## New draft of the TIM

I’ve just posted a new draft (draft 25) of the theory of incomplete measurements. I’m working on clarifying the text more than on the actual contents. There is one change in contents, however, which is to add a reference to Dr Charles Francis’ Relational quantum mechanics.

Of particular interest to me is his observation that space-time curvature normally attributed to mass can also be seen as a proper-time delay between absorption and emission of a photon. This seems to work well for one particle. I’m still struggling to understand how this would work for multiple masses. I’m going to ask ;-)

## Skolem’s paradox

Today, I learned about Skolem’s paradox, which I find pretty interesting. Here is a rough overview:

- Georg Cantor demonstrated in 1874 that there are sets that are not countable. An example is the set of real numbers. Such sets are also said to be uncountably infinite.
- But mathematics can be represented as a countable language. Such a technique was used by Kurt Gödel to prove his famous incompleteness theorem.
- This leads to the Löwenheim-Skolem theorem, which essentially shows that you can enumerate the propositions in such a mathematical system. In other words, the propositions in the system are countable. For example, there must exist a countable set that obeys all the relationships defined on the uncountable set of real numbers.

This leads to Skolem’s paradox. You cannot count real numbers, but you can count the mathematical propositions that define them… The Wikipedia page indicates that some do not see that as a paradox because even if there is no bijection within the mathematical model, there may be a bijection outside the model. My “perception” of the paradox is that this means that our mathematical model cannot define all real numbers. There are real numbers that are not the subject of any theorem, that are not the limit of any “suite” (i.e. some expression written with a finite number of symbols, like “limit of 1/n”).

And this is only the beginning. We can keep building sets, such as surreal numbers, which are even larger than the set of real numbers.

For those interested, Paul Budnik created a video discussing these topics. I contacted him today regarding his Quantum Mechanics Measurements FAQ, since I honestly believe that I have answered several of these questions with my theory of incomplete measurements.

## Life and Death of a photon

There was recently an article in Nature about how a team of French physicists managed to observe a single photon. The trick, of course, is to observe the photon without destroying it. Apparently, the trick is to use an interaction between atoms of rubidium and photons, that causes them to tick a little late. There are a few more details in this article (in French), but not much.

Now, I wonder how one can talk about “a single photon” for a particle that has been bouncing around and interacting with all sorts of particles. If a photon is absorbed, and then re-emitted, is this the “same” photon? At least, that’s the meaning I believe was given to the term “the same”. In any event, a photon with identical properties.