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Bertrand Russell's Set Theory Paradox

Posted January 25, 2017 5:31 PM by amichelen
Pathfinder Tags: education Logic math paradox

Bertrand Russell

Gottlob Frege

In June 1902, Bertrand Russell, the great British mathematician and logician, sent the statement of a paradox to his friend Gottlob Frege, a German philosopher, logician and mathematician. Frege had been working for more than 10 years writing his monumental work “The Foundations of Arithmetic” and was finishing the final chapter of the second volume of this two-volume treatise. Russell and Frege had been friends for many years and Russell encouraged his friend to write a book about the mere foundations of arithmetic based on the set theory of Cantor. Frege obliged, but one day Russell found a contradiction of the work Frege was proposing in his book. This contradiction simply destroyed the very principle of his friend’s logic. Frege was devastated; his work of over ten years was simply irrelevant. Russell’s letter to Frege terminated the labor of more than ten years. Frege sank into a deep depression, while Russel tried to repair the damage by constructing a new theory of logic that would be immune to the paradox. He couldn’t. The paradox appeared again in his new theory.

The Russell paradox has been popularized in many ways. One of the best known of these was given by Russell in 1919 and concerns the plight of a barber of a certain village who has enunciated the principle that he shaves all those persons of the village who do not shave themselves. The paradoxical nature of the situation is realized when we try to answer the question: “Does the barber shave himself?” If he does shave himself, then he shouldn’t according to his principle; if he doesn’t shave himself, then he should according to his principle.

Since the discovery of the above contradictions, additional paradoxes have been produced in abundance. These modern paradoxes of set theory are related to several ancient paradoxes of logic, such as:

  1. Eubulides, of the fourth century B.C. is credited with making the remark, “The statement I am making is false.” If Eubulides statement is true, then, by what it says, the statement must be false. On the other hand, if Eubulides’ statement is false, it follows that the statement must be true. Then Eubulides’ statement can neither be true nor false without entailing a contradiction.
  2. Epimenides, who himself was a Cretan philosopher of the sixth century B.C., is claimed to have made the remark, “Cretans are always liars.” A simple analysis of this remark easily reveals that it, too, is self-contradictory.

The existence of paradoxes in set theory, like those described above, clearly indicates that something is wrong. Since their discovery, a great deal of literature on the subject has appeared, and numerous attempts at a solution have been offered. For some mathematicians there seems to be an easy way out. One has merely to reconstruct set theory on an axiomatic basis sufficiently restrictive to exclude these known antinomies. This is simply a procedure that avoids the paradoxes (putting your head in the sand.)

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#1

Re: Bertrand Russell set theory paradox

01/25/2017 6:02 PM

But a person who grows a beard doesn't shave himself, and the barber doesn't shave him either. Or herself. So, maybe the barber has a beard?

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#4
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Re: Bertrand Russell set theory paradox

01/25/2017 9:00 PM

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#11
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Re: Bertrand Russell set theory paradox

01/26/2017 5:41 PM

Ok, then tell me this 'who is the dentist, dentist' ?

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#2

Re: Bertrand Russell set theory paradox

01/25/2017 7:01 PM

"One of the best known of these was given by Russell in 1919 and concerns the plight of a barber of a certain village who has enunciated the principle that he shaves all those persons of the village who do not shave themselves."

Actually, the way it is stated, there is no paradox. I don't see the word "only" in there, so he could shave the persons that don't shave themselves and also shave one that does, namely himself. But if you do add the word "only", either he is either claiming something he cannot do, or he has a long beard.

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#8
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Re: Bertrand Russell set theory paradox

01/26/2017 10:00 AM

The couple of words "all those" is equivalent to the word "only" in this context.

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#18
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Re: Bertrand Russell set theory paradox

01/29/2017 7:56 AM

No. 'All those' is strictly inclusive. What is missing/needed for the paradox is an exclusion.

As it currently reads, the barber may also shave those who shave themselves.

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#10
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Re: Bertrand Russell set theory paradox

01/26/2017 2:55 PM

The guy could try to shave the infidels. I do not know if he could do that.

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#3

Re: Bertrand Russell set theory paradox

01/25/2017 8:46 PM

"For some mathematicians there seems to be an easy way out. One has merely to reconstruct set theory on an axiomatic basis sufficiently restrictive to exclude these known antinomies. This is simply a procedure that avoids the paradoxes (putting your head in the sand.)"

There are systems of arithmetic, for example, in which division-by-zero is perfectly legitimate, but you always pay for it somewhere else.

TANSTAAFL.

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#5

Re: Bertrand Russell set theory paradox

01/26/2017 8:37 AM

To avoid the side issue of bearded barbers or lady barbers, a simpler example is "This sentence is false". If it's true, it's false, and if it's false, it's true. It is neither true nor false, but something else.

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#6

Re: Bertrand Russell set theory paradox

01/26/2017 9:14 AM

Kurt Godel also pointed out that due to such self-referential systems, Mathematics is incomplete.

The way I see it, self-referential systems, or in this case self-referential sets, are the equivalent of dividing by zero. Mathematics disallows dividing by zero. I suspect that if self-referential sets are disallowed, Mathematics would be complete - or at least, more complete.

Similarly the statement “The statement I am making is false.” is self-referential and fails (my) rules of logic.

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#7
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Re: Bertrand Russell set theory paradox

01/26/2017 9:28 AM

If you want to model it in electronic logic, you get an oscillator.

A schematic of a simple 3-inverter ring oscillator whose output frequency is 1/(6×inverter delay).

https://en.wikipedia.org/wiki/Ring_oscillator

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#9

Re: Bertrand Russell's Set Theory Paradox

01/26/2017 2:53 PM

Is Rusell the guy from the movie "The beautiful mind"?

