empty set

The empty set {} contains no element, nothing whatsoever.

Next we consider the set {{}}, which contains the empty set as its only element.

Then the set { {}, {{}} } which contains two elements and so on and so forth.

We assign the symbols 0, 1, 2, ... to these sets for convenience.



This is of course the standard definition of the natural numbers N as given by von Neumann; Once we have N then Z, Q, R, C etc. follow from N more or less in the usual manner.



I only mention it because some people believe that all physics is really just math.

But if "external physical reality is assumed to be purely mathematical" then all reality is based on the empty set.

7 x 6 = 41 you little sh**

Nowadays, Scott A. rarely writes blog posts, so I really enjoyed his latest entry and the discussion which followed. (By the way, Moshe refers to his own blog post which is here.)

It seems that Scott is worried about hyper-computation: "Yes, doing an infinite amount of computation in a finite time using exponentially-faster steps certainly does seem like a cheat to me!"

But I think he should be less worried about the discreteness of space-time and more about the question if one can create artificial black holes and baby universes. Of course, what appears as baby universe from one side, looks like a normal universe from the inside and we don't know if one could create a whole universe just to solve a math problem. (As long as we are not sure about quantum gravity, we are not sure about anything.)

Obviously, the Bekenstein bound would not limit the complexity of problems one could solve using baby universes (the volume of a universe is not bounded) and the only question is if/how one could get the answer out of the universe (perhaps using time travel?).



There are of course several indications that our own universe was indeed created as such a computing device:

1) Our universe seems to be fine tuned for the existence of math teachers.

2) We have reached some sophistication in our studies of math.

3) Life in this world seems to lack a deeper meaning and has a certain tendency towards the boring, uninteresting and annoying.

3b) The creator of this universe seems indifferent to the pain and suffering of its inhabitants.

4) One could resolve the Fermi paradox by assuming that the universe is fine tuned for its inhabitants to hang out at MathOverflow but prohibiting unnecessary inter-galactic travelling.



The only remaining question is this. If our universe was really created as some sort of computation device, is it at least part of a grand scientific project, some kind of ultimate mathematical inquiry?

Or did some ET kindergartener 'borrow' the baby-universe-computation-device of his older sister to solve the home work problem 7x6=?