"Is there a way to describe various finite quantities without going right to numbers?" - answer, yes there is: we can talk about 'lots', 'few', 'big', 'small' etc. It's the same with infinities. We can talk about 'endless', 'unbounded', for example. Is there anything that is endless or unbounded? I think so. E.g. a race round a standard athletic track that ends only when there is no more track ahead of us. But is that what you are thinking about?

Ad nauseam seems, is rather, related or is the same as ad infinitum.

Next time you're intoxicated by alcohol and feel the irresistable urge to hurl, you're, intriguingly it seems, experiencing the qualitative side to infinity.

Eeeeew! Disgusting! :vomit: = $\infty$

Infinite means in-finito, not finished, never to be finished. What in life never finishes? The finish line can be pulled away from you indefinitely. Indefinitely=infinitely? On can tell a never ending story, play infinite games. The universe goes on forever, as life in it. It never ends. Infinite!

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

Yes, we constantly refer to infinity in various ways without referencing math. Eternal, forever, and immortal are just some of the ways we express infinity.
I'm not sure if this is what you're looking for. Maybe you could explain more?

Spinoza (arguably) has a concept of a qualitative infinity right at the beginning of his Ethics, propositions 1 through 8 introduce it. Infinity as the total lack of limitations or constraints.

while(true){
print("I am an infinite loop")}

Infinite means in-finito, not finished, never to be finished. What in life never finishes? The finish line can be pulled away from you indefinitely. Indefinitely=infinitely? On can tell a never ending story, play infinite games. The universe goes on forever, as life in it. It never ends. Infinite! — AgentTangarine

Little kids love this. No reason philosophers won't too.

The interpretation of infinite I use as a mathematician (complex analysis) coincides with one employed in the physical world: unbounded. Were I to be a set theorist this probably would not be the case.

Is Infinity of running around a track indefinitely differ from running straight and having road added in front of you indefinitely?

Any examples of Infinity other than the suggestion that the universe may go on forever since we can't prove otherwise?

Any examples of Infinity other than the suggestion that the universe may go on forever since we can't prove otherwise? — TiredThinker

You were already given some examples in this thread. How about the Successor function as another example?

Is it a coincidence that the symbols for infinity and zero are similar? You can run around on both of them forever. But so you can on 8. On 4, 6, and 9, you can at least take a break on the the side.The infinite can never be reached. Every time you think you've reached it almost, it has resided to... infinity. So infinity is the unreachable. Can zero be reached? You can pass it on the way to negative, so I had exactly zero dollar, like I have right now. No dollar to be found in my pockets. Can you actually have zero dollars if you don't have them? Can the nothing be realized? You can take out all apples out of the back, but what if we dìvide them up? We can't do that indefinitely. But what about the vacuum? You can break that up infinite times. It consists of infinite points with an infinity of points in between them. Do you take points away when you divide it? Will dividing it up indefinitely leave you with nothing? Or with an infinite collection of points? If we assign real numbers to the points, how can two points touch? By bringing them infinite close to one another, so the decimal numbers coincide? The infinite line of real numbers needs an infinity of the infinity of the natural numbers to describe it. The continuum line can be said to have cardinal number $\aleph _1$, while the collection of natural numbers has cardinal number $\aleph _0$. The two-dimensional continuum has cardinal number$\aleph _3$, and the 3D continuum corresponds to$\aleph _7$.
The subscripts are ordinal numbers and they correspond to the number of times an infinity of the infinite is needed needed to specify the elements, which in the case of discrete points is one (so the ordinal in the cardinal number becomes 0). For the continous line, the number needed is two (ordinal subscript 1), for the 2D continuum it's four (corresponding to subscript 3), and eight will do for 3D continuous space (ordinal number 7, so $\aleph _7$).
Now we can play the same game with with the aleph numbers. For discrete aleph numbers, $\aleph _n$, the natural alephs, we can assign super aleph number $\aleph _{S_0}$, when the number of natural alephs needed is one. When 2 natural alephs are needed, for a continuous line of alephs, we get $\aleph _{S_1}$, for a 2D continuous plane $\aleph _{S_3}$, and for an n-dimensional continuous volume, when $2^n$ natural alephs are needed we arrive at $\aleph _{S_{n^2-1}}$.
Now we can play the game again for hypersuper alephs, $\aleph_{{HS}_n}$. For an infinity of inf... well, ad inf.


