Musings on set creation and a question about it

[nQue]

Hello! :)
My background:
MSc in engineering (up to, but not exceeding, multivariable calculus) and the rest is just free time hobby research. I ponder things because it's fun.

My musings:
I enjoy the idea of sets living within some kind of "universe", so that before a set is used or referred to it must be created. I have not seen anyone refer to set creation before, so let me briefly explain what I mean:

I dislike the phrase "Let S be a set, where blablabla..."
solely because it conjures up the set S out of thin air.
I prefer the phrase "Set S (created from set Q by algorithm blablabla)"
because it provides a solid grounding point for the existence of that set, thus decreasing the number of assumptions made! :)

Basically, I want to change all "If we blindly assume S exists then..." into "In case Q exists then..." by always requireing all sets to be created before they're referred to.

For this, I envision two different ways to create sets: By adding elements to another set (the additive way), or by removing elements from another set (the subtractive way). This creates a chain of dependencies, where the existence of the tail depends on the existence of the head. I have a vague idea that any set should be able to be created by adding elements to the empty set, or by removing elements from the infinite set. Since the additive way should be impossible due to the non-existence of the elements, this leaves us with the subtractive way. The infinite set should be able to refer to parts of itself when listing what elements to remove, to create the new set. Thus the subtractive way should be possible, while the additive should not.

The reason I want this is because I have a gut feeling that it should provide me with better insight into how to resolve certain paradoxes.

My question:
Does this sound like anything you're familiar with? Is there a jargon term for what I'm thinking about? Do you know of any research paper or book that I may read about this?

 

[Fredrik]

Every set theory has axioms that tell you in what ways you are allowed to construct new sets from the ones you already have. I'm somewhat familiar with the most popular set theory, called ZFC set theory. Its axioms have a few things in common with what you're describing, but are pretty different in some ways.

One of the axioms guarantees the existence of an infinite set. This is necessary if we want to incorporate the mathematics of integers and other types of "numbers" into set theory. And we do of course. What else would be the point of set theory?

There's at least one axiom that allows you to take a set and define one of its subsets by specifying a "property" that some of its elements has. For example, you can define "the set of all integers n such that n is odd". In this case, we start with the set of integers and keep only the odd ones.

But there are also axioms that allow you to take a set and construct a much bigger one. The most extreme one is the power set axiom. It says that for any set X there's a set P(X) such that the elements of P(X) are precisely the subsets of X. So P(X) is "the set of all subsets of X". This set is called the power set of X.

Hrbacek & Jech may be the best place to read about ZFC set theory. If you are especially interested in definitions of "numbers" in the framework of set theory, Goldrei is an alternative.

 

[nQue]

Oh thank you!
I had came upon it before but the article was so dense that I could not determine of it contained anything I was looking for. This time I re-read the summary (which suddenly seemed less dense) and then skipped straight to the axiom of infinity, which leads me to believe that this is very close to what I was looking for! :D

Also, I just found out the proper jargon for my "universe" = "Domain of discourse"

 

[Dmobb Jr.]

One other thing to consider is that you can not simply refer to THE infinite set. As I am sure you know there are many infinite sets. If you just choose one of them to be your main infinite set and take away from it you will never be able to make its power set (which surely exists).

It is also important that you must have at least one set which just exists. In ZFC this is the empty set and the set that you get from the axiom of infinity.

You said "I dislike the phrase "Let S be a set, where blablabla...""
Usually people are using valid sets and most classes would not get anywhere if they had to prove that every set existed. However, you can always verify if they exist if you do not trust your proffessor.

 

[nQue]

Thanks for pointing that out, when I had the first rough idea that was not clear to me; I thought there was only one. After having sketched with it for a while i realized there were many infinite sets (of greater and greater magnitude), but I didn't go back and incorporate that into my basic ideas. Thanks for reminding me I have thinking left to do.

I have no professor to refer to - I take no classes. This is a free time hobby, way out of my league really. But fun.

Two more things; Firstly I should point out that, to me, it appears there may be many different null sets as well. Some are "even more null". I get this result by starting with some finite set A and subtracting the infinite set (ω), yielding the empty set, and then comparing that to starting with the same set A and subtracting the superset of the infinite set (ω^2). On first glance this should just yield the same null set as the first operation, but on second glance... do a size comparison, and it appears... TO ME AS AN IGNORANT HOBBYIST... that the first operation yields a set that is smaller than all other sets, which we normally call the empty set, but that the second operation yields a set that is even smaller... urging me to conclude that neither of them actually arrived at the true zero, just a degree of zero. From the viewpoint of the set A that we started with, both of them should appear to coincide, but from the viewpoint of one of the nulls, they differ by "an order of magnitude". Anyway, that was just a side note and I thought you might be interested. I am interested if you have a reply ;)

Secondly; Is it really sane to let empty set be a set which "just exists", while the infinite set have to be introduced via an axiom? On first glance I would find it equally valid to let the infinite set be the one that "just exists" and then state the "empty set axiom". But it is way past my bedtime and I have not had time to think about this properly.

