Basics of set theory, part 1: lists, sets, universal set, zero set and elements of a set
Set theory is the base of modern mathematics. It has been developed by Cantor in the 19th century. Later Bertrand Russell and Whitehead, developed axioms for the set theory. Set theory uses some simple concepts of daily life and refines them.
Let us start with an example:
When a housewife goes shopping, she at first will prepare a purchase list. To avoid walking back and forth in the supermarket, she will arrange all items in the same order, as the products are arranged in the super market, from the entrance to the exit.
In mathematics this purchase list is also called a list. However, a set is even simpler than a list. The order of the items is of no concern and each kind of product is put only one time into the set.
A purchase list could be, for instance: [butter, bread, milk, sugar, salt]
A set would look similar: {butter, bread, milk, sugar, salt}
For a set we use curly brackets, to stress, that a set is not a list. Because the order of the items is not important in a set, all sets with another order, are the same, like {butter, bread, milk, sugar, salt} = {bread, salt, butter, milk, sugar} etc.
What is called an item in a list, is called an element in a set. To save writing and make reading of sets easier, we agree to use:
* small letters for the elements of a set
* large letters for the sets
We write, for instance: A = {b, 3, c, g}
The symbol = shall only mean, that every time, we use A in a calculation, we could also use {b, 3, c, g} in the calculation.
Now we can define a set precisely, having 3 qualities:
1. A set is a collection of elements; all the elements of a set are written between curly brackets.
2. Each element is different from all other elements in a set.
3 The order of the elements does not matter.
What is the largest imaginable set? Let us assume, that we have a really large bag. So large, that we can put all into it, what exists: our universe, all other universes, all ideas and thoughts, which ever existed or will exist, all feelings all numbers etc.
We call this set the maximal universe set. Smax. Now we can ask, how many elements has Smax?
To count the elements, we start numbering all elements of Smax. The first element gets the number 1, the second 2, etc. , until we have given a natural number to each element of the set.
However, that procedure is impossible, as Cantor has demonstrated. Remember, we have put all into the large bag, also all real numbers and the real numbers cannot be counted, using the natural numbers. There exist infinite more real numbers, than natural numbers. Therefore, Smax is called a non-countable set.
By the way, the bag itself is also an element of Smax, otherwise there would exist something, which is not in Smax. This would be a contradiction to our construction process of Smax.
Now we apply following procedure: We take all elements out of the large bag, until it is void. Then we have a large void bag. This voidness we call the zero set.
The zero set is also an element of Smax; it must be, otherwise there would exist something, which is not in Smax and this would be a contradiction to our construction of Smax.
Now, we know at least two elements, which must be elements of Smax: Smax = {zero-set, Smax, all the others}
However, our analogy with the large bag runs into a logical problem: How can the large bag be in itself ?
# # #
Basics of set theory, part 2 : Paradoxes in Cantor`s set theory and how to avoid them
We found in part 1, that we run into a logical problem, when we try to construct the maximal universal set Smax. And we had no answer, how we could put a bag into itself.
These logical contradictions are called Paradoxes of Set Theory. Logical Paradoxes exist also in philosophy. The German philosopher Kant for instance found logical problems, when he tried to proof and disproof, that God exists. He found, that one can argue in both ways.
He called that an Antinomy, not a Paradox. But both words mean the same.
Let us check now in some detail, why this paradox occurs:
1. as long, as we construct a set, using only elements, all runs well. Because, then we cannot put the set into itself.
2. the problem occurs, when we add Smax to itself. For some reason it must be forbidden, to add Smax to itself.
Let us look at an example, what happens, when we construct a set, using elements and sets: S = {a, b} (the order of the elements in the set is not important, as we know, but it looks nicer for our eyes).
Now let us form S1 = {a, b, S} , as we know, the sign = only means, that we also can write: S1 = {a, b, {a, b}}
Is S = S1 or are that different sets ?
S1 looks similar to {a, b, a,b} and because we have defined, that only different elements are allowed in a set, we get:
S1 = {a, b, a, b} = {a, b} = S or: S1 = S
A very nice result, because no paradox would occur, when we construct the universal set Smax. But is it allowed, to let off the curly brackets inside S1 ?
There is a difference between an element a and it`s set {a}. he equation a = {a} is not valid; there is only one exception:
zero-set = {zero-set}
From our definition of a set we cannot derive a criteria, what to do with the inner brackets. We have to add a rule, which describes, what we have to do. Depending on this rule, we will get different results.This rule shall describe, how we have to construct sets with a mixture of elements and other sets and how we have to get rid of the curly brackets inside the set.
We try it with the preliminary rule: If we put a set into another set, which also has simple elements, then it must be possible, to omit the inner curly brackets and reduce the set to its standard form. This rule describes, what we have done above. (With standard form we mean, that all elements are to be different.)
Therefore S1 is not a set, as long as it has inner brackets; t gets a set, after we have omitted the inner brackets and reduced it to it`s standard form. With this additional rule to our definition of a set, also the exception:
zero-set = {zero-set} is not really an exception, because when we use the formal definition for the zero-set: zero-set = { }, we get
zero-set = { { } }
and applying our rule, we get back the definition of the zero-set: { { } } = { } = zero-set
Looks like a simple playing with symbols. And that is, what mathematics is all about. In principle, all mathematics has to be reducible to such simple playing with symbols, which in practice would not be possible. Nobody could handle it.
Now, we should define this additional rule exactly and give it a nice name, to remember easily. We cannot derive this rule from the definition of a set.Anti-Paradox Rule: If a set has elements and sets, then the set must be reducible to the standard form, either with only elements or with only sets.Remarks:
1. If a set contains only sets, then we consider these sets to be elements and we have no problems. Problems can only occur, if we have a mixture of elements and of sets.
2. Reducible means, that the curly brackets of these sets are omitted and from all elements, which show up several times, only one is taken as a representative of all of them. This construction step is required by our definition of a set.
With our Anti-Paradox Rule, we run into no problems at all, when we construct the set Smax. After we have added Smax to itself, we apply our Anti-Paradox Rule and are left with Smax.
Our analogy with the large bag, which is contained in itself, has following solution: We cannot put the bag into itself; because we have only one bag. No paradox anymore. Using inner curly brackets is the same, as if we had a second bag or even more bags.
We can define the Anti-Paradox Rule even simpler: A set has only one pair of curly brackets. But now we don’t need the Anti-Paradox Rule anymore, we just add it to our definition of a set:
1. A set is a collection of elements; all the elements of a set are written between curly brackets.
2. Each element is different from all other elements in a set.
3 The order of the elements does not matter.
4. A set has only one pair of curly brackets. If a set has only sets as elements, then we just consider these sets to be elements. But if the set is in itself, then we must open it, omit the curly brackets and reduce the set to its standard form.
