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Infinite sets

The idea of infinity (in the Latin infinitas means "unboundedness) had been the subject of deep controversies least from the time of the ancient Greeks. One of the most known is  Zeno of Elea who  around 450 BC, with his paradoxes connected with the infinity, made one of the earliest known major contribution. His paradoxes were never satisfactorily resolved.

By the Middle Ages discussion of the infinite had led to comparison of infinite sets. At this time many believed that infinite sets could not exist.

Nicholas Cusa (1401-1464) contributed to the study of infinity, studying the infinitely large and the infinitely small.

The modern phase begins with Galileo, who observed that There are just as many integers that are perfect squares. This discovery shows that a small subset is actually just as large as the original. Later many mathematicians (and philosophers) revisited concepts involving infinity. Standard fall-back is to follow the Greeks in not allowing a "completed infinity," but only an "uncompleted infinity .

Bolzano defended the concept of an infinite set, he gave examples to show that, unlike for finite sets, the elements of an infinite set could be put in 1-1 correspondence1 with elements of one of its proper subsets, an idea to be used in the definition of a finite set.

A set typeset structure is called infinite if it is not finite , or equivalently if  its elements can be put in an  injective (one-to-one) correspondence with the elements of some its proper subset. This definition was used by R. Dedekind [1] , § 64.   Note that R.Dedekind  [1] , § 66 “proves” here that there exist infinite sets. He  says that his arguments are similar to those used by B.Bolzano in §13 of his Paradoxien des Unendlichen. This is Dedekind’s “proof”:

My own realm of thoughts, i.e., the totality typeset structure of all things, which can be objects of my thought, is infinite. For if typeset structure signifies an element of typeset structure, then is the thought typeset structure, that typeset structure can be object of my thought, itself an element of typeset structure. If we regard this as transform typeset structure of the element typeset structure then has the transformation typeset structure of typeset structure, thus determined, the property that the transform typeset structure is part of typeset structure; and typeset structure is certainly proper part of typeset structure, because there are elements in typeset structure (e.g., my own ego) which are different from such thought typeset structure and therefore are not contained in typeset structure. Finally it is clear that if  typeset structure are different elements of typeset structure, their transforms typeset structure are also different, that therefore the transformation typeset structure is a distinct (similar) 2  transformation. Hence typeset structure is infinite, which was to be proved.

The existence of at least one infinite set must be guaranteed. It is so done by the so called Axiom of Infinity , one form of which says:

∃ _ H (H != ∅ ∧ ∀ _ A (A ∈ H => ∃ _ B (B ∈ H ∧ A ⊂ B)))

Frege thought he could do without an axiom of infinity, but his system turned out to be inconsistent. In 1931 Kurt Gödel wrote an important paper in which he showed that an axiomatic system which contains the axiom of infinity could not be proved to be consistent or complete.

Note that the Axiom of Infinity can be proved in Quine’s set theory NF (New Foundations) .

Notes

1 The idea that size can be measured by one-to-one correspondence is today known as Hume's principle , although Hume, like Galileo, believed the principle could not be applied to infinite sets.

2 Here the similarity of two sets means that their elements are in  one-to-one correspondence.

References

[1]  Dedekind, R. (1888). Was sind und was sollen die Zahlen? (German). Braunschweig: Vieweg & Sohn..

Cite this web-page as:

Štefan Porubský: Infinite Sets.

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