Excuse me for this question, it continues this one.
I wonder if there is a text where it is written explicitly that the old paradoxes (like the Russell paradox, and the others) do not appear in the modern axiomatic set theories?
I need this as a reference to convince people in (Russian) Wikipedia that this is indeed so.
EDIT. From the downvotes and from Noah Schweber's answer I deduce that I have to explain my question in more detail.
Dear logicians, I do not need an explanation of why the paradoxes of the beginning of 20th century do not appear (or do not manifest themselves) in modern mathematics. I understand more or less clearly why it is so. I am asking about other thing:
I need a book (or a paper) where it is written explicitly that those paradoxes indeed do not manifest themselves now in mathematics and do not pose a problem. I do not need details, just statement.
I need this reference for showing it to people who are not mathematicians, and who still believe that those old paradoxes still pose problems for mathematics (and still make it inconsistent).
I think, it is difficult for you to imagine this, but such people exist. Moreover they have power as administrators of Russian Wikipedia, what allows them to publish absurd articles about mathematics, containing lots of false propositions, and because of this I have to argue with them, and I believe that you, my colleagues, could help me in this. Instead of downvoting and sneering.
Hope this makes my question more clear.
EDIT 2. I see votes to close and I deduce from them that further explanations are still needed. I believe, my comment, addressed to Carl Mummert, will be mobilizing: this
who still believe that those old paradoxes still pose problems for mathematics (and still make it inconsistent)
-- means that some people believe, that all modern mathematics, including modern axiomatic set theories like ZFC, NBG, MK, is inconsistent and exactly because of the old paradoxes like Russell, Cantor, etc.
And I now understand why they think so: because there are no sources where it is written that this is not so. And because when you ask specialists about this, they theatrically do not understand what you are talking about, they sneer and they persist in talking about something else.
I want to add here that it is not enough for me to refer to a book where the author explains why one of those old paradoxes (say Russell, or Cantor), "being translated into the formal language of the modern axiomatic set theory" that you consider, don't lead to a contradiction in this theory. Because this does not resolve the problem that I face: my opponents will say that this is the statement only about one paradox, not about all known paradoxes. And logically they will be right. Even the statement about several paradoxes, like in the reference that Noah Schweber gives (Evert Beth's book, page 495) does not resolve the problem. There must be an explicit statement about all known paradoxes in a publication that will be accepted in Wikipedia as a reliable source.
Does it exist?
EDIT 3. @AndrésE.Caicedo, @user21820, @Leucippus, @ChinnapparajR, @amWhy: could you please explain your reproach clealy? From the instructions for "put on hold questions" I see that there is only one detail that can be considered missing in my post -- the description of what I was trying to do myself. I don't understand how this can help, but here are my attempts. I am not a specialist in logic, my field is analysis. I was trying to find the answers in encyclopedias. I did not succeed. The Encyclopedia of Mathematics does not contain the article "Foundations of Mathematics". The article Consistency says nothing about what I need. The same for the "Mathematical Encyclopedic Dictionary"of 1988, which I have in Russian. The Wikipedia article does not contain reference that I need as well. I also asked several logicians in Moscow, again without success. That is actually all what came to my mind.
EDIT 4. This chat shows me that despite my numerous explanations some people believe that I am seeking a confession of consistency of mathematics. I never said this. What I need is a confession that no contradictions were found yet in ZFC, NBG and MK. Please, think about the difference before reacting.