Understanding the Basics of Set Theory and Nul-ary Operators for Self-Education

[Bob3141592]

My efforts at self-education aren't going well. In trying to read that paper on wheels, I kept having to look up terms, which takes me to all sorts of other topics, and I lose focus. I also realized I didn't sufficiently understand the ideas I was reading about. So I kept going back and ended up with the basics of set theory. Am I on track with these ideas here?

A set is a collection of elements, and at its lowest level these elements don't even have to be of the same type. We don't know anything about these elements except that they are contained in S. Even if they are of completely different types, they must have at least one attribute in common, that attribute which qualifies them for inclusion in S. At least this should be true if there is a rational reason for any object to be in S.

Can I say that this inclusion attribute need be no more than the ability of S to name $$\epsilon$$. Does it have to be anything more? We can name any $$\epsilon$$ by a unique nul-ary operator. There have to be as many such operators as there are elements in S. I suspect a nullary operator can't do anything more than name an element, so there should be no others. It's an identification operator - and I guess as far as operators go it's an identity. But it kind of an awkward operator, in that its not a generalization in any way. For example, we cannot label that operator with a subscript like $$O\sub{i}$$ unless we know the elements of S can themselves be sorted. Does the ability to be sorted imply at least one other attribute common to all elements of S? Does it sound right that the index of the nul-ary operator is the name (or in a sense the value) of the element? Can there be any other kind of nul-ary operators?

 

[CRGreathouse]

Even if they are of completely different types, they must have at least one attribute in common, that attribute which qualifies them for inclusion in S. At least this should be true if there is a rational reason for any object to be in S.

But there doesn't need to be any rational reason for the composition of S. Maybe I have an uncountably infinite number of monkeys, one associated to each real number, and I have them each flip a coin. Heads, their number is in; tails, out.

Can I say that this inclusion attribute need be no more than the ability of S to name $$\epsilon$$. Does it have to be anything more? We can name any $$\epsilon$$ by a unique nul-ary operator.

I don't think anything is gained by giving each element its own 0-ary operator. Now you have a whole bunch of operators... so what? I'd prefer a relational predicate P(s) that is true if s is in S and false otherwise.

Does the ability to be sorted imply at least one other attribute common to all elements of S?

A set can be well-ordered ("sorted") if it is countable, and some uncountable sets may be well-orderable as well. Under the Axiom of Choice, all sets can be well-ordered.

 

[Bob3141592]

First, thanks for your reply. Please accept that I'm not trying to be argumentative. I'm trying to find a way to ask the right questions.

But there doesn't need to be any rational reason for the composition of S. Maybe I have an uncountably infinite number of monkeys, one associated to each real number, and I have them each flip a coin. Heads, their number is in; tails, out.

I agree there doesn't have to be a rational reason for what elements are in S, and those sets are likely to be impossibly troublesome to work with. I've tried to think about this, and can't figure out how to even describe or identify the elements of such a set. In the example you provided, there is a rich structure underlying the elements of the set which the set takes for granted, and although it's weird, it's hardly irrational. This is a human failing - people are notoriously bad at inventing really random or irrational sequences.

I don't think anything is gained by giving each element its own 0-ary operator. Now you have a whole bunch of operators... so what? I'd prefer a relational predicate P(s) that is true if s is in S and false otherwise.

In my original question, I was trying to wrap my head around the notion of a nul-ary operator. What that might mean in a higher context I don't know, since I don't understand how the notion of a nul-ary operator works. In your predicate, I think you have a unary operator P acting on a nul-ary operator which returns s. P then identifies s as an element of S or not. Otherwise without the nul-ary operation defined on P how can you know anything at all about s? How can you even know it's legitimate to take s as an argument in the predicate relation? What could make P defined on every conceivable and inconceivable s? How could P be expressed except circularly?

A set can be well-ordered ("sorted") if it is countable, and some uncountable sets may be well-orderable as well. Under the Axiom of Choice, all sets can be well-ordered.

I must admit the Axiom of Choice makes my makes my head hurt. Perhaps a better understanding of nul-ary operators will give me a place to see the Axiom of Choice from a more stable perspective.

Sorry if my questions aren't very clear. I don't even know what I'm trying to get at yet. I only hope I'll recognize it if I get there.

