18 Mar 2009

Vergauwen, A Metalogical Theory of Reference, 2.1 Sets: Finite and Infinite

[The following is summary. Paragraph headings are my own.]

Roger Vergauwen

A Metalogical Theory of Reference: Realism and Essentialism in Semantics

Chapter 2: Reference and Theory of Reference

Chapter 2 will approach references from a twofold point of view:
1) it is based on set theory as well as general principles and theorems from the study of formal languages, and
2) it is also based on the concrete theory of meaning for natural languages: Montague grammar.

2.1 Sets: Finite and Infinite

In set theory, a set is "a group of entities in which all that matters is which entities are part of the group." (14cd) We are not so much concerned with how the group was formed or what reasons these particular entities are found together in the set. An example of a set would be, 'the set of Belgian Prime Ministers' or 'the set of natural languages.' To designate that specific element x belongs to set A, we write:

x ∈ A

Whole sets themselves can also belong as subsets to other sets:

a set A is a subset of a set B, represented as A B, if every element of A is also an element of B.

A set may contain several, one, or no elements. If there are none, it is an empty set, represented with .

A set may be finite or infinite. Georg Cantor offers a precise criterion for distinguishing infinite sets into two types: denumerably infinite and non-denumerably infinite sets.

We take for example the series of natural numbers. We may find correspondences between specific subsets of numbers to other sets. Such a correspondence would allow us to compare sets based on their number of elements. Consider if the sets correspond one-to-one. And, to each element of one set we assign precisely one element from the other set. This would then be a one-to-one correspondence. By this means, for example, we might take the set of 1, 2, 3 and correspond to them the set of Benelux countries

1 Belgium
2 Netherlands
3 Luxembourg.

Such one-to-one corresponding sets are called equivalent sets:

two sets are equivalent if they are in a one-to-one correspondence with each other. (15)

We will now define a finite set. We take N to represent the set of natural numbers. We will want to speak of a set of natural numbers with a finite number. That means there will only be so many numbers going-up to a particular terminating number. We will call that terminating number n, and the set containing that many numbers is

Our formulation will say that there are n numbers in the set, and that 1 and n are included.

We now definite finiteness for a set:

A set S is finite if either it is empty, or there exists a natural number n such that between the elements of S and the elements of the set

there exists a one-to-one correspondence.

We now consider the concept of cardinal number, which is the number of elements which the set contains, and it is that property which this set has in common with all equivalent sets, or:

The cardinal number of a set is the set of all sets that are in one-to-one correspondence with this set. (16b)

We might consider sets that are not finite. For example, consider that in set N, we have the natural numbers. Corresponding to each number in the set is its double in Set E.

and so on...

We see that set E contains the set of even natural numbers. But set N also contains within it all the even natural numbers. And yet we see there is a one-to-one correspondence between all the elements of N to all the elements of one if its subsets. Set E is called a "proper subset" of N, because there is at least on element of N which does not belong to E; while yet, E is still a subset of N. This property typifies infinite sets. Thus as Cantor discovered:

a set is infinite if it is equivalent to one if its proper subsets.

Hence also, a finite set is one that is not equivalent to one of its proper subsets.

We now ask if all infinite sets are equally big as each other. At first Cantor was derided for speaking of types or sizes of infinity. Are they all not the same size? Infinite?

Cantor will establish that not all sets have the same cardinal number. To do so, we first must define denumerability or denumerable infinity.

A set is denumerably infinite if it stands to a one-to-one correspondence to the set of natural numbers. (17b)

We consider the set of natural numbers as denumerably infinite, and we represent its cardinal number with the Hebrew letter 'aleph' with subscript 0.

So, a set is denumerable if it has the cardinal number

The question is, do all infinite sets have the same cardinal number

? In other words, are all infinite sets denumerable? Cantor proves that in fact not all infinite sets are denumerable. He does so by means of the diagonal method. We now consider it in more depth.

Vergauwen, Roger. A Metalogical Theory of Reference: Realism and Essentialism in Semantics. London: University Press of America, 1993.

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