How does one prove the formula for entropy $S=-\sum p\ln p$?
Obviously systems on the microscopic level are fully determined by the microscopic equations of motion. So if you want to introduce a law on top of that, you have to prove consistency, i.e. entropy cannot be a postulate.
I can imagine that it is derived from probability theory for general system. Do you know such a line?

Once you have such a reasoning, what are the assumptions to it?
Can these assumptions be invalid for special systems? Would these system not obey thermodynamics, statistical mechanics and not have any sort of temperature no matter how general?

If therodynamics/statmech are completely general, how would you apply them the system where one point particle orbits another?


Question: How prove this equation $\alpha=\aleph_{0}\alpha$
Answer: First of all take well-ordered set — every non-empty subset has a least element.
Then, for every (with the exception of the greatest element) element $x$ there is the "next" element $x+1$ — the least of all which are greater than $x$.
For some elements there may be no predecessor — let us call them limit elements (or points).
And every element of a well-ordered set has a form $x=a+n$ where $a$ is a limit element, $n$ is a natural number and $+$ means the same as above.  

OK, now Zermelo's theorem, which is equivalent to an axiom of choice, every set can be well-ordered.  

Also, $|\mathbb N \times \mathbb N|=|\mathbb N|$.

And we want to prove that for infinite $A:\;A\times \mathbb N$ equinumerous to $A$.
$A$ could be well-ordered, then $A$ is equinumerous to $B\times \mathbb N$ where $B$ is the set of all limit elements.
And we have $|A\times\mathbb N|=|(B\times \mathbb N)\times\mathbb N|=|B\times(\mathbb N\times\mathbb N)|=|B\times\mathbb N|=|A|$.