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Admiral Coeyman VS. Russel's Paradox


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Admiral Coeyman VS. Russel's Paradox

 

Me VS. Russel's Paradox

 

BY:

 

Admiral Coeyman

 

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Simply put, The Russel Paradox is an old riddle which exemplifies one of the shortcommings of mathematics in science without even knowing it. We define a set, R, which contains all sets which do not contain themselves as members. Should R contain R? If yes, it shouldn't. If no, then it should.

 

Commonly, this is expressed as the index catalogs of several libraries. Some librarians considered the catalog to be a book in the library and listed it in the catalog. Others did not think that the catalog should list itself and left it out. All the catalogs are then taken to a central library where two additional catalogs are prepared. One, which we are not interested in, lists all catalogs which report the catalog as a book in the library. The other is the Russel catalog.

 

If the Russel catalog does not list itself, then it is a catalog which does not contain its own listing and therefore it should. Once it contains this listing, it is a catalog which contains its own title and should be listed in the other catalog instead. This paradox killed set mathematics.

 

The second thing that I noticed, I'll show you the first later, is that the two catalogs in the central library aren't like the other catalogs. Each of these catalogs contained lists of books in other libraries, where each was prepared. R, our paradox catalog, actually contains lists of books which are not stored in the central library. They were simply there to be indexed. For reasons unrelated to the paradox, R should not contain itself.

 

My first observation is that there is a version of the Russel catalog which contains itself and one which doesn't. Commonly these would be R (which will contain itself since it's the original state) and R' which should not contain R. Think about a set, L, which is a set of all lights which are on. You will notice that L is only definite in the moment it was made. For instance, the hard drive light on your computer may flash. Have you ever used your caps lock key? That has a light.

 

I saved that for here because it is the scientific problem mentioned earlier. Any time you reduce a 'real world' problem to math, you lock it to a specific point in time-space where that measurement was made. This is not a good way to model a kenetic universe, N-Space. N-Space is vulnerable to the Heisenburg uncertainty principle both is structure and momentum.

 

Einstein wrote," As mathematics applies to reality, it is not certain. As math is certain, it does not apply to reality."

 

All of what I have written above would be good if we were allowed to add this R' (R-Prime) principle to set mathematics. It's a perfectly valid move which is actually required to reduce the theory to practice. Doubtlessly, I am not the first to come up with this, yet, the Russel Paradox is still considered unsolved. Maybe we want some uncertainty in our world...

 

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I'll start by explaining what sets are like to me. It would be irrational for me to state that mine is the exact nature of sets since I'm far from perfect in my understanding of anything. Only when the Russel paradox is solved, and all takers backed down, will the nature of sets be locked into a definite form. For now, I'll simply try to solve the paradox with my understanding, limited by the fact that I may not alter the premise of sets at the level of practice. There can be no R' additions in my attempt.

 

A set is a list of things which are defined by compliance with a rule. Anything which does not conflict with the rule may be included in a set, including other sets. The rule alone discriminates which elements are in which set or sets. And, this is important enough of a concept to restate twice.

 

Using this simple idea, and the Russel paradox itself, I can only achieve a definite solution by showing that R is always or never an element of R. I firstly tackled the always proposition with the idea that all sets must contain themselves by the identity property. Any two sets which contain exclusively the same elements are the same set. Since R contains all the elemtents of R, as all sets must, then R must always contain R. This is indirection since R would contain R by containing R's elements, but not R by name.

 

Russel would not be amused.

 

The idea of the Russel paradox is recursion. R must be included directly as an element of R to always satisfy the condition of the rule. This is not the case since R is indirectly included and therefore included only to one level of depth. Look into the R inside of set R and you will find that this copy of R would contain an additional copy of all the elements of R. That set will contain an R set with all the same elements again.

 

I went on to create an additional set which contained R and was an element of R because it did not contain itself. Y is the set of all sets other than set Y. R contains Y which contains R which contains Y... That is subscripting which technically works. But then again, R contains Y -- not R directly.

 

My final attempt was to create a set, Q, which contained the elements of set R. It was a cute trick, since Q is the same set as R, however, it was also invalid. Q contains R by indirection and is R by identity, yet Q contains Q the moment it is an element of R. We're then stuck in the same paradox with a new letter.

 

"Curses--foiled again."

 

This leaves us with proving that R can never become an element of R. It is also the reason I restated my view of sets in the beginning. For R to be ineligable to become an element of R, there has to be something to block membership. It is like the book example I cited earlier. What would it take to keep a set from becomming an element of a set?

 

As I repeated, only the rule which defines a set can set limits on membership. I cannot rewrite the rule to exclude R from consideration since the original R set will still exist to keep the paradox going. Even as it is strongly implied, I could not set R to The set of all sets which always contain themselves as members.

 

My attempt at this is to say that R is not an element of R because it would cause a paradox. This rule neither states nor implies that preventing a paradox is important to membership in the set, R. All the rule says is that no set which contains itself may be a member of R. This rule then forbids R to ever be an element of R.

 

How do we get around the fact that R, which never contains R is not in the set R? Static. Once the rule is examined to determine membership, no element is re-examined. Since the rule forbids R to be an element of R, it does not matter than R later does not contain the element R.

 

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L is the set of all lights which are on. I turn off a light. Given this data, has the set L been destroyed or changed? If the elements of a set are more important to a set than the rule which defines it, then turning off a light would destroy L as easily as turning on a light. Yet, L continues to exist in this ammended form. This is an important concept which simply restates what I started with. Sets are defined by their rules and not their elements.

