The Axioms of Set Theory

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Contemplating the theory of sets puts us at the very foundation of Mathematics. Or, to use a more suggestive metaphor, it puts us at the very roots from which the great tree of Mathematics grows. And set theory provides to us a gateway into an amazing city of the mind. We may well take the set and identify it as the most fundamental concept in mathematics, the sine qua non of a properly logical mathematics.

We define the set to consist of a collection of elements (or members) that we have assembled in accordance with some rule. We have, for example, the set of the natural numbers, which comprises all of those names and only those names that fall within a fixed sequence of our own making; thus, "twelve" lies in the set of the natural numbers but "automobile" doesn't. Mathematics, then, consists of assigning entities to sets and then defining functions that assign some elements to other sets. In that view we can paraphrase a famous syllogism as:

1. Sokrates is an element of the set of Men (inclusion rule = human, male, adult);

2. the set of Men is a subset of the set of Mortal Beings (inclusion rule = living thing of limited lifespan ending in death);

3. therefore, Sokrates is an element of the set of Mortal Beings.

And of course we have a very special set, the one whose elements are the axioms of set theory.

1. Axiom of the Empty Set

There exists a set S for all possible elements E such that no E belongs to S.

Mathematicians usually depict the empty set, the set containing no elements, as {Ø}. Another way of looking at the empty set is to take it as a membership rule that nothing satisfies. Thus, for example, the phrase "nuclear-powered automobile" gives us an empty set.

2. Axiom of Extension

For all sets S and T and for all elements E, is we make the statement "E is an element of S is equivalent to E is an element of T" , we imply that S is identical to T.

We say that two sets are identical to each other if and only if they have the same elements, which means that a set is uniquely defined by its elements (and, by implication, by the membership rule that makes them members of the set). So if set S consists of six-packs of X and set T consists of half-dozens of X, then we know that S and T are identical to each other.

3. Axiom of Unordered Pairs

For all sets S and T there exists a set Z for all elements E such that saying E is an element of Z is equivalent to saying that E is an element of S or E is an element of T.

That statement means that for any two sets we devise there exists a set of which they are both members. Thus, for example, saying that a person belongs to the set Humanity is equivalent to saying that the person belongs to the set Boys and Men or that the person belongs to the set Girls and Women.

4. Axiom of Union

For any collection of sets S there exists a set T for all elements E such that the statement that E is an element of any set S is equivalent to saying that E is an element of T.

Mathematicians also call this the sum-set axiom. It means that for every collection of sets there exists a set whose members are exactly the members of those sets. Thus, for example, every member of any of the sets citizens of the State of California, citizens of the State of Illinois, Citizens of the Commonwealth of Massachusetts, etc. is also a member of the set citizens of the United States of America. I used this axiom in step 2 of the syllogism above when I noted that an element of the set of Men is also an element of the set of Mortal Beings.

5. Axiom of Subsets

For all sets S there exists a set T for all elements U such that saying that U is an element of T is equivalent to saying that for all elements E the statement that E is an element of U implies that E is an element of some S.

That axiom is just a fancy way of saying that we can distribute the elements U of any set T into new sets S that then become the subsets of the original set T. In accordance with that axiom we can take the chemical elements from Dmitri Mendeleev's Periodic Table and put lithium, sodium, potassium, rubidium, cesium, and francium into the subset of the alkali metals, put fluorine, chlorine, bromine, iodine, and astatine into the subset of the halogens, put helium, neon, argon, krypton, xenon, and radon into the subset of the noble gases, and so on.

That axiom also allows us to rearrange the subsets within a set. If we make the membership criterion in T "railroad rolling stock" , then we have subsets that we can label locomotives, tank cars, gondolas, boxcars, and so on. From those subsets we can take elements and create new subsets that we call trains.

6. Axiom of Specification (or Replacement)

For all sets T that are elements of the set A such that there exists a set Z satisfying the relation α(T,Z), there exist sets S such that Z is an element of S. Or, given a set A and a one-place predicate P(X) that is either true or false of each element X of A, there exists a subset of A whose members are exactly those for which P(X) is true.

This axiom tells us how to create sets, which we define as collections of particular things. We use predication to create membership rules that separate the objects in an overarching set into appropriate subsets. Ultimately, in the realm of Mathematics or Physical Reality, we begin with the set of everything and, by asserting predicates, subdivide it into a hierarchy of subsets. We may then focus our attention upon any of those subsets in order to explore certain objects in the realm and their relationships with other objects. Thus, for simple example, for the set "chickens" and the predicate "lay eggs" we have the subset "hens".

7. Axiom of Powers

For every set S there exists a set T such that for all sets X the statement that X is an element of T is equivalent to saying that X is a subset of S.

For each and every set S there exists a set T, the power set of S, whose members are exactly the subsets of S. If we have S={1,2,3}, then we have the power set T={{1,2,3}; {1,2}, {2,3}, {3.1}; {1}, {2}, {3}; {i }}. If S contains N elements, then its power set contains 2N elements.

8. Axiom of Infinity

There exists a set S such that the empty set is an element of S and that if X is an element of S, then the successor of X is also an element of S.

This axiom merely tells us that in the realm of sets there exists something called infinity and that it possesses the defining property of endlessness. Creating an actual infinite set remains contingent upon a succession rule that has no limit. We have the example of the set of the natural numbers: the succession rule is simply "add one" , which converts any element N of that set into its successor N+1. Because the succession rule has no limit, the natural numbers constitute an infinite set.

9. Axiom of Foundation (or Regularity)

The existence of a set S satisfying the relation F(S) implies the existence of a set S such that F(S) holds and for all X that are elements of S the relation F(X) does not hold.

Whatever membership rule, F(S), that we predicate to put S into some other set, we cannot devise it in a way that would put S into itself as a member. This rejection of self-inclusion is necessary to avoid the creation of paradoxes, such as the well-known Cretan who asserts that all Cretans are liars.

10. Axiom of Choice

For all X that are elements of S there exists a relation F(X, Y) implying that there exist sets Y for all X elements of the set S defined by the relation F(X, Y(X)). Conversely, given a disjoint set T whose members are non-empty sets, there exists a set C which has as its members one and only one member from each member of T.

A disjoint set is one whose subsets have no elements in common. As an example we can consider T to comprise the electorate of the State of California. The subsets comprise the voters in each congressional district and the set C that we make by selecting one element from each subset is the California Congressional Delegation. There we have a good example of the converse version of the axiom of choice.

Now turn that example around. For all X (representatives) that are elements of S (the California Congressional Delegation) there exists a relation (legislative representation) implying the existence of sets Y (voters in California's congressional districts) for all representatives in the California Congressional Delegation defined by the relation "X represents the voters in California Congressional District X".

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Axioms 1 through 9 above comprise the axioms of Zermelo-Fränkel set theory (ZF). Addition of the Axiom of Choice turns that into ZFC.

Those axioms may seem less than profound B too simple and trivial to carry the full weight of Mathematics. We must remember that axioms, in order to display the property of self-evidence, are supposed to appear too simple and trivial. As in any other game, these simple rules support some extremely complicated plays. In this case we call the game Mathematical Logic. In the following essays we will see how that game plays out.

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