Is the full set possible (logic based question).

[lawtonfogle]

Being that my college puts logic as a subset of philosophy, I will post this question here.

In logic, you have both the full set and the null set. They are opposites of each other. The null set is a set containing nothing, and the full set is a set containing everything. So my question is this. If the full set contains everything, shouldn't it contain either a sub-set A or object a that requires the exclusion of a different sub-set B or object B? If it includes A or a, then it cannot include B or b, and is not the full set. Otherwise it contains B or b but not A or a, and once again is not the full set. So my question is this, if the above is true, how can the full set exist?

 

[WingsOfTheNeophyte]

I dont quite understand the question at hand, from the point of clarity

Is the question: Can the full set contain two mutually exclusive subsets

Or is the question, Should not the full set contain the null set within itself?

Ill do my best to answer upon clarification

 

[lawtonfogle]

It would be the question: Can the full set contain two mutually exclusive subsets?

I only mentioned the null set to give the opposite of the full set.

 

[elcapitan]

Being that my college puts logic as a subset of philosophy, I will post this question here.

In logic, you have both the full set and the null set. They are opposites of each other. The null set is a set containing nothing, and the full set is a set containing everything. So my question is this. If the full set contains everything, shouldn't it contain either a sub-set A or object a that requires the exclusion of a different sub-set B or object B? If it includes A or a, then it cannot include B or b, and is not the full set. Otherwise it contains B or b but not A or a, and once again is not the full set. So my question is this, if the above is true, how can the full set exist?

I am by no means an expert in set theory, but it seems to me that you have to establish the existence of a set or object that requires the exclusion of another set or object.

 

[lawtonfogle]

I am by no means an expert in set theory, but it seems to me that you have to establish the existence of a set or object that requires the exclusion of another set or object.

I have, and my question is how can the full set exist if it must contain both of these, but cannot.

 

[WingsOfTheNeophyte]

Logically, I cannot see any way the full set can contain two mutually exclusive ones, Im quit stumped here, will think a bit more though

 

[lawtonfogle]

And that is my problem. If it cannot include both, it is not the full set. So my question is, is there a full set?

 

[DJPavel]

It would be the question: Can the full set contain two mutually exclusive subsets?

I only mentioned the null set to give the opposite of the full set.

Sure it can. humans and lions are two mutually exclusive sets (human if, and only if not lion). The set "mammals" contains both.

The null set is not the opposite of the full set. The null set is a set with no members, and the full set is the set with all members of a given class, that's it. You might say that a null set is mutually exclusive with a full set in the set of all sets, but then again, it's not really "opposite". I wouldn't use that term, it's too vague.


I sense that what you were trying to get at in your original post is what is called "Russel's paradox" - Is the set of all sets that are not members of themselves, a member of itself. Answering "yes" or "no" creates a contradiction either way. To translate the paradox into a more natural language, would be to ask for example, In a village where everybody, who does not shave himself is shaven by the barber, who shaves the barber? If the barber shaves himself, then somebody who does shave himself is also shaven by the barber - contradiction. If somebody else shaves the barber, then it's not the barber who shaves somebody who doesn't shave himself - a contradiction again.

There have been a few "resolutions" to the paradox, including Russel's "theory of types" which has been considered more of a kludge than a resolution.

I personally go with a pragmatic view on this paradox - the setup in the paradox doesn't exist, it's logically impossible to begin with. If I asked you, Is the king of France alive, how would you answer this simple 'yes' or 'no' question? You can't, for it presupposes a setup that doesn't exist. While this analogy is not precisely the same beast as the Russel's paradox, it illustrates that a lot of paradoxes only appear to be paradoxes because the language they are formulated in disguises an inconsistency in the setup to begin with.

DJP

 

[lawtonfogle]

Humans and lions are not two mutually exclusive sets, otherwise mammals could not have both. By definition, if mammals has both of them, then the sub sets cannot be mutually exclusive. Also, the null and full sets are opposites. I am not talking about any set of objects when I mention the full set, I am instead talking about all objects. In which case the inverse of one is the other, and they are opposites.

Also, I am not asking the question you purposed.

 

[DJPavel]

Humans and lions are not two mutually exclusive sets, otherwise mammals could not have both. By definition, if mammals has both of them, then the sub sets cannot be mutually exclusive. Also, the null and full sets are opposites. I am not talking about any set of objects when I mention the full set, I am instead talking about all objects. In which case the inverse of one is the other, and they are opposites.

Also, I am not asking the question you purposed.

"mutually exclusive sets" does not mean two sets cannot be both members of a superset. It simply means that members of one set cannot be members of the other, that's it. Let me offer you a different analogy. You flip a coin 4 times. What is the set of all possible outcomes? It's 2 (head or tail) to the power of 4 (tries), which is 16. So, the set of ALL outcomes has 16 members in it. Now, you can divide this set into the set of "all heads" and "all tails". These sets are mutually exclusive, as a tail toss cannot logically belong to the set of all "all heads". But both "all heads" and "all tails" sets belong to the set of "all possible outcomes". Did that make sense?

Sorry about the Russel's paradox, I thought that's where you were going

DJP

 

[peano123]

Another interesting theorem is Godel's Incompleteness Theorem.

Mathematics is based on a finite set of axioms or propositions. The combinations of the axioms create more propositions (which is, essentially, what algebra is). However, there are some mathematical represented propositions within, that cannot be concluded by any combinations of the original set of axioms.

 

[Hnefi]

Lawtonfogle, you seem to use the term "mutually exclusive set" in the meaning of "if set A and B are mutually exclusive, they cannot both exist at the same time (in the same system)". As far as I know, no such sets exist in logic. If you have shown the contrary, would you mind demonstrating your proof?

 

[lawtonfogle]

Sorry for the long break in post...

Anyways, I am saying that would not they have to exist. Only if one would say that two fully mutually exclusive sub-sets (or objects) cannot exist in logic do we get around the problem, and therefore this may be the answer, that no such sub-sets/objects can exist.

 

[necroforest]

Maybe I'm misinterpreting the question, but it seems pretty easy to me.

Let A = {1,2,3} and B={a,b,c}, two mutually exclusive sets (A \intersect B = null)

Let F (the full set) = {..., A, B, ....}, or, equivalently,
F = {..., {1,2,3}, {a,b,c}, ...}

 

[variant]

Being that my college puts logic as a subset of philosophy, I will post this question here.

In logic, you have both the full set and the null set. They are opposites of each other. The null set is a set containing nothing, and the full set is a set containing everything. So my question is this. If the full set contains everything, shouldn't it contain either a sub-set A or object a that requires the exclusion of a different sub-set B or object B? If it includes A or a, then it cannot include B or b, and is not the full set. Otherwise it contains B or b but not A or a, and once again is not the full set. So my question is this, if the above is true, how can the full set exist?

Russell had a simmilar issue with a set of all sets that don't contain themselves.

http://en.wikipedia.org/wiki/Russell's_paradox