The Linguistic Limitations of Sets

saw038 September 17, 2016 at 05:29 3200 views 6 comments

If I say there exists an inside (i);

I am also saying there exists and outside (o).

I could then come up with a third category - a boarder one:

The set that contains o and i.

The definition of inside is: "the inner side of a thing."
So, let us presume that the broader category including i and o is called t (things).

But, we have produced a useless statement - a trap!

Because, the definition of a thing is "an object that one need not, cannot, or does not wish to give a specific name to."

The problem is we just gave this overarching class a name: thing; which is nonspecific..
But, I wanted to be specific with my naming, I wanted the name to only include inside and outside.

I wanted the entire set to only be comprised of (i,o).

But, what I am neglecting, is the fact the () carries some significance to it.

I don't have to define it as a 'thing' because the moment I put () around i,o...it becomes a singular entity.
What I am highlighting is a predicated relationship:

An outside is defined as "the external side of something," while an inside is: "the inner side of a thing."
The common factor being a 'thing'.

My point being is this:

The definition of inside is predicated by an outside and and vice versa;

Furthermore, they also share the existence of a 'thing'.

But, this 'thing' must be nonspecific, but my use runs contrary to its definition;

I want the set to only include inside and outside.

So,

these become my questions:

what would be the above (overarching) class of a inside/outside?

Even if the overarching set could be categorized by a word, would it not be a tautology?

If anything ever takes upon the existence of all, can 'all' not only be defined by the parts that make up all?

Comments (6)

Barry Etheridge September 17, 2016 at 13:15 #21783 0 likes

You've started with the premise that inside and outside are sets which they clearly are not. They are descriptions of the areas which become distinct only by the definition of a single set. Inside the set S are all the members of S. Outside the set S are all the non-members of S. The Universal Set (your Thing) contains the subset S. But of course it also contains every possible subset which will include subsets of S, subsets of Not-S, and subsets which have members from both S and Not-S making the whole concept of inside and outside as you originally posited it meaningless. As for the Universal Set itself, it, by definition, has no outside for nothing is Not-U and if U is infinite then it has no inside either for there is no limit (no border, if you like) to be inside of.

So to answer your questions directly ....

The Universal Set usually but Everything will do just as well as long as it literally means literally everything

I see no reason to think that at all. You've set up a false paradox.

No! Universe is a completely coherent a priori concept in and of itself, for example, and it's a set that we have far from completely defined the population of to date.

saw038 September 17, 2016 at 16:22 #21794 0 likes

Reply to Barry Etheridge I understand what your saying about the universal set and the idea of inside and outside was simply an example to illustrate a deeper point. I see the flaws you have pointed out in this example, but what I am trying to ask still remains.

I think my question comes back to the incompleteness theorem in a way and the notion that within any system there can be something that is true but unprovable.

My main point was if you get something that is all inclusive -whether all refers to the universal set or all refers to a specific subset that is just (I,O) - that word we use to define this all will only be definable by its parts.

And, if that is the case, this singular notion of all - universal or a subset - it really refers to something that is not singular. It in a sense becomes a failure of our language because if I was asked to define life?

The correct answer would be to include everything that I've ever had be a part of my life, I am asked to do so within the confines of language and all my words would fail me because I could never describe the entire human experience using simply words to express it.

So, it becomes somewhat of a meaningless question because no answer will suffice because any answer would require the use of words.

Hoo September 17, 2016 at 20:57 #21850 0 likes

Quoting saw038

My main point was if you get something that is all inclusive -whether all refers to the universal set or all refers to a specific subset that is just (I,O) - that word we use to define this all will only be definable by its parts.

And, if that is the case, this singular notion of all - universal or a subset - it really refers to something that is not singular. It in a sense becomes a failure of our language because if I was asked to define life?

I actually got into math by contemplating the idea of unity. Every thing as thing is an intelligible unity. Whatever else it has, it is wrapped up in a 'singular' concept. Look around, and you see "circles inside of circles inside of circles." The dog is a unity of its nose, tail, fur, etc. These are unities can that be broken down. The big unity, the totality from which nothing is excluded, is problematic. We can't explain it, since there's nothing outside of it to use as an explanation. Anything nice little idea we have about is instantly part of this therefore unstable totality.

You might also be interested in axiomatic set theory. Naive or purely intuitive set theory generates contradictions, so an attempt has been made to apply just enough rules so that no contradictions are easily found (though they could be hiding in there somewhere.) It's beautiful stuff. We have the ideas of containment or unity in relationship to ideas of order. All of math can be built out of a single "circle," the empty set, which is like "pure" unity itself. Being == nothingness.

saw038 September 17, 2016 at 21:26 #21853 0 likes

Reply to Hoo Very interesting and a lot of food for thought! The idea of unity and the totality of all things is really what spurred me to post my question. I think that there could be a limitation to simply the words we assign to such grand or abstract concepts. I mean, after all, words are used to differentiate this from that.

But, there are some words so broad and inclusive that they become almost impossible to define. For instance, trying to define consciousness seems somewhat problematic to me because I am using consciousness to define it; therefore, implicitly within my definition of consciousness is consciousness which seems like a tautology to me.

Either way, I will have to read some about the axiomatic set theory as you mentioned. Your comment was extremely helpful and interesting!

Hoo September 17, 2016 at 21:33 #21855 0 likes

Reply to saw038
Thanks!
Being is the limit concept. Everything is a being. But being has no particular content, for exactly that reason. I think consciousness is like this, at least from the first-person perspective. Being and consciousness are highly elusive! As I see it, being is also one or the one or the unit. So it's as if we were born with this "digital" or "atomic" insight for making sense of the world. All of math is built on this enclosed nothingness. But I'll stop there! I should actually do my homework.

(I jumped in because I had a hunch that you were processing totality, being, consciousness. Those are indeed the mysteries. )

saw038 September 17, 2016 at 21:44 #21856 0 likes

Reply to Hoo These are surely the great mysteries, and topics that can be discussed at great length. But, you're probably right that doing your homework is the more practical decision. :-}