 

[Fredrik]

Firstly I should point out that, to me, it appears there may be many different null sets as well. Some are "even more null".

The term "null set" should be avoided. It's not used in set theory, and has a different meaning in measure theory.

One of the ZFC axioms says that two sets are equal if they have the same elements. To be more precise: For all A and B, we have A=B if and only if every element of A is an element of B, and every element of B is an element of A.

An immediate consequence of this is that A and B are empty, then A=B. So there's at most one empty set.

What if B is empty and A is a subset of B? By assumption, every element of A is an element of B. And since B is empty, every element of B is an element of A. Now the axiom implies that A=B.

Is it really sane to let empty set be a set which "just exists", while the infinite set have to be introduced via an axiom? On first glance I would find it equally valid to let the infinite set be the one that "just exists" and then state the "empty set axiom".

Many presentations of ZFC set theory start with an axiom that says that there's a set that has no elements. But we don't actually need this axiom, because it can be proved as a theorem, using the axiom of infinity (#7 in the Wikipedia article) and the axiom of separation (#3).

 

[Dmobb Jr.]

I think you do not have a very good definition for subtracting one set from another. Typically we define A-B = The set of x in A with the property that x is not in B. With this definition your two empty sets should be the same size.

Also you need a definition of size comparison which for infinite sets is not not as straight forward as "which has more?" Look up cardinality of sets to learn more about size comparison.

Many presentations of ZFC set theory start with an axiom that says that there's a set that has no elements. But we don't actually need this axiom, because it can be proved as a theorem, using the axiom of infinity (#7 in the Wikipedia article) and the axiom of separation (#3).

In fact we do not even need the axiom of infinity. We just need an axiom that says there is at least one set. If there is a set then by the axiom of separation (subset) the empty set exists.

 

[Fredrik]

In fact we do not even need the axiom of infinity. We just need an axiom that says there is at least one set. If there is a set then by the axiom of separation (subset) the empty set exists.

This is true, but the only axiom that guarantees that there's at least one set, in Wikipedia's version of ZFC, is the axiom of infinity.

You (Dmobb Jr) probably know this already, so I guess this is for nQue: Some authors like to take advantage of the fact that any theorem that can be derived from the axioms can be used as an additional axiom without changing the content of the theory. What they do is to add an axiom at the start of the list that says "there's a set with no elements" or just "there's a set", and the reason they do it is that this makes it easier to start explaining the theory to people who have no previous experience with the language.

 

[nQue]

Aha, I see your objections. Thanks, I will think about them. Oh and I did know that about theorems and axioms actually, but thanks anyway.

I just have one more idea I want to poke you guys with:
You seem to have a different idea compared to me, about when one set is equal to another. So let me ask this:

If I have a function F that does something to a set B, so that:
A = F(B)
i have the strong feeling that A is never the same thing as B, even if the function performs no change.
Is there ANY way to view this where my feeling would be correct?

Or do I have to redefine "equality" for that?

 

[Fredrik]

If the function "performs no change", then we certainly have F(B)=B.

For all sets A and B, A=B if and only if every element of A is an element of B, and every element of B is an element of A.

 

[Dmobb Jr.]

Suppose your function F had an inverse which we will call G (There are certainly functions with inverses). Now consider the composite function GF. G(F(B)) is certainly equal to B yet is B != B.

(!= means not equal to)

 

[Dmobb Jr.]

Also if we are going to be talking so formally about equality we should also be talking very formally about functions.

A function is typically defined as a set of ordered pairs. For example the function f(x)=x^2 IS the set of (a,b) such that a^2=b.

Another thing I should not is that the ZFC axioms are used because mathemeticians believe that they represent their intuitive sense of sets. If your intuition is different then you should use different axioms. (Make sure your intuition is different though).

 

[Fredrik]

G(F(B)) is certainly equal to B yet is B != B.

I don't understand what you mean here. You seem to be saying that B≠B.

There's a "not equal to" symbol in the quick symbols list to the right when you type or edit a post.

 

[nQue]

I don't understand what you mean here. You seem to be saying that B≠B.

There's a "not equal to" symbol in the quick symbols list to the right when you type or edit a post.

I think he meant "yet is B≠B according to nQue" :)

 

[Dmobb Jr.]

Yes that is what I meant. By nQue's statement. F(G(B)) = B implies B != B.

Sorry about the !=. I'm on my phone