As of you may know, Gödels theorem states that no formal axiomatic system capable of represanting all the natural integers can be both consistent and complete, however a recent discussion regarding the concept of infinity made me wounder if this theorem has any relevance in practice.
Considder a finite space of the universe. It contains a finite number of particles, all of which has a finite energy. Thus there is a limit to the maximum entropy such a system may posess. As the entropy of the system is limited to a finite degree, it can only contain a finite number of micro states. And thus only a finite amount o information can be contained within it.
Thus no finite amount of space can contain an infinite amount of information. This limits the axiomatic formal systems we may create inside such a system to only contain a finite number of theorems (Since when we run out of places to store information, we will be forced to delete an old theorem in order to store the new one). Therefore, no man made formal system can ever produce enough theorems for Gödels theorem to be relevant. Therefore, it seems, a finite system could (in principle ) be deterministic.
However, the limitation of information storage hit us once again. Since our brains are certainly a part of the system which we try to predict the behaviour of (remember that Heisenbergs uncertainty principle prevents us from observing a system without at the same time disturbing it) the system will not have enough possible microstates to store all the information about the system and at the same time perfomr the calculation predicting the behaviour of the system storing it in a format we may undertsant. Thus predicting the future with infalable accuracy seems to be theoretically impossible.
information can be stored in photons (and other bosons, like the graviton and the gluon), which do not have a limit on how much can be in a single point in space (i.e many photons can occupy the same position in space - the principle of superposition) and thus it is possible for there to be an infinite amount of particles in a finite space.