So maybe what you mean by "0.1-1" is [0.1 1]?

If you're going to ask me to explain why you have no proof, then it would help for you to use recognizable notation, and would be more courteous too.

Now you say

Between 0.1 and 0.99999.... you use all numbers of N — AgentTangarine

What does "use the numbers mean"?

Do you mean they are all in the range of a function from N to some set? WHAT function? You have not adduced any function. For any given k in N, what is f(k)?

1,2,3,....9999999.... — AgentTangarine

So maybe you just mean that all the natural numbers are in the domain? Well, duh, yeah. So what? The domain is not at issue. It is the range that is at issue. You need to prove that there is a function whose domain is N and whose range is R (i.e. every real number is in the range).

So I ask:

Are you clear that your task is to prove that there is a function whose domain is N and such that every real number is in the range of the function?

Okay:

0.1 connects with 1
0.2 connects with 2
0.3 connects with 3
.
.
.
0.14 with 14
0.15 with 15
.
.
0.53 with 53
0.54 with 54
.
.
0.768 with 768
0.769 with 769
.
.
0.99998 with 99998
.
.
ad 0.9999999999......

Do I have to list all N numbers?

The bijection between R and RxR is not continuous.

This is getting painful to watch. — jgill

This is the same crank whose banning you were lamenting earlier because (he says) he is a physicist and we should be grateful for him being here to educate us... Be wary of unhinged bullshitters confidently throwing around specialist terminology.

Take the square {(x,y):0<x<1,0<y<1} and map it one-to-one to the line {r:0<r<1) by using the procedure implied by the simple example (.329576914..., .925318623...) <-> .39229537168961243...

You can figure it out if you stay off the Xmas grog long enough. Although you are a smart physicist and may be pulling our legs. You and Agent Smith can work this out. It cropped up in the course I used to teach in Intro to Real Analysis.

Hence, there are exactly the same "number" of points in the (section of) the plane and on the unit interval. Same cardinality.
5h — jgill

Yes. I get that. Still... something is nagging. If I map all naturals on [0.1-0.999999] (there you go...) what do I leave out? [0-0.1] seems to contain more numbers than [0.1-0.99999...]:

0.01-0,0999999...
0.001-0.009999...
0.0001-0.00099999...
.
.
.


Which is absurd. Still... On each of the intervals (including [0.1-0.99999...]) you can map the set N directly. Can you break up a continuous interval, like [0.1-1], up in real points? Like 0.1, 0.2, 0.3,...,0.91, 0.92,...,0.110, 0.111,...,0.1222, 0.1223,..., 0.2111, 0.2112,..., 0.24444, 0.24445, ...0,2023432, 0.2023433, ..., 0.655555, 0.655556,.......,0.999999999999....

Every cardinal is contained once. Or is one left out? The diagonal? But how can that be? Say that number is 1000000023432500876.... Isn't that contained in N? But not in R?

If so, can you represent x by 0.5878900... and y by 0.197867....(to name two arbitraries) to map on r?

Or is continuum just continuum, no matter the dimension? No points attached?

@AgentTangarine, @TonesInDeepFreeze

Can you knock it off please? The thread title is "Infinites Outside of Math". You've turned it into a factual discussion about math. That Agent has... spicy takes of dubious correctness... on how infinite sets work is besides the point. Please feel free to take it to personal messages, though.

Agent - would it take that much effort to try to fact check what you've written if you're genuinely interested in it?

Tones - horse is dead now.

0-0.1] seems to contain less numbers than [0.1-0.99999...]: — AgentTangarine

EG: you'd find out that those repeating 9s after the decimal point become the number above in the limit. So if you write 0.999... that already equals 1, or if you write 0.0999... that already equals 0.1.

Can you break up a continuous interval, like [0.1-1], up in real points? Like 0.1, 0.2, 0.3,...,0.91, 0.92,...,0.110, 0.111,...,0.1222, 0.1223,..., 0.2111, 0.2112,..., 0.24444, 0.24445, ...0,2023432, 0.2023433, ..., 0.655555, 0.655556,.......,0.999999999999.... — AgentTangarine

There you've provided at most a countable set of countable sequences, which together turn out to at most countable.

