Important implications of Russell paradox? All of us, maybe in our first incursions on pure mathematics or going further in logic or set theory, need to face the rare and "pretty easy" to understand Russell paradox, but maybe the further implication of this encounter is that there is not a set such that contains the sets that do not be log to it selves . But for it importance i have thought that this paradox should imply further meditations. I mean i am not going seriously in logic yet, but i need to know why this paradox is so important for mathematicians, or may be if it is an actually easy to understand first encourage with math or set theory.
When i first read the paradox and the proof i understand it as a demonstration of impossibility of the existence of a biggest class or the biggest set, besides the normal implication that all of us know, so i don´t understand it as well as i should understand it, and now i don't know if out of logic, basic analysis and basic topology the Russell paradox is that important, are there any interpretations in real life, applied math, physics of RP. 
 A: I was told that at the time many mathematicians didn't really care about Russell's paradox. They believed that most sets you would encounter in everyday mathematics are too small to be pathological like that.
Much like nowadays many mathematicians don't really pay attention to large cardinal research for the same reason (and then there are those guys who just use them regardless).
The main implication from Russell's paradox is that not every definable collection makes a set. This is quite a shock to the naive approach taken at the end of the 19th century. 
There are two major results (both, I believe, are due to von Neumann). The first is the development of the concept of a class, namely a definable collection. This meant that we can talk about nonexistent collections as long as we can describe them by a formula. This allowed in some sense "virtual sets" like the Russell class, or other paradoxes, to virtually exists as classes. We could refer to them, manipulate them, but they are not objects in the semantical universe (and so they don't really exist).
The second result is the idea of describing the universe as an increasing union of sets, i.e. the von Neumann hierarchy. This allows us to avoid the paradox in the sense that whenever we want to describe some collection, we can limit ourselves to a certain point in the hierarchy and see how the collection looks like at that point. In fact a definable collection is a class if and only if it is bounded in the hierarchy.
Outside of set theory, the Russell paradox doesn't have much impact, or rather we cannot see it nowadays when ZFC (a set theory in which the paradox is resolved) is the de facto meta-theory of mathematics. However you can see remnants of the paradox in every self-referential definition, like the category of all categories.
As for your last question, Russell's paradox (as most set theory) has little (or virtually no) relation to the physical world, as much as we know. Applied mathematics, everything physicists do, and real world-expressible shenanigans are usually very very small in terms of sets (I mean most are bounded below the power set of the real line).
A: I think the main effect of the discovery of Russell's paradox on mathematicians (outside the field of logic) was to make the axiomatic foundations of mathematics slightly more complicated than was previously hoped.  Since most mathematicians do not argue directly from the axioms of set theory anyway, this has a minimal effect on mathematical practice.  The main effect is that definitions of the form $\{x : P(x)\}$ must be replaced with definitions of the form $\{x \in Y: P(x)\}$ where $Y$ is a set large enough to contain all the $x$'s you want.  It is usually not hard to show that such a set $Y$ exists.
I do not think that Russell's paradox shows up in real life, applied math, or physics, although it would be pretty neat if it did and I hope to see another answer proving me wrong on this point.  At the core of Russell's paradox is the much older "liar paradox" which shows up in several other places in mathematics: notably in the halting problem, which arguably does have practical implications.
