat

As has been typical, there's a reading comprehension problem. By my very own argument, the concept is quantifiable whereas that which the concept refers to is not.

but I think I'm out. — Real Gone Cat

That makes the two of us.

Okay. I certainly don't understand what your stance is on whether or not infinite lines are countable. — javra

Are you talking about a single infinite line being somehow countable? Like the points on the line?

Or are you talking about the set of all infinite lines being countable?

Neither are countable. Countable means this is #1, this next is #2, the next is #3, etc. It means some sort of algorithm for actually counting.

Maybe you are using the word differently. Like "I can be counting on you to do the best you can." Rather than counting 1, 2, 3, ...

Real Gone Cat is a fellow mathematician.

Are you talking about a single infinite line being somehow countable? Like the points on the line?

Or are you talking about the set of all infinite lines being countable?

Neither are countable. Countable means this is #1, this next is #2, the next is #3, etc. It means some sort of algorithm for actually counting.

Maybe you are using the word differently. Like "I can be counting on you to do the best you can." Rather than counting 1, 2, 3, ... — jgill

In presuming that - unlike at least the initial posts of Real Gone Cat - you’re not posting this to have “a good time” at the expense of a poster you assume to be stupid (because laughing at retards is such an admirable trait in today’s world):

I’m sincerely bewildered at the chasm of understanding (your previous unwarranted rudeness aside).

Nowhere did I state either possibility you offer.

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

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.

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.

As it is I'm wanting to bail out. But on the possibility that your latest post was sincere in its questions, I've answered.

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

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

Certainly you have two infinite lines. And each line has the cardinality of the reals. Also, both lines, together, have that cardinality. And so on. But the "three numerically distinct infinite sets" are not really distinct, in that the set of integers is a subset of the set of reals, etc. So the cardinality of the integers is "less than" that of the reals.

The real numbers are quite complicated. And I didn't intend to belittle your efforts, but it sounded to me like you were thinking it is possible to count all lines in the plane through some sort of algorithm.

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.

Ontic determinacy, or the condition of being ontically determined, specifies that which is determined to be limited or bounded in duration, extension, or some other respect(s) - this by some determining factor(s), i.e. by some determinant(s). . . . Mathematical infinity specifies a state of being. This state of being is defined by the lack of limits or boundaries. — javra

Not familiar with that technical term, I googled "ontic determinacy" and found an article on "ontic vagueness"*1. Mathematical Infinity is vague only in the sense that it is off-the-map of real numbers. For example, in Fractal Graphics places where the computer encounters infinities, it stops calculating and renders the area as black, signifying merely "undefined" or "unbounded" or "indeterminate". However, physicists studying sub-atomic particles, also encounter off-the-map Math. Although that aspect of reality is beyond our ability to comprehend or to define, Heisenberg labeled it as an essential feature of the quantum level of reality : "Uncertainty" or or "Indeterminacy".

Unlike ancient mapmakers, who put warnings on the uncharted areas of their maps "here be dragons", Heisenberg merely warned that empirical quantum physics faded into theoretical metaphysics at the margins. So, he defined the ontically indeterminate zones in terms of "the old concept of 'potentia' in Aristotelian philosophy*2" (i.e. metaphysics). Then he bracketed that mysterious unknown zone as "something standing in the middle between the idea of an event and the actual event, a strange kind of physical reality just in the middle between possibility and reality". In Statistics, its called "Probability", and in Metaphysics, it's known as "Potential"

He didn't go so far as to label the state of Superposition as supernatural or unreal or nonbeing. Instead, he placed it in the statistical realm of infinite Possibilities, within which some things are more Probable than others. This is just a different way of looking at what we think of as Reality. So, he rejected the option of throwing-hands-up and declaring "absurd". "The other way of approach was Bohr's concept of complementarity . . . . as two complementary descriptions of the same reality". Two different states-of-being.

Unfortunately, to some of his colleagues that compromise smacked of equating empirical Physics with spooky Metaphysics. Nevertheless, in the 21st century, physicists have been forced to accept the unacceptable position of Overlapping Magisteria. Reality, on the quantum scale, is a vague gray area that is neither fully real & physical, but also not-yet-real (possible) & meta-physical (ideal or mathematical). :smile:

*1. Ontic Vagueness :
A note about terminology: for this paper, I‟m using "ontic vagueness‟ because that‟s been perhaps the most common term in the literature on the subject. "Metaphysical vagueness‟ is probably the better
term (see Williams (2008)b for discussion). Perhaps even better would be to stop talking about
vagueness altogether and just talk about metaphysical indeterminacy.
https://elizabethbarnesphilosophy.weebly.com/uploads/3/8/1/0/38105685/ontic_vagueness_final.pdf

*2. Physics and Philosophy, by Werner Heisenberg, 1958


THE BLACK AREAS OF A FRACTAL ARE INFINITE, HENCE INDETERMINATE

No, I don't think this is correct.

