# Resolution of Russell's and Cantor's paradoxes

1. Aug 9, 2011

### DanTeplitskiy

The link below is to the paper, that is, in my opinion, contains resolution of one of the most interesting problems of math logic and set theory – Russell’s paradox. As Cantor’s and Russell’s paradoxes is a “paired” problem you will find resolution of both in it.

The paper is now in pre-print phase. I need the folowing kind of help on it: your comments and questions.

The paper is written in a clear organized way – I do not think it will take more than 35-40 minutes from a person knowing the very basics of set theory to get it all.

The paper is in English (according to a professional mathematician having position in USA - quite readable English), though, of course, it is not English of an English speaking person.

Dan

2. Aug 9, 2011

### Citan Uzuki

You really didn't need anywhere near that much verbiage. You could have summarized the entire paper by saying "All Russel's paradox shows is that the set $\{x:x\not\in x\}$ cannot exist, but it says nothing about the set of all sets." And this is actually true! The contradiction only comes from the generally accepted principle in naive set theory (which is formalized as an axiom schema in ZFC) that given any set x, and any predicate φ, there exists a set consisting of all the elements of x satisfying that predicate. In ZFC, this allows you to show that if there was a set of all sets, then the set $\{x:x\not\in x\}$ would exist, which is plainly absurd, so in ZFC there can be no universal set. However, in a more general setting, you can keep the universal set and instead weaken the axiom of specification so that it only covers certain classes of formulas, which leads you to set theories like new foundations instead.

The problem with NF and related theories which keeps them from being more widely used is that because they do rely on restricting specification, you have to constantly keep checking every time you want to make a new set whether the set you're making can be specified by one of the allowed types of formula. In contrast, ZFC allows you to use any formula you like, even nice diagonalizing ones that would have the power to create paradoxes if applied to a universal set, provided only that you first verify that there is some set that contains the set you wish to create. This is much more flexible and easier to work with, and this is generally more important than keeping a universal set around.

3. Aug 9, 2011

### DanTeplitskiy

Dear Citan Uzuki,

Sorry, it seems to me that you missed the point of the paper. It is not on keeping the set of all sets. Nor on any alternative formal system keeping it some way or another (like NF and the like).

It is on resolution of the paradoxes within non-axiomatic approach. And on the methods of using the classical logic (classical logic itself, not in the context of some formal system) to complete the task.

Could you please read the part of the paper named "analysis of contradictory equivalence" and further to the very end of the paper? I hope you will be able to get the point. I would be very obliged if you do and write some comments then.

Thanks a lot in advance, Citan!

Yours,

Dan

4. Aug 9, 2011

### micromass

Staff Emeritus
You are working with some kind of "replica" of the set of sets, but a replica that apparently doesn't equal the set of sets.

Can you show that such a replica in fact exists?
What do you mean with "replica" anyways, can you give a definition of it?

5. Aug 9, 2011

### micromass

Staff Emeritus
There is no difference what so ever between R as an element and R as a set. It is the same R.
And if we say $R\in R$, then it is always the same R. R will still always equal $\{x~\vert~x\notin x\}$. You will have to give a convincing logical arguement as to why these R's are different. Defenitions in mathematics are not dynamic!!

As before, R is defined as $\{x~\vert~x\notin x\}$ so R wil always equal that set. Whether $R\in R$ or not, is of no importance.

You assume somehow that definitions change one way or the other. But they don't.

6. Aug 9, 2011

### DanTeplitskiy

Dear Micromass,

Sorry, but the definition of the word 'replica' is given right where it is used.
From the very definition it is clear that I do not have to prove that it exists.
The whole point is that if use the diagonal argument "some way" we can prove that some set is bigger than itself. Please, try to read the whole paper (or at least the whole part of it ).

As for "dynamic" definitions there is an explanation below that you might want to undestand.
The explanation starts with "Or, if you prefer seeing..."

Anyway thanks a lot for even trying to help! I really mean it, pal.

Yours,

Dan

7. Aug 9, 2011

### micromass

Staff Emeritus
I did not see any definition. From what I can read, you use the word replica just too distinguish between the uses. So the replica of the set of sets is exactly the set of sets. Or am I wrong??
But this does not make any sense, since later you go on about how there is no bijection between the replica and the set of sets. It's extremely confusing. What are you really trying to say?

Yeah, that parts makes no sense at all. You say that R becomes like $R\neq \{x~\vert~x\notin x\}$, which makes no sense. And then you say that we have to see the formula dynamically. Uh well, I hate to break it to you, but "dynamic formula's" are not part of mathematics. I don't even know what a dynamic formula is supposed to mean...