If you're interesting in in these things, I'm sure someone involved would be happy to provide you with study materials.

That's exactly what I have done. And didn't agree with. But it's clear now. Thanks to my opponent. I don't agree with him though.

confidently throwing around specialist terminology. — SophistiCat

That's exactly what I not do. I question it. That's all. I have been given no satisfactory answer though. Jgill came close. But the bijection he prescribes is discontinuous and suffers from the same problem as the number of points on the interval [0.1, 0.999999....). You just can't make points touch, or break the continuum up in points. How many points lay between 0.1 and 0.999999...? Do all these numbers constitute the continuous interval [0.1-0.99999...)? (What is continuous? Undivided.) You can assign natural number to each of these numbers. Is a number left out, by the diagonal argument? If so, isn't that a new natural number, contradicting that you left one out?

What if I asked the question, as a new thread, if the continuum can be broken in parts? It's maybe a math question. Maybe not. It will not alter the essence though of this thread, which is life-related.

I think that would be fine. There's been plenty of discussions before about the nature of the continuum. Just try and keep it away from the mathematical equivalent of pseudoscience.

I think that would be fine. There's been plenty of discussions before about the nature of the continuum. Just try and keep it away from the mathematical equivalent of pseudoscience — fdrake

The "point" is that constructing a continuum out of points seems like a pseudoscience to me. The fact that the line, plane or volume have the same cardinality is because of the attempt to reduce them to points. I'll give it a try later.

This is the same crank whose banning you were lamenting earlier because (he says) he is a physicist and we should be grateful for him being here to educate us... Be wary of unhinged bullshitters confidently throwing around specialist terminology. — SophistiCat

I'm well aware of the situation. :cool: Having communicated with him I believe he has a graduate degree in physics. However, his concerns are with the "physical" or interpretive aspects of the science rather than mathematical descriptions. Even advanced math in QT seems not to have reached axiomatic set theory to a noticeable extent. Kenosha Kid is another genuine physicist. To the best of my knowledge neither of these gentlemen work in the profession at present.

The fact that the line, plane or volume have the same cardinality is because of the attempt to reduce them to points — AgentTangarine

Ignoring the references to cardinality and accepted theory , it does seem like the unit cube has more of "something" than one of its edges.

The bijection between R and RxR is not continuous — AgentTangarine

Really?

Regarding a remark in a previous post, yes I am happy to suggest study materials that would provide understanding of these points:

(1) A bijection between a proper subset, or even the entire set, of terminating decimal expansions (which reduces to the set of finite sequences of natural numbers) and the set of natural numbers is not a bijection between [0 1] (which is represented by the set of denumerable (not terminating) expansions) and the set of natural numbers.

(2) Among the members of [0 1] not represented by terminating expansions are both some of the rational numbers in [0 1] and all of the irrational numbers in [0 1]. Those numbers are not represented in the domain of the bijection.

(3) To answer "What are examples of members of [0 1] that are left out of the domain of the bijection?", we could mention for example: .333.. (i.e. 1/3) and .14159265358979323846264338327950288419716939937510... (i.e. the decimal portion of pi).

Also, the domain of the bijection is only a proper subset of the set of terminating expansions, as for example, not even .01 is a member of the domain. However with a more careful construction, of course, we can show a bijection between the entire set of terminating decimal expansions and the set of natural numbers.

(4) To answer "What natural number is not in the range?", we say that no natural number is not in the range. But so what? Uncountably many members of [0 1] are not in the domain, so it is not a bijection between [0 1] and the set of natural numbers. Of course, any mathematician knows that there is a bijection between the set of terminating expansions and the set of natural numbers. Indeed it is a basic tool of formal languages and computing that the set of finite sequences on a countable set is countable. And, indeed, Cantor proved that there is a bijection between the set of rational numbers and the set of natural numbers, thus, a fortiori, there is a bijection from the set of terminating decimal expansions into the set of natural numbers. But Cantor also proved that there is no bijection between the set of denumerable (non-terminating) expansions and the set of natural numbers. Versions of the proof may be found in any good textbook on set theory.

