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## Thought Club

Started by The Wizard Joseph, February 03, 2020, 01:43:50 AM

#### altered

Quote from: chaotic neutral observer on March 10, 2020, 12:33:52 AM
Quote from: altered on March 10, 2020, 12:12:15 AM
I think the closest you could get to ~0 is 1/inf. It's not quite right, you might have to do something like 1/omega. Omega is the term for power towers of inf right?
Infinity isn't a number, though; you can't divide 1/∞.  It would be like 1/grapefruit, or 1/ .

This whole thing has been done to death over the years.

http://mathforum.org/library/drmath/view/62486.html
https://math.stackexchange.com/questions/1372459/one-divided-by-infinity-is-not-zero

As the first link says, it's a short hand for the limit: you get asymptotically close to having an infinitely small fraction. TWJs ~0 is the value of one of those pieces, at the limit.

I think the functional value is not what TWJ thinks it is, though, given we don't actually need a link between 0 and 1.

I was considering more along the lines of, "what is the thickness of a line in two dimensions?" Normal answer: zero. Problem: you can have infinite unique line segments of identical length, position and so forth, literally no identifying properties to separate them. "Trust me, they're all there" can't be disproved. (Similarly, infinite points at the same location, except even harder to fix because they could all have distinct labels.)

Having a "~0" unit value to the size of a point (and thus a line and etc) could be valuable in such cases. Less likely to be the case for theoretical mathematicians than for computer scientists, but that's not "useless".
"I am that worst of all type of criminal...I cannot bring myself to do what you tell me, because you told me."

There's over 100 of us in this meat-suit. You'd think it runs like a ship, but it's more like a hundred and ten angry ghosts having an old-school QuakeWorld tournament, three people desperately trying to make sure the gamers don't go hungry or soil themselves, and the Facilities manager weeping in the corner as the garbage piles high.

#### rong

the natural numbers (1,2,3,4 . . .) are countably infinite - this "size" infinity (cardinality) is called Aleph0

the real numbers 0, .00000000001, pi, 23, 1234.56789, 123456789, and a bajillion-bajillions, literally every number (and every number in between) on the continuous number line are uncountably infinite - it's cardinality is Aleph1

I think the idea of ~0 being a "smallest possible non-zero number" tries to reconcile the paradoxical uncountably infinite places to stop on your way from 0 to 1.  sort of like a Planck's length in physics.

However, I would argue that there would be a one-to-one mapping from the number line constructed from multiples of ~0 and the natural numbers.  so this number line would also be countably infinite and really, no different or any more useful than the natural numbers already are.

there could be a possibility of some sort of fractal-countability here (sort of like how certain objects are fractally dimensional - more than 1, less than 2) where a number line might be constructable that has cardinality between Aleph0 and Aleph1

If there is, that's above my pay grade, but 0 and infinity have this sort of black hole type property when it comes to countability and things kinda get pulled one way or another so I don't think this is really a possibility.
"a real smart feller, he felt smart"

#### chaotic neutral observer

Quote from: rong on March 10, 2020, 03:10:23 AM
there could be a possibility of some sort of fractal-countability here (sort of like how certain objects are fractally dimensional - more than 1, less than 2) where a number line might be constructable that has cardinality between Aleph0 and Aleph1

If there is, that's above my pay grade, but 0 and infinity have this sort of black hole type property when it comes to countability and things kinda get pulled one way or another so I don't think this is really a possibility.

While googling whether the rational numbers were uncountably infinite or not (spoiler: they're countably infinite), I ran across the Continuum Hypothesis, which postulates that there is no intermediate cardinality between that of the integers, and that of the real numbers.  Apparently it's a hard problem, since the hypothesis hasn't been proven (or disproven) since it was introduced in 1878.
Desine fata deum flecti sperare precando.