To show that a set has the same cardinality as the set of natural numbers, you only have to show that the elements of the set can be placed in one-to-one correspondence with the set of natural numbers. You don't actually have to complete the set. To show that an infinite set has a greater cardinality, you only have to show that at least one element exists that cannot be placed in the one-to-one correspondence (ala Cantor's diagonal proof). No actual infinities need be assumed.

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.

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.

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.

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.

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.

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.

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.

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?

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.

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, [...] — TonesInDeepFreeze

Hey, though I hope I'm wrong, your forgone conclusions regarding me and what I was addressing, your indignation, and your seeing of red is evident to me.

But so it’s said: There are at least two distinct senses of “mathematical”.

I have nowhere stated - nor to my mind insinuated - that I am addressing infinities as they are defined by schools of mathematical thought, i.e. as they are defined by mathematics.

I have instead used “mathematical infinities” in layman’s terms from the get-go, as I initially thought (mistakenly) the OP made clear by the way terms were defined in relation to each other: in the sense of infinities that can be quantified and that furthermore pertain to quantities, and which are thereby, in this sense alone, mathematical. “Mathematical” in the sense that if a bird can count to ten, then this bird holds a respective measure of mathematical skill - despite this bird having no cognizance of any theoretical underpinnings devised by humans for the quantities it can count. In the sense that any appraisal of quantities, such as 1 + 1 = 2, is a mathematical ability that makes use of mathematical notions - irrespective of how these notions are established (hence, with or without the symbols that we humans use to express 1 + 1 = 2).

The thread was in part because of this placed in the category of “General Philosophy” rather than “Logic and Philosophy of Mathematics”.

Nor do I personally take established concepts in mathematics to the foundational cornerstone of what "infinity" at large can signify.

Notwithstanding, for my part, I have learned from this thread not to term quantifiable infinities that pertain to quantities “mathematical” - nor “countable” for that matter. 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.

I have no interest in addressing the many comments made in your many posts. And presently intend to let you have the last word, laugh, insult, or what have you.

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

Nor do I personally take established concepts in mathematics to the foundational cornerstone of what "infinity" at large can signify. — javra

I hate to forestall this thread's death, but I am curious about this.

I looked back at the OP yet again, the centerpiece of which is this question:

Can mathematical infinities (e.g., geometric lines, infinite sets, and so forth) be ontically determinate? — javra

You're talking specifically about the mathematical versions of concepts in wider (and vaguer) use — and that wider use is what some of us assume lies at the foundation of mathematics, our intuitions about shapes, collections, counting, patterns, all that. I gather it's something like those intuitive, pre-theoretical ideas you really wanted to address, not their mathematical axiomatization.

Which is fine, and I can imagine doing a phenomenology of boundedness and unboundedness, that sort of thing. No doubt that would be interesting.

But there is still something odd about your decision — though maybe I've misunderstood you — to exclude mathematics. After all, we've had a few thousand years now of people thinking about just these things, and some of that thinking is what we call mathematics. The history of mathematics is far messier and various than your grade school textbook led you to believe, precisely because it's the history of people thinking about the sorts of things you've expressed interest in. Mathematics as it is now may strike you as somewhat rigid and narrow, and therefore of no use to you, but it is still a body of serious, rigorous thought about things like the infinite, so even if there's more to say than you can get out of established mathematics, it is surely the natural starting point, not the natural body of work to be excluded.

Maybe this thread would have gone differently if I had asked you directly to explain this:

And my stumbling block is that by defining determinacy as I did in the OP (i.e., having limits or boundaries set by one or more determinants), I run into this stubborn paradox of having to differentate semi-determinacy from what I've so far termed "mathematical infinities" ... which are, again, only partly infinite in some respect while yet being finite in others. — javra

Looking back at our exchange, I realize I hoped that what you're talking about here would become clear as we worked through some examples, but it didn't.