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#13
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Re: Bertrand Russell's Set Theory Paradox

01/27/2017 9:10 AM

A Beautiful Mind is the story for John Nash a Nobel Prize Laureate in Economics.

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#12

Re: Bertrand Russell's Set Theory Paradox

01/27/2017 3:10 AM

A very good (and fun) reading is Logicomix, by Apostolos Doxiadis (https://www.amazon.com/Logicomix-search-truth-Apostolos-Doxiadis/dp/1596914521) The events described in this post - and not only - are contained and very well put forward even for a layman to understand. What I liked most in this comic-book, is that it submerges you in the atmosphere of that romantic era when all centuries-old mysteries seemed that could be resolved... in the end, of course, the mysteries took their revenge, especially when Godel - with his "incompleteness theorem" - demolished the hopes for a universal axiomatic system that could put logic in safe mathematical grounds.

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#14
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Re: Bertrand Russell's Set Theory Paradox

01/27/2017 9:18 AM

I agree, the book "Logicomix: An epic search for truth" is a great book about the story of Bertrand Russel spending his whole life looking for the truth and trying desperately to postulate the foundations of mathematics ... to no avail ...

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#15

Re: Bertrand Russell's Set Theory Paradox

01/27/2017 12:52 PM

Might I suggest that the whole paradox premise is stupid? "Given the Set S{} which contains the things it does not contain." Is little more than linguistic nonsense.

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#16
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Re: Bertrand Russell's Set Theory Paradox

01/27/2017 1:05 PM

This paradox is more than a "nonsense". It is at the most fundamental basis of the foundation axioms of mathematics. Many mathematicians "quit" after learning about this paradox. Bertrand Russel spent many years trying to understand it.

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#17
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Re: Bertrand Russell's Set Theory Paradox

01/27/2017 2:52 PM

It is no more a paradox than √2 not appearing in the set of rational numbers or √-1 not appearing in the set of Reals. What is does is point out a shortcoming of the language used to define set theory.
FWIW I am a member of that unfortunate generation whose 3rd grade math textbook was titled "Sets & Numbers". In 1972 we were the experimental group for the new math, which if you'll allow me to say, was an abject failure!

As one of the kids chosen to participate in the early academically talented program, I can say with conviction that just because we capable of doing more than the average student, some of us saw no benefit with being tasked with what was essentially just more busy work.

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#19
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Re: Bertrand Russell's Set Theory Paradox

01/29/2017 2:32 PM

I am not very familiar with the 1972 "new math", but I agree with you that sometimes we need better usage of the language not only for set theory, but for many other things.

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#20

Re: Bertrand Russell's Set Theory Paradox

01/31/2017 12:38 AM

I do not agree that something that is not true must be false. Something I state that is false can still be true.

I am I right?

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#21
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Re: Bertrand Russell's Set Theory Paradox

01/31/2017 7:59 AM

'I am I right?'

Roughly half. The other half is left.

Reminds me of a koan (sorry in advance, Del) from the Mumonkan called:

'Nansen cuts the cat in two'

'Nansen saw the monks of the eastern and western halls fighting over a cat. He seized the cat and told the monks: `If any of you say a good word, you can save the cat.'

No one answered. So Nansen boldly cut the cat in two pieces.

That evening Joshu returned and Nansen told him about this. Joshu removed his sandals and, placing them on his head, walked out.

Nansen said: `If you had been there, you could have saved the cat.' '

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#22

Re: Bertrand Russell's Set Theory Paradox

02/02/2017 3:35 AM

Sometimes I compare of self-referencial logic paradoxes with some other paradoxes that appear in maths. Like for example: What is the sum of the infinite sequence: +1 -1 +1 -1 +1... (Grandi's series) Apparently it is oscillating. Exactly like some self-referencial logic problems. One could say that there is no meaning to talk about sum in this case. Another, could use the formula to calculate the sum of a geometric progression (ratio = -1) and come to the result: 1/2 (or -1/2). Wow! A value that never appears no matter how many _finite_ terms you take to calculate the sum. Is the formula not applicable in this case? If not why not? Or is there some "truth" that is hidden in here and pops up only when you are patient enough and follow the sequence up to infinity? One could similarly imagine that if you let the preposition: "this phrase is not true" oscillate till infinity, the truth of it would settle down to 1/2! Maybe for maths to work OK at all cases, the notion of "absolute truth" and "absolute falsity" should be somehow revised.

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#23
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Re: Bertrand Russell's Set Theory Paradox

02/02/2017 5:56 AM

Wait, wait, before you go revising the notion of absolute falsity, could you review the accepted notion?

That seems like a slippery one. It suggests something more than simply incorrect. Yet if all aspects are purely false it would seem to have lost reference to what might be true and in doing so it would have lost basis on which it could be false.

Absolute falsity seems a paradox unto itself.

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#24
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Re: Bertrand Russell's Set Theory Paradox

02/02/2017 7:43 AM

I was only focusing about the truth of falsity of a specific proposition, and whether this could be absolute this way or another. I'm not talking of course about the truth or falsity of mathematics in general. Nevertheless, Godel's theorem shows that there is something wrong with what we think about this matter. We either need to leave some prepositions to be neither true or false (as they cannot be PROVEN to be this or that), or we have to accept that the axioms we have chosen to build our theoretic edifice are not necessarily true (whatever "true" means in this context). Of course Godel used self-referencial paradoxes again: there is no reason to believe that there are inconsistencies beyond such special cases (hmmm, can somebody prove it?) I'm not expert in the matter, so I don't really know whether mathematicians really try to figure out what's wrong, or they just accept the fact that maths is a non-perfect tool (although probably the "most perfect" one can find) just to do the job.

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