Okay, I think I understand what you're asserting well enough to offer some constructive advice now. It's a long post, so grab yourself some tea before you start reading.
Let's start with a few organizational points. Since the point of your paper is to argue for the utility of a non-axiomatic approach, you should state that fact up front. Ideally in the abstract. Something like "We analyze the paradoxes of Cantor and Russel from the standpoint of naive set theory and discuss possible resolutions. We then argue that these paradoxes enhance, rather than diminish, the utility of a non-axiomatic approach to set theory."
Second, actually give an argument for why these paradoxes are okay, and
make it the bulk of the paper. This is after all, your main point, and the only one that is specific to your paper. Right now the only mention of this point is a few sentences buried in the conclusion, and it's easy to miss. Conversely, spend a lot less time talking about the paradoxes themselves -- analyses of why the paradoxes of Cantor and Russel exist are commonplace, and will be boringly familiar to anyone familiar with the paradoxes themselves. Also, you should avoid saying that you "resolve" these paradoxes -- such language is usually used by people trying to claim, in plain contradiction of the obvious facts, that the arguments are invalid or otherwise do not constitute paradoxes in the first place, and this
will set off every crank detector that gets anywhere near your paper.
Next some linguistic points. If you don't intend to assert the mathematical existence of two distinct "copies" of the same set, you should avoid talking about a replica of the set. I know that you're trying to make the distinction between the role of the domain and the codomain clearer, but you actually just end up confusing people (cf. micromass's post). Likewise, referring to the set used in the proof as the "proof" set (quotes in original) is also confusing to English readers. It would be better to just give the set a name, and refer to it by that name.
Next an academic point. In American universities, Wikipedia is not considered a reliable source, and should NEVER be cited in an academic paper, unless the subject of the paper IS the Wikipedia article in question (e.g. how accurate it is). I would be very surprised to learn that Russian universities work any differently. Further, if you must cite Wikipedia, you should link to a stable version of the page, so that if it is revised in the meantime others can still access the version of the article you were working from. The current stable link to the Russel's paradox article is
http://en.wikipedia.org/w/index.php?title=Russell's_paradox&oldid=440736446
All right, now on to the logic of the paper itself. I don't think your argument in the section "analysis of contradictory equivalence" actually holds. As micromass stated, there is no such thing as a dynamic formula in mathematics. Once a definition such as R={x:x∉x} is established, it does not "become" anything else. We consider different possible assumptions about what relationship R might have to itself, but we are always reasoning about the same object {x:x∉x}. It is precisely because we do this that we can turn it into a proof by contradiction that the set {x:x∉x} doesn't actually exist. Using a fixed symbol to refer to an impossible object in order to derive a contradiction is utterly commonplace in mathematics, and completely logically valid. My favorite example is this proof that there is no greatest integer: "Let n be the greatest integer. Then n+1 is greater than the greatest integer. Q.E.A." The very first thing we do is assign n to be an impossible object! Another example is the proof that 2 has no rational square root. Actually, I'd like to emphasise this point, because you say in your paper:
Though just in case: reasoning on square root of 2 is about some property of the object, not about the existence of the object.
Actually, the proof that √2 is not rational
is about the existence of a mathematical object -- namely, integers n and m such that (n/m)^2 = 2. You can prove that 2 has no rational square root even if you have not yet proven that √2 exists at all (a useful thing, if you haven't yet done the construction of the real numbers). It's exactly the same as the proof that {x:x∉x} cannot exist, so why is one a paradox needing "resolution" and the other just a straight proof by contradiction?
Finally, I would like to say a few words in defense of the axiomatic method itself. You seem to be under the impression that the axiomatic approach is used only because it's "good enough" and out of tradition. This is not true. Mathematics is a unique discipline amongst the sciences. In literally every other scientific discipline, the accumulation of large amounts of empirical evidence is considered more than enough reason to accept something as scientific truth. Only in mathematics do we do something silly like insisting on a proof of, say, the Riemann hypothesis, when it has already been confirmed numerically up to the first ten trillion zeroes. But in these other disciplines, our models are only approximations to the underlying physical reality. We live with an unavoidable possibility of error. So we learn very quickly not to do anything that might magnify these errors to the point where they have practical consequences. We use the simplest principles possible and usually in straightforward ways. And when we do construct something theoretically using complicated arguments, we build an actual physical model and field-test it against reality before we do anything critical with it. And very frequently, we find that http://news.yahoo.com/contact-lost-hypersonic-glider-launch-163016325.html" due to small unexpected problems.
Mathematics is different, because the mathematical objects we're dealing with are either creations of our own mind or living in some platonic ideal realm (depending on your philosophy), and in either case are completely specified by the properties we define them to have. This gives us the possibility of error-free reasoning, which we regularly exploit to give
http://planetmath.org/?method=l2h&f...remAndThatEAndPiAreTranscendental2&op=getobj", which take the tiniest of subtle differences and slowly build up their consequences until at last you are able to derive an absurdity, thus showing your initial assumption wrong. But the same arguments that amplify subtle differences between objects also amplify subtle mistakes, so if you try an argument like that one with a physical object, the difference between theory and reality will get blown up to the point where there is no obvious resemblance between reality and your conclusions. Thus not only does mathematics make use of error-free reasoning, but it
depends on it. The smallest of mistakes, the most subtle confusions, will eventually be magnified to the point where they ruin your whole conclusion.
It was ultimately this necessity, combined with the illustration given by Russel's paradox of just how easily mathematicians could get themselves into trouble, that led to the development of axiomatic treatments of set theory. By formalizing logic and confining our arguments to those that could be derived using rules of inference known to be sound, we could reduce any possibility of error to a small number of places, which could then be very thoroughly inspected to ensure that there were no fallacies lurking about. We wanted to go even further and prove using purely finitistic methods the consistency of set theory, but unfortunately Godel showed that to be impossible. So we're left with having to take some set of axioms on faith. Still, reasoning from the axioms of ZFC or some other similar system is the closest we can get to "there is no possibility whatsoever of error". That is why we accept the axiomatic method. A system in which paradoxes are accepted would in every reasonably complicated argument (which is most of them nowdays) leave us wondering "Is the theorem I proved really true, or did I just magnify a subtle paradox to the point of absurdity?" Unless you can find a way to resolve that fear -- and I don't think that you can -- your suggested approach, while perhaps a nice exercise for philosophers, is inadequate to the task of actually doing mathematics.
Nor on any alternative formal system keeping it some way or another (like NF and the like).
