If we construct axiomatic system of physical laws that are independent one another as in axioms in mathematics, what should they be? Can there be such a finite system of physical laws that can explain every physical phenomenon? Or is it impossible to have such finite axiomatic system in physical laws?
Not a complete one.
Kurt Gödel proved this was not possible by proving his "Incompleteness theorem". It turns out that in any axiomatic system (whether or not these axioms were to do with physical laws) we must select either consistency, or completeness, but not both.
Basically the "Incompleteness theorem" says that any 'computable axiomatic system' will have the following properties (such as one containing physical laws):
As a corollary of 1, any axiomatic system that is consistent cannot be complete (the system you describe would hopefully be consistent). So if you want your physical laws to be internally consistent you have to accept that there will be true (observable) physical laws that cannot be proven.
With respect to infinite systems, there are two types, countable and uncountable. The Incompleteness theorem has also been proven true for countable infinite sets. In the case of infinite axiomatic systems dealing with physical laws, they would be countable since each axiomatic law could map into the set of natural numbers. Even here Gödel's conclusions hold; either this system would be consistent but incomplete, or complete but inconsistent.
Apparently, we cannot get around the fact there are unprovable truths. Gödel provided a simple example.
Let S be the statement "This statement is unprovable."
If S is true, we cannot prove it since it is unprovable. However, if we can prove S true, the statement is self contradicting, so inconsistent.
Notice from the answer above J. Bell eloquent quote, "Quantum Mechanics, together with a finite cut-off QED, explains all of chemistry and NEARLY EVERYTHING in Physics." Unfortunately for Bell, Gödel has shown that as long as Quantum Mechanics seeks to be internally consistent, it will only ever be "NEARLY EVERYTHING" and not actually "EVERYTHING". If Quantum mechanics does actually achieve the ability to explain everything, Gödel shows us we have good cause to look for its self-contradictions (inconsistencies)
This is an experimentalist's answer:
I do believe that an axiomatic model , note "model", of nature can be found, but as an experimentalist I am wary of claims that "we have now wrapped up physics and only details have to be mopped up" which was the claim before quantum mechanics rocked the science in the beginning of the twentieth century.
One should be open to the possibility that as we delve further and further into experiments with new technologies, and understand more and more of the cosmos, the axioms might have to be changed. Otherwise physics will become fossilized.
Yes it is possible. As J. Bell eloquently wrote, Quantum Mechanics, together with a finite cut-off QED, explains all of chemistry and nearly everything in Physics. It was axiomatised by Weyl and Dirac by 1930.
There are only six axioms, which is certainly a finite number. Five would be better still..since most physicists no longer believe in the literal truth of the sixth axiom.
There are notorious problems with this axiomatisation, but they can certainly be fixed, although physicsists are not in agreement on how to fix them. The problem was analysed most logically by Wigner and, later, by J.S. Bell, in his "Against Measurement", I have posted a copyright-free copy at http://www.chicuadro.es/BellAgainstMeasurement.pdf. That is, the first three axioms apply to all physical systems, the second three axioms apply only to measurments, but surely measurements are carried out by measurement apparatuses which are physical....unfortunately the answers given by the first three axioms applied to the interaction of a microscopic system with a measurement apparatus are different from the results given by applying the second three axioms to the same physical setup. Not contradictory, but so different that there has been no agreement on how to compare them.
Most physicists now feel that the measurement axioms are only approximations, and ought to be derivable from the first three axioms as approximations. H.S. Green, under (I think) Schroedinger's influence at Dublin, published an extremely important paper analysing the physics of the measurement process as a phase transition, and there has been more recent work as well. See my own http://arxiv.org/abs/quant-ph/0507017, for example.
The only remaining difficulty is to either define the concept of ''probability'' as it occurs in these axioms, or to formulate a few more axioms to connect it with the other axioms. For the quantum case this was done in the paper referred to, and something similar can be done in the Classical Case.