 

[CRGreathouse]

In my original question, I was trying to wrap my head around the notion of a nul-ary operator. What that might mean in a higher context I don't know, since I don't understand how the notion of a nul-ary operator works.

A 0-ary operator is as simple as you think it is: a constant. Don't overthink it to the point you don't understand!

In your predicate, I think you have a unary operator P acting on a nul-ary operator which returns s.

Too complicated. My function maps an element of 'something' (the domain of discourse) to an element of the set {true, false}. So f, the indicator function of {0, 1}, returns results like this:

f(0) = true
f(9) = false
f(apple) = false
f(0 + 1) = true

I must admit the Axiom of Choice makes my makes my head hurt. Perhaps a better understanding of nul-ary operators will give me a place to see the Axiom of Choice from a more stable perspective.

Within the context of axiomatic frameworks, you can't just assume that something you can describe is actually a set or an object. You have to show that it is. So you can't just say that {0, 1} is a set; you get it from applying the Axiom of Pairing to 0 and 1. You can't just assume that 0 (defined as {}) exists; you know it exists by the Axiom of the Null Set. You can't just assume that 1 (defined as {{}}) exists; you know that it exists by applying the Axiom of Pairing to {} and {}.

Similarly, the Axiom of Choice says when you're allowed to say that a certain function exists.

 

[Bob3141592]

This isn't making things any easier! Let me ask two distinct questions, and hope for two distinct answers.

First, is what I said in my first post actually wrong? Useless, over thought, pointless, whatever, was it simply wrong? Was it wrong from the beginning?

Sure, we can call a nul-ary operation a constant. OK, but it sounds so atomic. A constant of what, any thing? Nothing? Whatever kind of mathematical objects they are, from the perspective of S they are constant. From the perspective of some other set, T, they might not be constants. They might have a lot of structure, just about stuff that S doesn't care about. Perhaps in some higher function all that structure in T just maps to a single point in S. Nothing wrong with this, is there?

OK, second question. These constants that you speak of, 0 and 1, they are not the same ones that I knew from kindergarten, are they?

See, I thought that every set contained the empty set. Not so, I've learned, it's the power set that contains the empty set. I'm trying to understand the difference, then you go and define {} as 0 and {{}} as 1. It's enough to make me dizzy! And, it makes me feel insecure that I know anything about the set operators, particularly intersection. I've seen that simply defined as all the points that belong to both of the sets. No qualifications. This is a total binary operation. But now let's take two sets, A as the set of all even natural numbers, and including the null set. And B as the set of all odd natural numbers, also including the null set. The intersection of those two sets is the null set. Now we change B to exclude the null set. The intersection of the two sets is still the null set, right? It's as if the exclusion never occurred. Either every set contains the null set, or the intersection of two sets isn't always something that can be found in both sets.

This goes back to a question I asked in another thread, but I never got a clear answer. Is there a notion of an empty element? Does that make any sense? It clearly wouldn't be zero. If we try to say what it is, or even what it isn't, then it becomes something other than nothing.

Thanks for the help.

 

[CRGreathouse]

First, is what I said in my first post actually wrong? Useless, over thought, pointless, whatever, was it simply wrong?

I'm going to go out on a limb and suppose you mean post #3.

"In the example you provided, there is a rich structure underlying the elements of the set which the set takes for granted, and although it's weird, it's hardly irrational."

I'm not sure that there is a rich structure. A set is 'just' a set -- the way that it was produced doesn't matter. (This actually has a name: the axiom of extensionality, one of the basic ZF axioms.)

"In your predicate, I think you have a unary operator P acting on a nul-ary operator which returns s. P then identifies s as an element of S or not."

I was describing a function $P:S\to\{\mathrm{true},\mathrm{false}\}$. You describe a function $P:(\{\}\to S)\to\{\mathrm{true},\mathrm{false}\}$. These can be thought of as the same, but strictly speaking they're different: mine takes a value, yours takes a function that maps to that value. This is a bit of a nitpick, but you asked...

"Otherwise without the nul-ary operation defined on P how can you know anything at all about s?"

Now we get to the part that I fail to understand you. I don't know what you mean by a null-ary operation defined on P (P is a function! I don't suspect you mean it in the literal sense of mapping from the set of ordered pairs defining it.) Further, I don't understand how lacking this object makes it hard to know things about s.

