B Infinite multiverse would contain the ridiculous?

[Andrew Wright]

Again I am a novice. Could an infinite multiverse contain an 'asymptotic density' of silly universes and what would that actually mean?

I read a bit about asymptotic density. Correct me if I am wrong.

An infinite multiverse could contain a subset of teapot universes that is also infinite. (Like square numbers appearing in a set of all integers).

You could even get a feel for the density of teapot universes in the multiverse. But the density could vary, like square numbers appearing more often in the lower integers. Our universe could be in a patch of other similar universes that contain life. Meanwhile, the overall density of universes containing life could be low, because the required physical constants could be very unlikely.

There could instead be a finite number of teapot universes in the multiverse, like primes appearing in an infinite set of numbers.

Teapot universes could be an empty subset in an infinite set of universes.
So even if there is an infinite multiverse, it may not contain chocolate teapots in silly places.

Edits: Assembling my understanding.

 

[mfb]

There could instead be a finite number of teapot universes in the multiverse

That has probability zero.

Teapot universes could be an empty subset in an infinite set of universes.

Same here.

As every universe (that someone has something like our laws of physics) has a finite probability to have a teapot orbiting a dwarf planet like pluto, if there is an infinite set of universes we expect an infinite set of universes with teapot.

 

[hilbert2]

As I understand it, mfb's post addressed the probability of obtaining a number with property π within a set of n numbers. In this case the property π is "being a prime number."

Yes, we were obviously talking about different things...

Curiously, if we choose a random natural number, with number $n$ given a relative probability $e^{-n}$, the likelihood of $n$ being divisible by $2$ is smaller than $\frac{1}{2}$:

$\lim_{k\to\infty}\frac{\sum_{n=1}^{k}e^{-2n}}{\sum_{n=1}^{2k}e^{-n}}=\frac{1}{1+e}\approx 0.269$.

If the relative probability of $n$ is $\frac{1}{n^2}$, we get:

$\lim_{k\to\infty}\frac{\sum_{n=1}^{k}\frac{1}{(2n)^2}}{\sum_{n=1}^{2k}\frac{1}{n^2}}=\frac{1}{4}$

(Mathematica was able to solve the exact values of those limits). The reason why the probability of odd $n$ is higher than $\frac{1}{2}$ in these calculations, where smaller numbers are given more weight, is obviously the fact that the first natural number (number $1$) happens to be odd.

And even though the harmonic series is not convergent, the limit

$\lim_{k\to\infty}\frac{\sum_{n=1}^{k}\frac{1}{2n}}{\sum_{n=1}^{2k}\frac{1}{n}}$ exists, and it has exactly the value $\frac{1}{2}$.

 

[hilbert2]

I read a bit about asymptotic density. Correct me if I am wrong.

An infinite multiverse could contain a subset of teapot universes that is also infinite. (Like square numbers appearing in a set of all integers).

You could even get a feel for the density of teapot universes in the multiverse. But the density could vary, like square numbers appearing more often in the lower integers. Our universe could be in a patch of other similar universes that contain life. Meanwhile, the overall density of universes containing life could be low, because the required physical constants could be very unlikely.

There could instead be a finite number of teapot universes in the multiverse, like primes appearing in an infinite set of numbers.

Teapot universes could be an empty subset in an infinite set of universes.
So even if there is an infinite multiverse, it may not contain chocolate teapots in silly places.

Edits: Assembling my understanding.

An easy geometrical example about infinities of different sizes:

Suppose we choose a completely random point $(x,y)$ from the rectangle that has one of its corners at $(0,0)$, the origin of $xy$-plane, and the opposite corner at point $(1,1)$. How likely is it that this point $(x,y)$ is on the straight line $y=x$?

 

[Andrew Wright]

"Teapot universes could be an empty subset in an infinite set of universes."
"That has probability zero."

OK, you mean there are universes with a chocolate tea pot near Pluto? So perhaps NASA put one up there. Or maybe our universe will turn out to be the teapot universe? NASA are said to have an odd sense of humour...

 

[Andrew Wright]

An easy geometrical example about infinities of different sizes:

Suppose we choose a completely random point $(x,y)$ from the rectangle that has one of its corners at $(0,0)$, the origin of $xy$-plane, and the opposite corner at point $(1,1)$. How likely is it that this point $(x,y)$ is on the straight line $y=x$?

This feels like the probability should be zero because x and y could be anything and what are the chances that they are exactly the same? On the other hand, the line y=x exists, so the probability can't be zero.

 

[hilbert2]

This feels like the probability should be zero because x and y could be anything and what are the chances that they are exactly the same? On the other hand, the line y=x exists, so the probability can't be zero.

Now, in this case, the line segment where $x=y$ is a one-dimensional subset of that two dimensional square. Because of this, the probability of "hitting" that line, when choosing a random number on the rectangle, is zero.

 

[Andrew Wright]

So even if there are an infinite number of universes like ours, the probability of our universe coming out at random could still be zero?