So I still don't have the faintest idea how what you call "mathematical infinities" inserted themselves into whatever you were working on, and why their arrival was such a problem.

If actual mathematics is no use in solving your problem, then presumably these "mathematical infinities" obtruded for non-mathematical reasons; but I can't figure out what sort of non-mathematical problem would drag in a bunch of — as a matter of fact, somewhat recondite, even for math — mathematical concepts.

If you're of a mind, and not burned out on the topic, take another swing at it. It is, after all, a philosophy forum not a math forum. Maybe if you could explain a little more clearly how your problem relates to mathematics without being a mathematical problem, we could make some progress.

If you're of a mind, and not burned out on the topic, take another swing at it. It is, after all, a philosophy forum not a math forum. Maybe if you could explain a little more clearly how your problem relates to mathematics without being a mathematical problem, we could make some progress. — Srap Tasmaner

To properly address this, I believe there first ought to be a commonly understood or accepted, philosophical (rather than one pertaining to established schools of mathematics) differentiation between types of concepts regarding the notion of infinity (again, infinity not as its is mathematically defined but as a general, sometimes philosophical, notion: commonly defined by the absence of (non-mathematically defined) “limits” to that addressed).

I don’t know. Once bitten twice shy. This thread’s issue can to my mind be easily overtaken by a broader philosophical issue and possible underlying stance. Namely, one of whether a) mathematics subsumes all reality (such as, for example, by grounding all of physics and, via further inference, thereby all of physicality … this being one example of what could be termed a “neo-Pythagorean” view) or, else, b) reality holds aspects which can be in part and imperfectly modeled by what we humans have devised - over the long course of history you address - as various schools of mathematical thought.

So, does the mathematician’s specialized definition of countability thereby take precedence over what layman understandings (such as the two Wiktionary definitions of infinity previously provided) of countability are? This on account of stance (a). Or else is the so here termed “mathematical” notion of “countability” a specialized understanding that is subsumed by commonly accepted every day notions of countability in general, namely "the capacity to count quantities"? As would naturally be the case in scenario (b).

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

But then, what on earth would “the ‘non-mathematical’ countably of infinity” signify to a general audience?! To be countable but not mathematical is a bit of a conceptual contradiction.

Yes, I’m a little frustrated, maybe blowing off my own steam. But help me out a bit if you can.

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?

All infinities defined by mathematics would then be stereotypical examples of “quantifiable infinity”.

In contrast, the infinity of nothingness would then be one example of “nonquantifiable infinity” - such as when nothingness is conceptualized to have once “existed/been/occurred” devoid of anything. Those who claim the possibility that before the big bang was nothingness (e.g., https://en.wikipedia.org/wiki/A_Universe_from_Nothing) can be ascribed to implicitly make use of such concept of infinity.

If we can’t conceptualize - and then properly term - this differentiation between species of infinity which humans at large can historically conceive of, then I don’t see much point in further addressing the topic of the OP and, by extension, in answering your inquiry.

And this, in part, because “nonquantifiable infinity” (if this term doesn’t get lambasted as well) can only be completely non-determined and thereby completely indeterminate ontically (though, again, it will be a determinate concept). The OP’s inquiry, however, applies only to those infinities that are “quantifiable” and thereby in some way definite due to some demarcation or other occurring in that being addressed. Furthermore, to simplify the variety of such, the OP limits itself only those quantifiable infinities that are themselves made up of discrete quantities (like a geometric line being made up of discrete geometric points, or an infinite set made up of discrete numbers ).

To further complicate matters, then there needs to be a commonly held understanding of what “to determine” signifies (when the term isn’t used to address psychological processes of mind or states that thereby result). There’s again been much criticism of how I’ve attempted to define it (in short, as “to set the limits or boundaries of” - a standard dictionary definition, spelled out by me in greater explicit detail in the OP for an intended greater accuracy); none of this criticism being constructive in offering any alternative definition.

… I’m not hopeful this can work out, but I’ll check back in some time. Thanks, however, for the offer.

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.

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 — TonesInDeepFreeze

:up:

It would be helpful if the philosophy of mathematics was ungraded to address the problem of abstract concepts. Information does not, can not, exist as an abstract concept. Information always exists as brain state.
Mathematics is mental content and cannot exist except in individual brains that physically exist in the physical present. It seems some hold a magical view of how mathematics is physically done. Yes, lots of good work over the millennia to build on.