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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

#141

Post by taneq »

pherq wrote: post #90
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…

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…

I didn't go far into set theory but, informally, it seems to me that for any given natural number N, there will be N natural numbers less than or equal to N (obviously) and P prime numbers less than or equal to N, and P < N. Doesn't taking the limit as N goes to infinity show that there are fewer primes than there are natural numbers? Where am I going wrong?

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

#142

Post by masterzora »

taneq wrote: post #141
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…

I didn't go far into set theory but, informally, it seems to me that for any given natural number N, there will be N natural numbers less than or equal to N (obviously) and P prime numbers less than or equal to N, and P < N. Doesn't taking the limit as N goes to infinity show that there are fewer primes than there are natural numbers? Where am I going wrong?

> Where am I going wrong?

In short, infinities are complicated and intuition doesn't work well.

In slightly longer, you're talking about the difference of the rate of growth of two functions rather than actually about the cardinalities of the sets.

We define two sets as having the same cardinality when we can create a bijection between them. We can list the primes in order from smallest to largest and number them with the natural numbers. So we'll have 2 match up with 0, 3 with 1, 5 with 2, 7 with 3, etc. Every single prime number will correspond with a natural number AND every single natural number will correspond with a prime number, no exceptions. So they must be the same size. They are also the same size as the integers and the rational numbers but the set of real numbers is a bigger infinity.

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

#143

Post by chimeracoder »

taneq wrote: post #141
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…

I didn't go far into set theory but, informally, it seems to me that for any given natural number N, there will be N natural numbers less than or equal to N (obviously) and P prime numbers less than or equal to N, and P < N. Doesn't taking the limit as N goes to infinity show that there are fewer primes than there are natural numbers? Where am I going wrong?

It might be easier to think of it this way: Are there more natural numbers than there are even natural numbers?

The answer is no, as illustrated by the Hotel paradox[0], in which we have a infinitely many rooms and want to accommodate a (possibly infinite) number of guests.. To summarize: For any finite number of guests, you can always find an even-numbered room to correspond to that guest (assume the guests are numbered sequentially, then double their number and put them in the room that has that number.). This creates a one-to-one correspondence, which means that the sets are the same size.

You might say, 'well, that only works because we're dealing with a finite number of guests. But we're talking about infinity'. There are a few different ways of answering that question. In my opinion, the easiest way to look at it is to remember that there is no such number as 'infinity' - when we say 'infinity', we're really trying to express the concept of growing without bound. So, the above strategy (double the person's number) works for any arbitrarily large group. At no point does it stop working, even as the group size grows larger and larger, so we can say that the two sets have the same size.

On the other hand, we have no such strategy for putting every irrational number in one-to-one correspondence with the counting numbers. That proof is a little harder, and the analogy with the hotel guests breaks down, unfortunately, so it's a bit tougher to explain.

[0] https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand...

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

#144

Post by eru »

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......

The last thing we can see with light in any direction is the background radiation. That's when light and matter separated.

That's why gravitational waves are such a big deal: they allow us to look further. (Not further than the limit imposed by the speed of light, though.)

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

#145

Post by vacri »

harryjo wrote: post #61
sametmax wrote: post #39

Earlier quoted context omitted.

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.

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.

Sametmax's point is that you're comparing a simple count (atoms) to a factorial (combinations of pieces). For example, it's hardly surprising that 6! is larger than 6 - factorials grow much faster than simple counts.

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

#146

Post by nhaehnle »

tromp wrote: post #112
dogecoinbase wrote: post #107

Earlier quoted context omitted.

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.

Though if it is dynamic programming, then it should be possible for you to answer dogecoinbase's question using not much more computational power than you used to count them in the first place, right?

If you think of dynamic programming as counting the number of paths in a directed graph (in this case, from skimming the paper, the nodes correspond to border states), then given a path number, you can trace the path backwards through the graph, as long as you remember the number of paths ending in every vertex.

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

#147

Post by JackFr »

Retric wrote: post #140
pcmonk wrote: post #106

Earlier quoted context omitted.

"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…

You say same number, you mean same flavor. Either the 'number' is not in R and thus it's not a number or you end up with a host of contradictions. But, I have had my fun poking people who don't really get set theory.

Best trolling on HN in weeks.

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

#148

Post by rrauenza »

yathern wrote: post #89
kagia wrote: post #79

If 12megapixels can produce 10 to the power 86696638 images, and we came up with a way of enumerating those images, could we then build a function that given anyone of those images return the index of that image within reasonable time with current hardware. ie. "you have just taken 3999999987493th image"?

Yes, but it wouldn't save any space. As a thought experiment, think of it this way: How would we enumerate all these several gazillion image possibilities? Well. Let's say number one is all black. Every pixels and every channel is all zero in its value. And let's say the last image to be enumerated is all white. 255 for each pixel and each channel. Every conceivable image is created in between these two ends. For exa…

This numberphile video discusses a similar concept:

The 'Everything' Formula - http://youtu.be/_s5RFgd59ao

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