Suggested study materials (studied in this order).

Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar. I consider this to be the best textbook for learning symbolic logic, which is extremely helpful (for me, crucial) for understanding set theory.

Introduction To Logic - Suppes. This has the best explanation (it is superb) of formal definitions that I have found.

Elements Of Set Theory - Enderton. My favorite textbook on set theory. It is beautifully written.

Axiomatic Set Theory - Suppes. Good as a backup to Enderton.

Philosophy Of Set Theory - Tiles. An overview of the intuitions and philosophy behind various views on set theory and mathematics.

The OP's main intention is to see if there's such a thing as nonmathematica infinity. Infinity, as far as I can tell, is an exclusively mathematical object (magnitude/number). So, to ask whether there's nonmathematical infinity is like asking if there's a red that isn't a color?

However, a question has been asked and our task is to answer it as best as we can. Where do we begin?

I was only there for seconds. It felt like an eternity. — Brazil nightclub fire survivor (2013)

A "few seconds" isn't infinity but this hapless burn victim experienced eternity (infinity). This can't be a mathematical infinity for only a "few seconds" had gone by. Ergo, this must be nonmathematical infinity. We're not out of the woods yet for eternity can only be described mathematically. I'm only offering a possible starting point for an inquiry.

It cause for misunderstanding to say that in mathematics infinity is an object. Granted, there are objects sometimes called 'infinity', such as points on the extended real line. But the more general set theoretic notion of infinity is not of the noun 'infinity' but of the adjective 'is infinite'. Overlooking that distinction often leads to serious misconceptions about how set theory and mathematics treat the subject.

'is infinite', as a set theoretic notion, is a 1-place predicate defined:

S is finite iff there is a bijection between S and a natural number

S is infinite iff S is not finite

S is Dedekind-infinite iff there is a bijection between S and a proper subset of S

Salient about infinite sets is that every Dedekind-infinite set is infinite (this is the other side of the coin of the "pigeonhole principle"), and, with an appropriate choice axiom, every infinite set is Dedekind-infinite.

And mathematical infinity is not just magnitude or number. Yes, there are infinite ordinals and infinite cardinals, which are called "numbers" (there is not, as far as I know, a mathematical definition of 'is a number') but there are infinite sets that are not ordinals or cardinals.

Then there's that hypothetical spaceship we are in, approaching a Black Hole at a constant speed. We sail in and if not annihilated continue our explorations. But a friend is watching our progress from a great distance and sees us going slower and slower until it appears we have stopped before penetrating the anomaly. To him, we will sit there for eternity, gradually inching towards the periphery - like a mathematical limiting process, never quite making it.

(I'm not a physicist, but I recall reading of this. Could be wrong)

This is getting painful to watch. A simple example shows that the "number" of points in the interior of a cube {p=(x,y,z):0<x<1,0<y<1,0<z<1} , is exactly the "number" of points on the line {r:0<r<1}:

1:1 correspondence demonstrated by r=.3917249105... <-> p=(.3795..., .921..., .140...)

Extending these ideas shows the cardinality of R^3 is the same as that of R. — jgill

Does this kind of reasoning really work? Since at the same time,
r=.xyzabc... <-> p=(.xyzabc..., 0, 0)
and
r=.xyzabc... <-> p=(.xyzabc..., 0, 0.0...01)
and so on?

And moreover,
r1=.xyzabc... <-> r2=(.0....1xyzabc...)
r1=.xyzabc... <-> r2=(.0....2xyzabc...)
...

Yep. It's an algorithm for you to guess demonstrated by the example. Standard stuff. :cool:

Tell me, I was never any good at math.

My point is, every point on the line can be mapped to an edge of the cube. What about all the points in the rest of the cube?

The interleaving algorithm looks good to me. But then, you can map those interleaved points onto a single edge as well. This can go on and on in a cycle. Why is there not therefore a paradox?