Other than that, the post has just questions and statements I'm fine with.

Sure, we can call a nul-ary operation a constant. OK, but it sounds so atomic. A constant of what, any thing? Nothing? Whatever kind of mathematical objects they are, from the perspective of S they are constant.

This depends on the framework we're in. If it's ZF, they would be sets. In naive set theory, they would be "any thing" in some vague, undefined sense. If that vagueness bothers you, join modern mathematics and choose some framework that gets rid of the problem!

Whatever kind of mathematical objects they are, from the perspective of S they are constant. From the perspective of some other set, T, they might not be constants. They might have a lot of structure, just about stuff that S doesn't care about. Perhaps in some higher function all that structure in T just maps to a single point in S. Nothing wrong with this, is there?

I don't understand you at all. Example?

I call f(x) = 7 constant because no matter what you do to x, f(x) stays the same. I don't call g(x) = x constant because you can modify g(x) by a suitable choice of x. But in both cases, the value of y is irrelevant (unless x is defined in a way based on y, of course).

OK, second question. These constants that you speak of, 0 and 1, they are not the same ones that I knew from kindergarten, are they?

The constants you knew from kindergarten were formulated in a naive kind of arithmetic: they just 'were'. To avoid various paradoxes and to simplify (!) the fundamentals of mathematics, various systems were developed that could show rigorous things about numbers (and other mathematical objects). In ZF, the only objects are sets. To work with numbers, you define them in terms of sets. Zero is the empty set (the one with no members: {}), and the successor of a number (the "next one") n is $n\cup\{n\}$. Addition is defined on the basis of this framework. This system, then, is a formalization of the system you learned as a child.

0 = {}
1 = {0} = {{}}
2 = {0, 1} = {{}, {{}}}
3 = {0, 1, 2} = {{}, {{}}, {{}, {{}}}}
etc.

Peano arithmetic is the general formulation of arithmetic, independent of the exact framework; this system above is the ZF formalization of Peano arithmetic.

See, I thought that every set contained the empty set. Not so, I've learned, it's the power set that contains the empty set.

The set {triangle, square, circle} has only three elements: triangle, square, and circle. If it contained the empty set as a member it would have at least four elements. Its power set is the set of some, none, or all of its members, to wit:

{
 {},
 {triangle},
 {square},
 {triangle, square},
 {circle},
 {triangle, circle},
 {square, circle},
 {triangle, square, circle}
}

which has eight members, one of which is the empty set.

And, it makes me feel insecure that I know anything about the set operators, particularly intersection. I've seen that simply defined as all the points that belong to both of the sets. No qualifications. This is a total binary operation. But now let's take two sets, A as the set of all even natural numbers, and including the null set. And B as the set of all odd natural numbers, also including the null set. The intersection of those two sets is the null set. Now we change B to exclude the null set. The intersection of the two sets is still the null set, right? It's as if the exclusion never occurred. Either every set contains the null set, or the intersection of two sets isn't always something that can be found in both sets.

This, at least, I can explain to you.

As a technical matter, let's agree to define the natural numbers in some way where {} is not a natural number. (This would obviously be a problem: if {} = 7, for example, then the intersection of sets containing only, respectively, the empty set and 7 would be {7}, which is confusing.)

Let's call the even natural numbers E, the odd natural numbers O, and the even and odd (respectively) natural numbers along with the empty set E* and O*. The intersection of E* and O* is {{}}: the set of the empty set, a set with one member. They have this one member in common, but nothing else. The intersection of E* and O is {}: the empty set, a set with zero members. The sets have nothing in common.

Is there a notion of an empty element?

Sure, why not? It's the set of elephants in my pocket.

 

[Bob3141592]

I'm going to have to think about all of this. I've printed the thread out and will try to address whatever points seems most troublesome to me. There are plenty of them. Plus details on some of the Wikipedia pages I've read make me uncomfortable. It's a busy week, but I'll try to get back to you on Thursday, or after the weekend.

Thanks.

 

[CRGreathouse]

I'm going to have to think about all of this. I've printed the thread out and will try to address whatever points seems most troublesome to me. There are plenty of them.

If it's any consolation, it took me a while to get used to all this stuff too. Take your time; we'll try to figure out what needs to be explained when you're ready.