8. Aug 10, 2011

### DanTeplitskiy

1. Yes, replica is a copy. What I am getting at is that if we use the diagonal method like in Cantor's paradox we finally get even bigger absurd - the set of all sets is bigger than itself.

2. The point is we eigher consider R as both R = {x: x∉x} and, at the same time R≠ {x: x∉x} which is absurd OR consider the formula dynamically: that is R≠ {x: x∉x} "replaces" R = {x: x∉x}.

9. Aug 10, 2011

### micromass

Staff Emeritus
Well, what is the definition of a copy?? And why does such a copy exist?? The same points remain...

No, $R=\{x~\vert~x\notin x\}$ always. No exception.
Dynamic formula's is not part of mathematics.

10. Aug 10, 2011

### pwsnafu

Is it just me, or does this read like a philosophy paper to anyone?

11. Aug 10, 2011

### micromass

Staff Emeritus
It's not just you.

12. Aug 10, 2011

### Citan Uzuki

Okay, I've read this paper a second time -- I think you're right, I did miss your point. You spent a lot of time trying to resolve the paradoxes of Cantor and Russel, so naturally I assumed that that was what the paper was about.

Assuming that I've understood you correctly this time, you're arguing for a non-monotone approach to mathematical reasoning, whereby naive arguments are accepted (even knowing that they might lead to a contradiction), up until a paradox actually occurs, at which point the paradox is analyzed to determine where a case of invalid set formation actually occurred, and the contradictory conclusion then disposed of. Your argument is that since such an instance can always be found, we will gain a better understanding of mathematics by finding such paradoxes and analyzing them than by trying to confine our arguments to a set of rules where such paradoxes cannot occur. Have I understood you correctly this time?

13. Aug 11, 2011

### DanTeplitskiy

Dear Citan Uzuki,

I do not state that the instance can always be found, sorry.

By the way, we usually find some approach useful if it gives us something at all - we do not need everything to be resolved by some approach in some area to find it useful. That is "...since such an instance can always be found, we will gain a better understanding..." does not seem to be logically grounded to me.

In my paper I determine where and how a case of invalid paradox formation actually occurred really - that is, metamath over the standard form of Russell's paradox. Then, separately but connected to, we get a proof that some collection of objects does not exist.

I hope the above will be helpful to you to get my point more correctly .

Thanks a lot in advance, Citan!

Yours,

Dan

Last edited: Aug 11, 2011
14. Aug 11, 2011

### DanTeplitskiy

Dear Pwsnafu,

In order to make it simpler a great amount of words used. Maybe (I mean maybe) because of this my paper might be perceived as "some philosophy". Actually it is all logics on math objects (sets, diagonal argument and the theorem usually called "Russell's paradox").

Yours,

Dan

Last edited: Aug 11, 2011
15. Aug 11, 2011

### pwsnafu

No, I mean the arguments and techniques that you use makes this a philosophy of mathematics paper, not a mathematics paper.

16. Aug 12, 2011

### DanTeplitskiy

Dear Pwsnafu,

As far as I can see, if it is logic over math objects the logic may be either correct or not.

Yours,

Dan

17. Aug 12, 2011

### DanTeplitskiy

Dear Citan Uzuki,

One more thing (just in case): the contradictory conclusion is disposed of not because we prove some non-paradoxical way that the set does not exist but from analyzing the Russell's paradox itself. Maybe you understood this - though I decided to point it out here in case you did not. Meant no disrespect by this.

Yours,

Dan

18. Aug 12, 2011

### Citan Uzuki

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."

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:

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.

Last edited by a moderator: Apr 26, 2017
19. Aug 12, 2011

### DanTeplitskiy

Dear Citan Uzuki,

Thanks a lot!

I know there should not be such a thing like dynamic formula! But Russell's formula actually is like that - I only write about it! I write that we have to consider the formula dinamically as the other choice is to consider R (unconsciously or not - after reading my paper) as both R ≠ {x: x∉x} and R = {x: x∉x} whithin one reasoning (I mean within standard formulation of Russell's paradox) - that is breaking the basic laws of logic. Maybe the word "dinamically" or the word "becomes" confuses people. Maybe.

It seems to me (maybe I am wrong) that you missed an important point: Analysis of contradictory equivalence R ∈ R ↔ R ∉ R (which I consider to be the resolution of the paradox) is not about whether the R exists or not. Not at all! It is quite a separate thing.

By the way do you understand what I mean by this: "If R includes itself, R is not that R = {x: x∉x}. What I mean is that under the assumption “R ∈ R” R
includes a member that is included in itself (R itself is such a member)."? Just in case: I am not asking if you agree or not.

Yours,

Dan

Last edited: Aug 12, 2011
20. Aug 12, 2011

### micromass

Staff Emeritus
And in this case I'm afraid that it's not correct.