So, the infinite can't be reached like zero can't be reached (if you don't include the negation of the positive real). Both can't be reached, and still some try to reach it while others, mostly unwillingly are pushed towards nothing. The desire to reach for the nothing is not so different from the desire to reach for the infinite, though the implementations of this in material life have quite different implications. Life and death, even.

Infinity is just as useless as nothing. In between is where the action is. Are the things in life that never can be reached infinite? Yes. My wife is one of them. And I have to admit, attempts made by pettifoggers and skirlers on their doodlesack never cease to bumfuzzle me.

Everyday examples of infinity. Maybe falling asleep and waking up. It seems that time tic-toc-ed infinitely fast in between. It seems nothing at all exists in between, another example that nothing and infinite have a close, if not intimate connection.

The continuum line can be said to have cardinal number aleph_1, — AgentTangarine

'the continuum' is probably most exactly defined as <R less_than>, but let's simplify here to just say it's R. It is the continuum hypothesis that its cardinality is aleph_1. It is not given or settled mathematics.

The two-dimensional continuum has cardinal number aleph_2, and the 3D continuum corresponds to alelph_7. — AgentTangarine

If by "the two-dimensional continuum" you mean RxR, then it is incorrect that its cardinality is different from the cardinality of R. If by "the 3D continuum" you mean RxRxR, then it is incorrect that its cardinality is different from the cardinality of R. For any natural number n>0, R^n has cardinality equal to the cardinality of R.

The subscripts are ordinal numbers and they correspond to the number of times an infinity of the infinite is needed needed to specify the elements — AgentTangarine

I don't know what you mean by "number of times an infinity of the infinite is needed to specify the elements" but the aleph notation is defined by transfinite recursion on the ordinals. For an ordinal k+1, aleph_k+1 is the least cardinal greater than aleph_k. For a limit ordinal L, aleph_L is the union of {aleph_k | k < L}.

I don't know what you mean by "number of times an infinity of the infinite is needed to specify the elements — TonesInDeepFreeze

If you need an infinity of infinites then aleph is 1. If you need an infinity of them, then 2, etc. The cardinal number of the continuum is defined on one dimension only. You really think the cardinality of the 2 or 3 dimensional space and the 1 d are the same? They're not. Aleph line is 1, aleph plane is 2 and aleph space is 3.

while(true){
print("I am an infinite loop")} — emancipate

lol man, that's not infinite because if computer loses power, your loop will end as well :razz:

.

It is a theorem that card(R) = card(R^n) for any natural number n>0. This is known by anyone who has read a basic textbook in set theory. Just read the proof for yourself.

Moreover, you keep claiming that card(R) = aleph_1, thus precluding that card(R) might be greater than aleph_1 and thus precluding that there might be cardinalities between card(N) and card(R) . That is only the continuum hypothesis, not settled mathematics.

Then you have to reread your proofs. Are you seriously implying that the cardinality of the 2-d continuous plane is the same as that of the continuous line? The cardinality of R is the power set of N. The cardinality of RxR is 2. There are obviously more elements in RxR than in R. That's why the cardinality of RxR is 2, of R it's one, and of N it's zero.

There could be cardinalities between 1 and 2. The fractals, with fractal dimension.

The points on the side of a square have c=1. The square has c=2, while a fractal curve in it has c between 1 and 2.

Are you seriously implying that the cardinality of the 2-d continuous plane is the same as that of the continuous line? — AgentTangarine

I have not mentioned continuousness. I have merely pointed out the utterly well known fact that it is a theorem that card(R) = card(RxR),.

There could be cardinalities between 1 and 2. — AgentTangarine

That is risibly wrong.

There are no cardinalities between 1 and 2. And there are no cardinalities between aleph_1 and aleph_2. However, without the continuum hypothesis, there could be cardinalities between card(N) and card(R), as, without the continuum hypothesis, card (R) could be aleph_x for some x>1, as I pointed out to you over and over in the other thread.