I'll jump in here for a moment, and jump out if I'm out of place. It appears you're saying that there are more points in a volume (of, say, 1x1x1) than in a 1x1 plane, and more in that plane than in a line segment length 1. Even I know that the cardinality of those sets of points is the same. Or in informal-speak, that they all contain the same number of points. And a simple proof, although probably not completely rigorous - on request. But that wasn't what you were saying, was it?

edit: what hypericin et al said, just above.

Why is there not therefore a paradox? — hypericin

Probably because points have no dimension. The real numbers are complicated and fascinating.

Look, suppose I pick at random a point in the interior of the cube: (.251956... , .629435..., .194735...) Then that point corresponds to a unique point on the x-axis edge: .261529194947...

You just alternate digits. Here, you do it by going the other way: find the point inside the cube corresponding to the point on the x-axis: .381402258639...

The process is continuous in the epsilon-delta sense.

Going back to the OP.

English is a real language. The range of sentences expressible in English is infinite. Is that an infinity in the world?

Of course, these sentences can not be enumerated, at least in the world.

I guess whether there are enumerated infinities in the world depends on the nature of the world. Is space-time closed or open? Is there really a unit length, the plank length? Or do lengths truly map to real numbers?

Well put.
Now that I think about it some more, the argument is convincing.

Outside of mathematics, I believe that some people do experience infinity in mediation, other spiritual practices, and with psychedelics. Also, one can experience the law of non-contradiction transcended, so that all dualities are one.

Here's another, albeit mathematical, way of experiencing $\infty$: Live! If one lives for even just 5 minites, $\infty$ moments from t = 0 to t = 5 mins would have to flow by. The same can be done with distance, mutatis mutandis.

Given a given line segment has exactly the same number of points as any other line segment, no matter their difference in lengths, and given lifespans can be mapped onto lines, we could say that

1. Everyone one of us - infant, child, adult, man, woman, young, old - has the same lifespan ($\infty$, measured in terms of moments/instants). We're kinda immortal mortals. The Greeks, it seems, refused to touch $\infty$ with a barge pole for a very good reason: paradox galore!

2.

I was only there for seconds. It felt like an eternity. — Brazil nightclub fire survivor (2013)

This blaze survivor was experiencing instants/moments ($\infty$) when the fire was raging around him and not the timespan ("seconds").

I propose a distinction betweem absolute and relative infinities.

As a kindergarten child, I couldn't count beyond 10; the numbers greater than 10 are beyond my ken. This, is relative infinity and since it's in fact not an absolute infinity (more on this in a while), it can be treated as a nonmathematical/qualitative infinity.

Now, as an adult, I could go on counting (the natural numbers), but of course I'll never complete the task; this, is absolute infinity and it's mathematical/quantitative.

Qualitative infinity has to do with limits (our own, our tools'); Quantitative infinity is, at the end of the day, a concept to which limits don't apply.

When I see your alias, I always wonder how or why you have chosen it ... Thinking should not be tiresome! :smile: Not if directed correctly and it is controlled, and one doesn't "torture" one's mind by chosing to talk about such subjects as infiniteness! :grin:

Is there a way to describe various infinites without going right to number lines? Anything in real life to reference in terms of the infinite? — TiredThinker

I'm not sure what you exactly mean by "going right to number lines", but if you imply that "infinity" and "infiniteness" are normally connected to numbers (Math), I think that this is too much restricting. There are a lot of things we call "infinite", even if actually they are not, but only very huge in size, from simple to complex: a line, a circle, love, time, space, the Universe (although its infiniteness is still debatable), ... That is, anything the limits of which cannot be determined.

“Two things are infinite: the universe and human stupidity; and I'm not sure about the universe.”
― Albert Einstein

Is there a way to describe various infinites without going right to number lines? Anything in real life to reference in terms of the infinite? — TiredThinker

An abstraction with utility, but not an actuality. All things are finite, including the universe. There has never been any reason to suggest otherwise apart from our inablity to formulate thoughts on the subject.

Anything in real life to reference in terms of the infinite? — TiredThinker

Unbounded spaces e.g. Earth's surface, etc.