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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

#101

Post by Retric »

JBiserkov wrote: post #99
Retric wrote: post #84

Earlier quoted context omitted.

The set of all prime numbers is contained within the set of rational numbers, but they are rational numbers that are not within the set of prime numbers. Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B. Just because you can map two infinity's to each other does not mean they are of the same size consider: Limit(0->inifinity) of (x - (x/2)) algebraical…

>Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B. If that were true, why go to all the trouble, just show 1/2 which is not a natural number, or sqrt(2) which is not a rational number. Cantor's diagonalization is proving that no mapping exists between the natural numbers and the real numbers in [0, 1]; that no matter what mapping you (try to) come up,…

So, you can show the Real numbers between 0 and 1 is larger than the number of Natural numbers.

There are an infant number of points between 0 and 1 and an infinite number of points between 0 and 2. The distance between 0 and 2 is larger. The number of points between 0 and 1 is smaller than the number of points on the unit circle AND they are a different class of infinity.

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

#102

Post by pherq »

Retric wrote: post #97
pherq wrote: post #90

Earlier quoted context omitted.

The existence of a bijection between two sets is what "same size" means in set theory. Yes, there are non-prime integers, but you can establish a bijection between the two, so their cardinalities are equal (both have a cardinality of aleph zero). The reals, on the other hand, cannot be placed in a bijection with the natural numbers, and there are therefore "more" reals than naturals (i.e. there is an injection from t…

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), and it covers every real number in that interval (so it is a surjection).

This gives you a bijection between points in (a) 2-dimensional space and a segment of the real line (which, in turn, has a bijection with the whole real line if you want to specify that).

Once again, cardinality in set theory is based on injections and bijections. If there is an injection from X into Y, then Y is at least as big as X. If there is a bijection between them (i.e. injections in both directions), then they are the same size.

(Also, bijections are inherently bidirectional.)

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

#103

Post by yongjik »

Retric wrote: post #100
pcmonk wrote: post #96

Earlier quoted context omitted.

That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very much not the case for infinite numbers. There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers. Try reading this: https://en.wikipedia.or…

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.

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

#104

Post by pcmonk »

Retric wrote: post #100
pcmonk wrote: post #96

Earlier quoted context omitted.

That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very much not the case for infinite numbers. There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers. Try reading this: https://en.wikipedia.or…

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...

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

#105

Post by deepnet »

sametmax wrote: post #39
boredguy8 wrote: post #29

I like how Ken Jennings dealt with the 'Go complexity' analogy: "Go is famously a more complex game than chess, with its larger board, longer games, and many more pieces. Google’s DeepMind artificial intelligence team likes to say that there are more possible Go boards than atoms in the known universe, but that vastly understates the computational problem. There are about 10^170 board positions in Go, and only 10^80…

Comparing combinations with numbers of items is unfair. In Go, the number of items is the number of pieces, and it's very small. In the universe, the number of combinations of positions of all the atoms is, well, wonderful.

Compared with a googol our Universe has negligible atoms , 10^100 - 10^80 = ~10^100

Compared with a googolplex (10^(10^100)) the entire Evrettian metaverse is negligible as (10^(10^100) - 10^80^2 * (average quarks in atom) * leptons(10^200) * dark multiplier(10^2) = ~1 googolplex

Has anyone ever used a googolplex for anything ?

[For ~ read approximately]

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

#106

Post by pcmonk »

Retric wrote: post #101
JBiserkov wrote: post #99

Earlier quoted context omitted.

>Cantor's diagonalization is simply demonstrating that same inequality by showing a number in set A is not in set B. If that were true, why go to all the trouble, just show 1/2 which is not a natural number, or sqrt(2) which is not a rational number. Cantor's diagonalization is proving that no mapping exists between the natural numbers and the real numbers in [0, 1]; that no matter what mapping you (try to) come up,…

So, you can show the Real numbers between 0 and 1 is larger than the number of Natural numbers. There are an infant number of points between 0 and 1 and an infinite number of points between 0 and 2. The distance between 0 and 2 is larger. The number of points between 0 and 1 is smaller than the number of points on the unit circle AND they are a different class of infinity.

"The distance between 0 and 2 is larger" is a question about the metric of the space, not size of sets. There are the same number of points in both sets, since f(x) = 2x is a bijection between them.

Try to make arguments from axioms and definitions rather than asserting things from intuition. Intuition is often a useful tool, but (1) it's not an argument, and (2) it's not very helpful once you step into the infinite realm. Incidentally, that's why I went for programming: it's like math, but with no infinity (unless you're using floats, but that's a much easier infinity).

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

#107

Post by dogecoinbase »

tromp wrote: post #64
harryjo wrote: post #61

Earlier quoted context omitted.

I don't think anyone believes that Go is somehow more complex than the universe it is a subset of. The point is that enumerating all cases of Go is impossible and always will be, so more sophisticated analysis is required.

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

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

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

#108

Post by jeff_tyrrill »

lisper wrote: post #11
nsxwolf wrote: post #9

Earlier quoted context omitted.

That actually makes the Earth seem small to me. I wouldn't have blinked if someone had told me the blueberries would fill up a sphere the size of the Solar System. Just shows how hard it is to visualize these numbers.

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.

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

#109

Post by Razengan »

DougWebb wrote: post #69
levemi wrote: post #67

Earlier quoted context omitted.

If that is the case was the universe once finite and then went infinite during the early (big bang) expansion? I don't understand how something could have expanded if it was always infinite in size. I'm not even sure the concept of expansion even makes sense. What is infinite + 1? It's just infinite. It seems more like the expansion is a distribution of internal things.

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......

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

#110

Post by Jyaif »

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?

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.

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