You don't know what you're talking about,.

Came here to say the same thing, but with JavaScript instead.

I have not mentioned continuousness. I have merely pointed out the utterly well known fact that it is a theorem that card(R) = card(RxR),. — TonesInDeepFreeze

Yeah, you have said that infinite times already. It's just not true. There are inf^2 points between 0 and 1. Aleph1. There are inf^4 of them in 2d. Aleph2. In 3d there are inf^8. Aleph3. I'm off. It's boring.

Well, it's not. Between aleph1 and aleph2 lies aleph1.5. A fractal line occupying half the square. There are inf^3 points for this figure.

"inf^2" is not a recognizable notion. Probably what you mean is 2^N.

And you ignorantly, wantonly persist about aleph_1

The claim that aleph_1 = 2^N is the continuum hypothesis.

It's boring — AgentTangarine

It's not so much boring as it is unfortunate that you persist to post misinformation while you won't even bother to look it up on the Internet.

I just mean inf x inf.

I just mean inf x inf. — AgentTangarine

Which has no apparent meaning.

I guess what you mean is SxS where S is infinite.

But then there are more than card(NxN) real numbers between 0 and 1.

card({x | x is a real number between 0 and 1:) = 2^N, which is greater than card(NxN),

You are abysmally confused.

The claim that aleph_1 = 2^N is the continuum hypothesis. — TonesInDeepFreeze

Then you have a different notion of aleph one. The ordinal in Aleph one is just related to how many times the infinity is present. For the naturals inf^1, so aleph 0. For the line inf^2, so aleph1.. For the 2d plane inf^4, so aleph2. For a volume inf^8, so aleph3. For a 1d fractal, say inf^3. So aleph1.6, approximately.

oh you got syntax highlighting and everything :)

Then you have a different notion of aleph one — AgentTangarine

Indeed I have a different notion from yours! My notion is the usual one in mathematics.

You, on the other hand, are unfamiliar with the ordinary mathematical definitions.

The ordinal in Aleph one is just related to how many times the infinity is present. — AgentTangarine

Aleph_1 is the least cardinal greater than Aleph_0. That is the ordinary mathematical definition.

inf^1 — AgentTangarine

Please stop writing 'inf' that way. It doesn't have any apparent meaning.

line inf^2, so aleph1 — AgentTangarine

No, bad Eliza, bad.

Anyway, you said you were off, Eliza. Too bad you didn't mean it.

As if you are a mathematician... Keep it up Cantor! Why I write inf^1 is to highlighten the concept of cardinality. FZ lived 120 years ago. The axiom of choice is based on finite sets. Inf^3 might sound nonstandard in your ears but it's just infxinfxinf. What's so difficult about that? You have to raise 2 to the power 1.6 or something to obtain 3. So the aleph is aleph1.6, approximately. The fractal line occupying part of the square has aleph1,6 multiplied by a factor to account for the degree of covering.

It's gotta be understood though that the cover of a square by a fractal curve is not the same as just cutting out that part of the continuous square. That would imply that the covered piece has the same cardinality as the whole square. The fractal line doesn't cover the whole part. Like a fractal collection of points doesn't cover a continuous part of an interval (it can be piecewise continuous though.).

Anyway, you said you were off, Eliza. Too bad you didn't mean it. — TonesInDeepFreeze

Why? It's not as boring as I thought. There are even alephs0.5 and alephs0.99 or alephs0.01. If there are alephs(sqrt2) remains to be seen. A closed interval on the real line, like [0-1] can be fit infinite times on the real line, so in fact the cardinality of the real line is 1.4. That of the 2d plane is about 2.6. That of the 3d volume is about 3.2. That of a fractal line, plane, or volume, lies between these. Cantor didn't realize this yet.

Coming to think about it, of course aleph(sqrt2) exists. And the aleph for [0-1] is in fact aleph1. Cantor overlooked one infinity! Which only shows his genius! Who can overlook infinity...?

That would imply that the covered piece has the same cardinality as the whole square. — AgentTangarine

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.