Does paradoxes in a theory mean that the theory is incorrect and should be discarded?

Question

I have two questions which relate to two different subjects, Science and Mathematics have different meanings of "theory", first is based on ever-growing scientific evidence while other is based on logical deductions following from axioms that are assumed to be true, Now paradoxes often lead us to develop our theory, As was the case for "Set theory" when Russel's paradox was proposed

Taking the above in Context, Here are My questions

1)Do paradoxes always indicate that there is something wrong in the theory, And New theory must be proposed? Does the existence of paradoxes itself act as evidence/proof(Science/math) for incorrectness of the theory proposed

2)Should a complete theory in specifically science ideally have no paradox?

Answer 1

A paradox is a situation which does not seems plausible, “a statement that runs contrary to one's expectation”. Further investigation may show that it is indeed an antinomy, i.e. it produces a logical contradiction.

Hence the classification as a paradox is subjective, while the classification as an antinomy is objective. Russell discovered even an antinomy in the concept of arbitrary comprehension in set theory.

My answer to your questions:

1. A paradox is always the prompt for further investigation to decide whether it turns out as an antinomy. In case of an antinomy the theory has to be changed, possibly limited or in the worst case dismissed. In the case of a confirmed paradox we have to correct our own wrong expectation.

   The non-invariance of Newton’s mechanics with respect to Lorentz transformations indicates that the theory has to be limited to classical mechanics, presupposing low energy, low velocities, …

2. A complete scientific theory does not exist. Said pointedly: Scientic theories are successful hypotheses.

Remark: Revised due the comment of @DheerajGujrathi

Answer 2

A theory that leads to predictions that contradict with each other, or with reality, is likely to be incorrect or incorrectly applied. A theory in science ideally should agree with reality, so if reality is not paradoxical the theory should not be.

Answer 3

The Barber Paradox in its various forms, for example, can easily be resolved using formal logic.

Version 1

The original form with an unspecified domain of discussion:

~EXIST(barber): ALL(a): [Shaves(barber, a) <=> ~Shaves(a, a)]

Where Shaves(x, y) means "x shaves y."

Version 2

We can also specify a particular domain:

ALL(barber): [Man(barber) => ~ALL(a): [Man(a) => [Shaves(barber, a) <=> ~Shaves(a, a)]]]

Where Man(x) means x is a man.

Version 3

More interestingly, by narrowing the domain even further, we can prove:

ALL(village): ALL(barber):ALL(men): [barber @ village & ALL(a): [a @ men => a @ village]

=> [EXIST(shave): ALL(a): [a @ men => [(barber,a) @ shaves <=> ~(a,a) @ shaves]] <=> ~barber @ men]]

Where men is a subset of village, and shaves is binary relation instead of a predicate as above.

In words: In an village with a resident barber, that barber can shave those and only those men in the village who do not shave themselves if and only if that barber is not a man.

See formal proofs here.