#### rong

oh, neat - i'd heard of the continuum hypothesis, but didn't really know what it was.

for the sake of saving face, i'd like to point out that the set of all real numbers contains (and is larger than) the set of all rational numbers.
"a real smart feller, he felt smart"

#### minuspace

Okay, I still don't get how infinities can be contained. But I really LIKE the idea that there are different infinities. It makes eternity seem less lonely.

#### rong

#140
what ever your concept of infinity is now, it will seem much smaller after watching this video

edit: watching the video has pointed out my error - I wrongly said the cardinality of the reals is Aleph1 and it is not.  At least, I don't think it is.  sorry for any confusion folks.
"a real smart feller, he felt smart"

#### minuspace

That was cool, the "skip ad" button on uTube had an infinity symbol instead of seconds countdown

#### Cramulus

Mathematician Georg Cantor was driven insane by the idea of a hierarchy of infinities.

He's proved that there are more real numbers (ie points on a number line, including fractions and pi) than natural numbers (ie 'counting numbers': 0, 1, 2, 3...). Which at first seems wacky, because there's an infinite number of both, right? But there are an infinite number of fractions between every counting number, so the number of fractions must be greater than the number of natural numbers.

Cantor developed these numbers called Transfinites, and I do not understand what they are, but the concept upset both mathematicians and the church. Which is a pretty impressive feat, if you ask me. Like making a 7/10 split.

The contemplation of infinity is a transcendental experience. I can understand why focusing one's powerful Lutheran mind on the concept of infinity over the course of decades could lead to a mystical experience. Cantor believed the concept of Transfinites came to him through mental contact with this infinity of inifinities, the set of all sets, which was also God.

Christian theologists accused him of pantheism. A devout Lutheran, he rejected this, but I think maybe he just chickened out.

#### rong

#143
i had my first (and only, i think) real epiphany while contemplating infinity.

i was thinking about how you can map an infinite number line onto a circle using a concept illustrated in this picture:

by drawing a line from the bottom of the circle to any point on the line, you can solve for a unique point (x1,x2) and therefore represent the entire number line on the circle.

you can map a plane onto a sphere using a similar technique

(this image shows the complex plane, but you can just as easily project any 2-D plane onto a sphere.

I always thought this was a cool technique because it collapses an unfathomably infinite concept down into fathomably finite one.

The epiphany was that the real trick was adding a dimension.  An infinite 1-D line can be completely shown on a finite 2-D circle.  An infinite 2-D plane can be completely shown on a finite 3-D sphere.

I don't know all the ramifications of this, but it feels important (why i called it an epiphany)

Extra Credit:
If you map ln(x) and ex onto a sphere, you will get a representation of the infinity symbol wrapped around the sphere.
"a real smart feller, he felt smart"

#### Doktor Howl

Quote from: Cramulus on March 16, 2020, 12:13:36 PM
Mathematician Georg Cantor was driven insane by the idea of a hierarchy of infinities.

He's proved that there are more real numbers (ie points on a number line, including fractions and pi) than natural numbers (ie 'counting numbers': 0, 1, 2, 3...). Which at first seems wacky, because there's an infinite number of both, right? But there are an infinite number of fractions between every counting number, so the number of fractions must be greater than the number of natural numbers.

Cantor developed these numbers called Transfinites, and I do not understand what they are, but the concept upset both mathematicians and the church. Which is a pretty impressive feat, if you ask me. Like making a 7/10 split.

The contemplation of infinity is a transcendental experience. I can understand why focusing one's powerful Lutheran mind on the concept of infinity over the course of decades could lead to a mystical experience. Cantor believed the concept of Transfinites came to him through mental contact with this infinity of inifinities, the set of all sets, which was also God.

Christian theologists accused him of pantheism. A devout Lutheran, he rejected this, but I think maybe he just chickened out.

It didn't take much to drive people nuts in the old days.
Molon Lube

#### The Wizard Joseph

Gaze not overlong into the depths...

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#### minuspace

Nah, infinity is "by definition" an insane concept, with crazy simply being the price of admittance.

#### axod

fine, in my limited experience though, there are at least two infinities. one of them is thicker.
just this