Skolem's paradox

So far I never mentioned how I understand
the famous Löwenheim-Skolem result (*):

"no matter how fancy your axiomatic system, which seems to talk about real numbers, complex numbers, geometries, fields etc.,

in the end, all it really does is talk about the countable natural numbers, nothing more and nothing less."



In my opinion it is the most shocking result of the Grundlagenstreit.

But does this tell us something about the true nature of physical reality?



(*) and let me be very clear that I am a layman who read exactly one book about number theory (and I understood perhaps half of it).

down to earth

I assume you will be relieved that
this blog post for once is not some crazy speculation about our universe and it does not contain empty pseudo-philosophical thoughts. It does not even count espressos.

Instead, it is about the down-to-earth topic of quantum gravity. Actually it is just a collection of links to pre-prints; In other words I am cleaning out my to-do list.



B.F.L. Ward: ".. by using recently developed exact resummation techniques ... we get quantum field theoretic descriptions for the UV fixed-point behaviors of the dimensionless gravitational and cosmological constants postulated by Weinberg. Connecting our work to the attendant phenomenological asymptotic safety analysis of Planck scale cosmology by Bonanno and Reuter, we predict the value of the cosmological constant ..."



U. Gursoy: "We propose a general correspondence between gravity and spin models, inspired by the well-known IR equivalence between lattice gauge theories and the spin models. This suggests a connection between continuous type Hawking-phase transitions in gravity and the continuous order-disorder transitions in ferromagnets. ..."



N. J. Poplawski: "The Einstein-Cartan-Kibble-Sciama theory of gravity provides a simple scenario in early cosmology which is alternative to standard cosmic inflation and does not require scalar fields. The torsion of spacetime prevents the appearance of the cosmological singularity in the early Universe filled with Dirac particles averaged as a spin fluid. Instead, its expansion starts from a state at which the Universe has a minimum but finite radius. ..."



A. Strominger et al.: "The problem of gravitational fluctuations confined inside a finite cutoff at radius r=r_c outside the horizon in a general class of black hole geometries is considered. Consistent boundary conditions at both the cutoff surface and the horizon are found and the resulting modes analyzed. For general cutoff r_c the dispersion relation is shown at long wavelengths to be that of a linearized Navier-Stokes fluid living on the cutoff surface. ..."



If this would be a better blog, each one would have its own blog post with interesting explanations etc. - some value added.

But by now you should know that with this blog you will have to make up your own mind about all this ...

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=?