 

[hilbert2]

So even if there are an infinite number of universes like ours, the probability of our universe coming out at random could still be zero?

If you allow some little "error tolerance" in how closely the universe has to be a replica of ours, then it can have a probability that is larger than 0.

If in the previous line-rectangle example we said that instead of $y=x$, we require $|y-x|<0.0001$, the probability would already be nonzero, because the set where $0\leq x \leq 1$, $0 \leq y \leq 1$ and $|y-x|<0.0001$, has a nonzero surface area.

 

[mfb]

OK, you mean there are universes with a chocolate tea pot near Pluto? So perhaps NASA put one up there. Or maybe our universe will turn out to be the teapot universe? NASA are said to have an odd sense of humour...

No. I mean "if a given set of premises are correct, then there are teapot universes". In particular, we asummed an infinite set of universes with laws roughly similar to the laws in our universe.

 

[Andrew Wright]

If you allow some little "error tolerance" in how closely the universe has to be a replica of ours, then it can have a probability that is larger than 0.

If in the previous line-rectangle example we said that instead of $y=x$, we require $|y-x|<0.0001$, the probability would already be nonzero, because the set where $0\leq x \leq 1$, $0 \leq y \leq 1$ and $|y-x|<0.0001$, has a nonzero surface area.

Right, but instead of variables x and y, we have c,G and h?

 

[Andrew Wright]

No. I mean "if a given set of premises are correct, then there are teapot universes". In particular, we asummed an infinite set of universes with laws roughly similar to the laws in our universe.

OK, but we do expect teapots in an infinite multiverse with physical laws like our own?

 

[mfb]

OK, but we do expect teapots in an infinite multiverse with physical laws like our own?

Yes, sure.

 

[Andrew Wright]

Is that just a consequence of chocolate being a valid configuration of matter?

 

[mfb]

Yes.

 

[hilbert2]

Is that just a consequence of chocolate being a valid configuration of matter?

When you say that you want a chocolate tea pot to orbit some planet in some universe, you leave a lot of variables undefined. The mass of the teapot could be anything from 100 grams to 2 kilograms, the percentages of trans fats and saturated fats in the chocolate could be any number in some acceptable interval, and the angular momenta of the teapot's rotation around its axis and it's orbital motion around the planet could be just about anything. None of these variables take values from a real continuum set, though, because you can add mass to the teapot only one molecule at a time, angular momenta are theoretically quantized even in the macro scale, etc.. However, if you leave enough room for different things to fit in your definition of what qualifies as a "chocolate teapot orbiting a planet", then probably it exists in some universe.

 

[EnumaElish]

Caveat emptor - all said, I think Bertrand Russell's argument about a china teapot orbiting in space still stands. I think he was specifically referring to this here our own universe, during approximately the first half of the 20th century, or thereabouts. I wouldn't have believed it then as I don't believe it now.

 

[EnumaElish]

Right, but instead of variables x and y, we have c,G and h?

You can read it that way. Or you can read it as "the probability of any object orbiting pluto over an analytically well-specified one-dimensional trajectory is zero."

 

[Hornbein]

So even if there are an infinite number of universes like ours, the probability of our universe coming out at random could still be zero?

It could be.

If you had an infinity of infinite universes I would expect each to be unique. Each would have probability zero of being alike to another.

There would also be no reason to think that every possibility would be reified. Even if the universes are finite it seems to me that there there is no reason to think that. There could still be many more possibilities than universes.

 

[Andrew Wright]

The mass of the teapot could be anything from 100 grams to 2 kilograms, the percentages of trans fats and saturated fats in the chocolate could be any number in some acceptable interval, and the angular momenta of the teapot's rotation around its axis and it's orbital motion around the planet could be just about anything. None of these variables take values from a real continuum set, though, because you can add mass to the teapot only one molecule at a time, angular momenta are theoretically quantized even in the macro scale, etc..

Basicly a chocolate teapot has loads of parameters. (There might be a supermarket somewhere selling a few varieties of them). So some parameters have finite configurations. Surely there are some parameters for a chocolate teapot that "take values from a real continuum set" - you mean like possible x and y values on the graph, perhaps?

 

[EnumaElish]

Yes.

Would the answer have changed if the expression were "chocolate crockery" instead?

 

[Andrew Wright]

What was that thing about spaghetti monsters? ;)

 

[EnumaElish]

This is not a teapot :D

 

[Andrew Wright]

I think that conscious spaghetti is an invalid configuration of matter. Someone back me up here!

 

[Andrew Wright]

But seriously, even quantum mechanics would have some invalid configurations, like the exclusion principle (or is that just an approximation?)

 

[mfb]

Would the answer have changed if the expression were "chocolate crockery" instead?

The answer would not change for anything you can assemble, and not even for some things you cannot assemble today.

But seriously, even quantum mechanics would have some invalid configurations, like the exclusion principle (or is that just an approximation?)

The exclusion principle is absolute. You cannot have anything that violates the laws of physics, by definition of laws of physics.