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On the (Small) Number of Atoms in the Universe

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Re: On the (Small) Number of Atoms in the Universe

#111

Post by Retric »

pcmonk wrote: post #104
Retric wrote: post #100

Earlier quoted context omitted.

We agree that {A,B,C} has a lower cardinality than {A,B}. Now, feel free to try and map the set of Real numbers to the set of irrational numbers. ex: e + ei.

{A,B,C} is of greater cardinality than {A,B}. However, the argument that "a rule works for finite numbers, so it must work for infinite numbers" is clearly false. For a your mapping, see: http://math.stackexchange.com/questions/512397/is-there-a-si...

The distance between 0 and 1 is smaller than the distance between 0 and 2. The number of points between 0 and 1 is larger than the number of rational numbers.

Re: On the (Small) Number of Atoms in the Universe

#112

Post by tromp »

dogecoinbase wrote: post #107
tromp wrote: post #64

Earlier quoted context omitted.

Indeed, enumerating all 208168199381979984699478633344862770286522453884530548425 639456820927419612738015378525648451698519643907259916015 628128546089888314427129715319317557736620397247064840935 positions in Go is impossible.

In your enumeration, what's the board look like at position 348277381979984699478633344862652779770286522453884530548425639456820927419612?

I can't say, because I didn't enumerate them. I only counted them. See

http://tromp.github.io/go/legal.html

for the method used, which is a form of dynamic programming.

Re: On the (Small) Number of Atoms in the Universe

#113

Post by pcmonk »

Retric wrote: post #111
pcmonk wrote: post #104

Earlier quoted context omitted.

{A,B,C} is of greater cardinality than {A,B}. However, the argument that "a rule works for finite numbers, so it must work for infinite numbers" is clearly false. For a your mapping, see: http://math.stackexchange.com/questions/512397/is-there-a-si...

The distance between 0 and 1 is smaller than the distance between 0 and 2. The number of points between 0 and 1 is larger than the number of rational numbers.

> The distance between 0 and 1 is smaller than the distance between 0 and 2.

This is correct. However, the number of real-valued points between 0 and 1 is the same as the number of real-valued points between 0 and 2.

> The number of points between 0 and 1 is larger than the number of rational numbers.

This is also true because there are uncountably many real-valued points between 0 and 1 and countably many rational numbers.

Re: On the (Small) Number of Atoms in the Universe

#114

Post by tromp »

Jyaif wrote: post #110
dogecoinbase wrote: post #107

Earlier quoted context omitted.

In your enumeration, what's the board look like at position 348277381979984699478633344862652779770286522453884530548425639456820927419612?

1/ Convert the number to base 3. 2/ Each digit represents an intersection on the goban, assuming the following mapping: 0 = no stone, 1 = black stone, 2 = white stone.

That would work for all positions regardless of legality, which are in 1-1 correspondence to {0,1,2}^(19*19).

The count above is for legal positions only, i.e. those where every connected group of stones is adjacent to an empty point.

Re: On the (Small) Number of Atoms in the Universe

#115

Post by Retric »

yongjik wrote: post #103
Retric wrote: post #100

Earlier quoted context omitted.

We agree that {A,B,C} has a lower cardinality than {A,B}. Now, feel free to try and map the set of Real numbers to the set of irrational numbers. ex: e + ei.

1. e + ei is not real. 2. f(x) = x + sqrt(2) if there exists an integer k>=0 such that x - k * sqrt(2) is rational; f(x) = x otherwise. This function maps all real numbers to irrational numbers, 1-to-1.

e is real, e + ei is irrational.

Re: On the (Small) Number of Atoms in the Universe

#116

Post by pcmonk »

Retric wrote: post #115
yongjik wrote: post #103

Earlier quoted context omitted.

1. e + ei is not real. 2. f(x) = x + sqrt(2) if there exists an integer k>=0 such that x - k * sqrt(2) is rational; f(x) = x otherwise. This function maps all real numbers to irrational numbers, 1-to-1.

e is real, e + ei is irrational.

I believe you mean "complex". You can form a bijection between the reals and the complex numbers by interleaving the digits, as any Google search can tell you.

Re: On the (Small) Number of Atoms in the Universe

#117

Post by lisper »

jeff_tyrrill wrote: post #108
lisper wrote: post #11

Earlier quoted context omitted.

The earth is small. If you build a scale model of the solar system the size of a football field, with the sun and one end and Neptune at the other (Pluto has been laid off as a planet) then the sun will be about the size of a ping pong ball and the earth will be the size of a poppy seed (and it will be about ten feet from the sun). Jupiter is about the size of a pea at this scale. Alpha Centauri is about four miles a…

> Alpha Centauri is about four miles away. Actually, around 500 miles. Distance to Alpha Centauri: 4.37 light years = 276,364 astronomical units. In your diagram, the Earth is 10 feet from the Sun. Multiply by 276,364 to get 2,763,640 feet, or 523 miles. The scale jump from distances around the Solar System to the next closest star is mind boggling.

Wow, you're right. That'll teach me to try to do the math in my head.

Re: On the (Small) Number of Atoms in the Universe

#118

Post by Retric »

pcmonk wrote: post #116
Retric wrote: post #115

Earlier quoted context omitted.

e is real, e + ei is irrational.

I believe you mean "complex". You can form a bijection between the reals and the complex numbers by interleaving the digits, as any Google search can tell you.

For every point on your mapimg I define two points X + 1 and x + 1i. You can assign infinity to one of them but not both.

Re: On the (Small) Number of Atoms in the Universe

#119

Post by Retric »

pherq wrote: post #102
Retric wrote: post #97

Earlier quoted context omitted.

You're confusing a classification system with size. Is the set of Real Numbers larger, smaller, or the same size as the set of points in a finite 2d object? Can you setup a bijection in either direction?

Assuming you consider the dimensions/axes/whatever of the 2d space to be indexed by reals (which is conventional), then yes one can construct a bijection: - normalise x and y coordinates in the shape into the interval (0, 1) - interleave the bits of the normalised x and y coordinates This gives a single real value in the interval (0, 1), which exists and is unique for every point in the space (so it is an injection),…

You used a countable infinity to tile an uncountable one. The set of 0 to 1 line segments being a countable infinity. 0 to 1 maps 1, 1 to 2 maps to 2 ect.

Re: On the (Small) Number of Atoms in the Universe

#120

Post by hellameta »

Razengan wrote: post #109
DougWebb wrote: post #69

Earlier quoted context omitted.

Off the cuff thought: the overall universe is infinite and not expanding, and it's only the visible universe that's expanding into that infinite space. Now try to wrap your mind around this: someone that's one light year to the left is going to see a slightly different visible universe, also expanding, into the same infinite space. But if we look in their direction, we see the edge of our visible universe expanding i…

I've always wondered; is there a "last" galaxy in any direction, such that for an observer in that galaxy, no further light or radiation can be detected from that direction? (outside that galaxy) That, must be a terrifying place to live in......

I don't see how our situation is any less terrifying haha
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