\section{Classical Thermodynamics: heat death}
Next let us carefully and rigorously recheck how the conclusion of
heat death of the universe is drawn. Basically, an \emph{isolated}
system with \emph{finite} volume is studied and it is discovered
that the entropy of the system always increases. Then it is
claimed that at some time point in the future, very very far
future, the universe will end up with a heat death.
There is a fundamental mathematical error in the above line of
logical inference. When a system is finite then we can find a
``bag'' and put the system completely into the bag, so that system
has no mass and energy interactions with the outside-bag world.
Then we do find an ever increasing of entropy which leads to the
heat death conclusion. But how about the universe? Can we find a
``bag'' at the first place to put the universe in?
In mathematics, there are fundamental differences among finite,
countably infinite, and infinite numbers. So we shall be aware
that conclusion drawn for a finite system may be completely not
applicable to an infinite system. The universe is very likely
infinite though some contemporary theories insist the finiteness
of the universe. In case the universe turns out to be finite
indeed, then the heat death conclusion can be drawn. However, the
heat death of universe assertion survived all those years when
there were no theories about the finiteness of the universe, and
today the heat death assertion is still presented without any
reference to the possible finiteness of the universe. In other
words, the bottom line is, the logical process of heat death
conclusion is fundamentally flawed.
As a matter of fact, if the time is infinite, then heat death
probably never will happen: since the time is infinite, if the
heat death could happen then it must have happened (then we would
not see such a wonderful world).
Actually the right above logical inference is clear-cut if the
time is infinite while the space is finite. However, when both are
infinite then it becomes difficult to judge. But if let me take a
metaphysical guess, I would like to pick that even for both
infinite time and infinite space the heat death will never happen.
