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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.

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    $\begingroup$ Nobody said that they were "overcame". But since they all came out of unrestricted comprehension, we removed it, and found no obvious way of repeating a proof of this paradox. So we're pretty happy with the results. $\endgroup$ – Asaf Karagila Nov 18 '18 at 8:44
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    $\begingroup$ Right. Have a nice day. $\endgroup$ – Asaf Karagila Nov 18 '18 at 8:51
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    $\begingroup$ @SergeiAkbarov I'll simplify history: before Cantor, everyone thought that naive set theory didn't lead to paradoxes. And yet it did. Who's to say that the next Cantor isn't about to prove ZF inconsistent? $\endgroup$ – Patrick Stevens Nov 18 '18 at 11:30
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    $\begingroup$ @AsafKaragila I don't think your behavior is appropriate as a moderator. $\endgroup$ – Hanno Nov 18 '18 at 12:58
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    $\begingroup$ @Hanno: Which part? Making comments on an answer, or making follow up comments on that? $\endgroup$ – Asaf Karagila Nov 18 '18 at 13:42
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There are many sources that claim no inconsistencies have been found in ZF (or ZFC, which is known to be consistent if ZF is consistent.)

  • Ershov and Palyutin on p. 92 of Mathematical Logic write: "What can be said about the consistency of ZFC? No inconsistencies have been discovered in ZFC itself this far."

  • Ebbinghaus, Flum and Thomas on p. 12 of Mathematical Logic write "Nevertheless, the fact that ZFC has been investigated and used in mathematics for decades and no inconsistency has been discovered, attests to the consistency of ZFC."

  • Bloch on p. 120 of Proofs and Fundamentals: A First Course in Abstract Mathematics 2nd ed writes "However, even if no one has definitively proved that the ZF axioms are consistent, we observe that these axioms have been designed to remove the known problems of naive set theory such as Russell’s Paradox. ... Ultimately, the ZF axioms seem reasonable intuitively; they work well in providing a framework for set theory as we would want it; the known problems with naive set theory have been eliminated by the ZF axioms; and experts in the field have not found any new problems that arise out of these axioms. Hence, we can feel confident that the ZF axioms are a very reasonable choice as the basis for mathematics. We simply cannot do any better than that."

There are many more texts which say essentially the same thing.

As many authors note, including all I mentioned above, the incompleteness theorem prevents us from having a proof of the consistency of ZF that is formalizable in ZF - which many authors feel precludes any sort of finitistic consistency proof.

So we are left with the situation that, although we can only prove the consistency of ZF by assuming sufficiently strong axioms, few mathematicians treat ZF as if they expect it to be inconsistent.

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  • $\begingroup$ Carl, I found Ershov-Palyutin in Russian. In which section do they write this? $\endgroup$ – Sergei Akbarov Nov 20 '18 at 21:00
  • $\begingroup$ Ah, yes, I found this! In Russian version this is also page 92. Thank you! $\endgroup$ – Sergei Akbarov Nov 20 '18 at 21:02
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    $\begingroup$ Here's another quote. Penelope Maddy, "Set Theory as a Foundation," in Foundational Theories of Classical and Constructive Mathematics (Springer, 2011) says in a footnote, "ZFC is not prone to Russell's (or any other known) paradox." $\endgroup$ – Timothy Chow Nov 21 '18 at 0:45
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Discussions of the resolution of Russell's paradox by restricting the comprehension axiom(s) can be found in most introductory set theory textbooks. For example, just search for "Russell": this works for Kunen's book (page $19$), Jech's book (page $4$), ... When in doubt, search!

As a specific example, page $20$ of Barr and Wells' freely-available book Category theory for computing science sums up the situation quite succinctly:

A simple way to avoid this paradox is to restrict x to range over a particular type of data (such as one of the various number systems – real, integers, etc.) that already forms a set. This prophylaxis guarantees safe sets.

For a discussion of the resolution of the Burali-Forti paradox, see Hellman's article On the significance of the Burali-Forti paradox (second paragraph).


Meanwhile, most other paradoxes (e.g. the Liar, Berry's paradox, Hilbert-Bernays' paradox, ...) have nothing to do with set theory per se, but are rather about logic itself: they essentially amount to the observation that if we could define a truth predicate in any context sufficiently strong to accommodate basic self-reference, we would get a contradiction. Viewed this way, these aren't paradoxes but rather theorems (e.g. Tarski or Chaitin) about first-order logic, and we don't look to set theory for their resolutions.


One takeaway from the above is that the resolutions are always the same: the obvious route to paradox is blocked right at the outset, either by not having the needed axiom (e.g. full comprehension vs. separation/restricted comprehension) or not having the relevant object "built in" to the logical framework (e.g. truth predicate). I don't know of a single paradox of naive reasoning whose resolution in ZFC with classical logic is nontrivial. Once one understands how to resolve the liar and Russell's paradox, everything else is an easy exercise. This is why the citations one finds are often perfunctory: there just isn't very much to say.

This isn't to say that the topic can't be interesting; there are definitely instances of systems avoiding a contradiction in clever ways. But the point is that classical logic + ZFC isn't one of those: it avoids the paradoxes in the most basic way possible.

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    $\begingroup$ @SergeiAkbarov But I have included an explicit reference stating that (and linked to three others). Is it not explicit enough, or not strong enough, somehow? $\endgroup$ – Noah Schweber Nov 18 '18 at 18:24
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    $\begingroup$ @SergeiAkbarov You're right about the first reference; Kunen mentions the role of not allowing $y$ to be free in the applying formula, which is important but not what you're asking. But re: "The same for others": quoting from Jech: "The safe way to eliminate paradoxes of this type is to abandon the Schema of Comprehension and keep its weak version, the Schema of Separation: If $P$ is a property, then for any $X$ there exists a set $Y =\{x\in X :P(x)\}$. Once we give up the full Comprehension Schema, Russell’s Paradox is no longer a threat." That seems pretty clear to me. $\endgroup$ – Noah Schweber Nov 18 '18 at 18:36
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    $\begingroup$ @SergeiAkbarov Similarly, from the other Kunen example: "Cantor’s advice was to avoid inconsistent sets (see [10]). This avoidance was incorporated into Zermelo’s statement of the Comprehension Axiom as it is in Section I.2. Namely, once you have a set $z$, you can form $\{x \in z : Q(x)\}$. You can still form $R = R_z = \{x \in z : x \not\in x\}$, but this only implies that if $R \in z$, then $R \in R \leftrightarrow R \not\in R$, which means that $R\not\in z$." $\endgroup$ – Noah Schweber Nov 18 '18 at 18:38
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    $\begingroup$ And I gave another example for Burali-Forti. And I explained why the others aren't about set theory. Do you want me to add sources on why they're not a problem for logic? Regardless, it seems to me - based on the ease with which I'm finding sources in standard textbooks - that you haven't really conducted a thorough search on your own, and I think it's this that's lead to a downvote. $\endgroup$ – Noah Schweber Nov 18 '18 at 18:40
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    $\begingroup$ @SergeiAkbarov "It can be made clear that those arguments which in Cantor's original version of set theory gave rise to paradoxes are eliminated in virtue of the restrictions implied by the axiomatisations which Zermelo and others have given." From Evert Beth's article The paradoxes of logic and set theory and their solution (page $495$). Better? $\endgroup$ – Noah Schweber Nov 18 '18 at 18:50

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