Author Topic: Thought Club  (Read 3608 times)

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Re: Thought Club
« Reply #135 on: March 10, 2020, 01:34:09 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/ :fnord:.

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”.
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Re: Thought Club
« Reply #136 on: March 10, 2020, 03:10:23 am »
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.
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Re: Thought Club
« Reply #137 on: March 10, 2020, 10:28:09 pm »
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.
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Re: Thought Club
« Reply #138 on: March 11, 2020, 01:05:06 am »
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. 
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Re: Thought Club
« Reply #139 on: March 14, 2020, 08:28:40 pm »
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.
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Re: Thought Club
« Reply #140 on: March 14, 2020, 10:23:46 pm »
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.
« Last Edit: March 14, 2020, 10:44:51 pm by rong »
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Re: Thought Club
« Reply #141 on: March 14, 2020, 10:30:48 pm »
That was cool, the “skip ad” button on uTube had an infinity symbol instead of seconds countdown  :lol:
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Re: Thought Club
« Reply #142 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.


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Re: Thought Club
« Reply #143 on: March 16, 2020, 04:38:48 pm »
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.
« Last Edit: March 16, 2020, 05:01:23 pm by rong »
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Re: Thought Club
« Reply #144 on: March 16, 2020, 08:16:25 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.
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Re: Thought Club
« Reply #145 on: March 17, 2020, 08:19:56 am »
Gaze not overlong into the depths...

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Re: Thought Club
« Reply #146 on: March 17, 2020, 09:22:11 pm »
Nah, infinity is “by definition” an insane concept, with crazy simply being the price of admittance.

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Re: Thought Club
« Reply #147 on: March 20, 2020, 10:25:08 pm »
fine, in my limited experience though, there are at least two infinities. one of them is thicker.
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