 

How do we apply this to the set R? We can no longer look at the elements of a set to define its membership in R. Other sets can exist which do not contain themselves, are non recursive, only some of the time. E is the set of all sets which have an even number of elements. Depending upon how many sets you have, E could itself contain an even number of elements at some time, thereby becoming recursive.

 

An additional paradox comes to bear here because E has to contain an even number of elements before it can contain itself and, once recursive, E will have an odd number of elements. Your set E could still contain an even number of elements and be recursive if you allow E to be counted when you add it. You know that, once E is added to the odd number of sets, E will be even. No law is violated by this fact.

 

This means that we have to look at the rule which defines a set to determine its inclusion in R. The Russel Paradox is caused by absolutising the set R. Any element which is contained in R must always be contained in R or there is no paradox. Static sets, those which require compliance with the law defining them at more than one point in space-time, can contain only elements which are defined by rules which require compliance with the rule of set R.

 

We would not have this problem if you did not look at the set R once to determin if it should be included in R (true), and then look again to see if it is still eligable for membership at a later time (false.) It is this static nature of R, where every element must be contained in R at all points of space-time, which gives rise to the Russel Paradox.

 

All sets which are not sets of sets are included in R because their defining rule forbids recursion. Only sets of sets which exclude the possibility of recursion may be included in R, thus R is never included in R. You could then say that the rule which defines R now forbids membership to R, thus R is included in R. This would be invalid because R is excluded because the rule defining R fails to disqualify R as a potential element in R.

 

You could now look at R and see that R is not an element of R. This would not allow you to list R as an element of the static set of R since static sets must not change. If R was a dynamic set, allowed to change, then R could sometimes contain R. R would contain a version of R which does not contain R and the recursion would abort there. Logically, there is no way of having a paradox without also allowing this solution.

 

All I have added to set mathematics is words. I did not add the concept of primes or limited recursion because these have been tried and rejected. 'Do not include themselves as members' means exactly that you can look in the set at any time and not see recursion. This must always be true or never be true; There is no allowance made for 'sometimes contain themselves as members.' My challange stands at this.

 

--Admiral Coeyman

 

May you live as long as you wish and age but a single day.

 

 

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RE: Admiral Coeyman VS. Russel's Paradox

 

Yay! A math post! Nice to see one before I go. It's been interesting watching this forum, I'm off to other interesting things. The lack of attention span that TV has infised me with strikes again. But I can't resist replying to this post...

 

I think that the 'solution' to Russel's paradox is not a solution to it at all - rather, it's a solution to a problem very much like Russel's, but differing with respect to the static/dynamic nature of the set R. The paradox itself applies to the absolute set R, that is the set that does not change with 'time'. To try and solve the paradox by resorting to making R static is not solving the original paradox, but a nearly parallel one. There's still no solution to the problem of the absolute set R. That shows a flaw in the foundations of set theory. Look at the problems with the Axiom of Choice. To solve a paradox like Russel's requires a jump into metamathematics - it can't be solved in the framework of the math it exists in.

 

Keep up the interesting conversation! One day, you guys might just achieve what you're trying to!

 

 

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RE: Admiral Coeyman VS. Russel's Paradox

 

This may mean that the russels paradox also marks a border in the mathematical universe and perhaps the solution to russels paradox is dividing all the sets in r an infinite number of times bringing the static set R outside the range of absolute R causing the counterdicting rule of the paradox unify into a single non counterdicting rule that states that R contains all sets that do and donnot contain themselves. For at infinitismal size the difference between the 'do' and the 'do not' portion of this paradox becomes zero in the same way that all angles of intersection between two parallel lines diminishes to zero at an infinite distance thus distuishing the former two parallel lines as one single straight line with an line seperating the single two paralell lines;of course since the infinitismal straight line that is seperating the two paralell lines the seperation is seen as in essance nonexistant and therefore the two parallel lines are seen as a single straight line. However I disagree with this line of logic for it would require the law thermal dynamics to be violated by destroying an entity that exists into noexistance. This is remedied by recognizing the flaw in the logic of the interpretation of the definition of infinitismal as being difined as nonexistant when in reality infinitismal is infinitismal as existant. In short something that is infinitely small is not nonexistant as we are taught in math, but instead something that is infinitely small is something that is infinitely small but still exists. Something that is negative is not something that is less than nonexistant but is something that is smaller than infinitismal. There are therefore lower levels of infinitismal just like there are higher levels of infinity as was proven by georg cantor and his transfinites. I believe zero to be a relative term that distinguishes the border of finite range of sets within that relative universe and marks the barrier between the lower levels mathematical domains which mark parallel universes sets which is the equivelant of parallel dimensions of the physical universe.

 

 

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RE: Admiral Coeyman VS. Russel's Paradox

 

This is somewhat of a general ussumption but it may answer on a broad spectrum the phenomenon of paradox: Time is the sum of two equal opposite forces acting in direct counteraction on objects in the spacetime continuum. For example if one were to pull on a rope with five pounds of pressure and another person was to pull on the other end of the rope in the opposite direction with five pounds of pressure then the one person would be applying force to the one end of the rope and the rop accelerates to the left at the same time the other person applies force to the other end of the rope and the rope accelerates to in the other direction.

 

 

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RE: Admiral Coeyman VS. Russel's Paradox

 

It's more of a paradox of the mind than anything else. A "what if" scenario if you will.

 

True recursion can be demonstrated in software. Problem is, you need infinite memory for it to recur infinetly.

 

Using recurision in software always involves the use of a "break" in the rules to end the recursion. Otherwise it will "hang" the computer. Eventually.

 

This is why Russell's Paradox is more an exercise of the mind than anything that really has a bearing on reality. Other than the fact that it has to be accounted for, it has no real pratical application.

 

 

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