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Foundation of mathematics
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Main Index
Foundation of mathematics
Set theory
Subject Index
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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 
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 
of all things, which can be objects of my thought, is infinite. For if 
signifies an element of 
, then is the thought 
, that 
can be object of my thought, itself an element of 
. If we regard this as transform 
of the element 
then has the transformation 
of 
, thus determined, the property that the transform 
is part of 
; and 
is certainly proper part of 
, because there are elements in 
(e.g., my own ego) which are different from such thought 
and therefore are not contained in 
. Finally it is clear that if 
are different elements of 
, their transforms 
are also different, that therefore the transformation 
is a distinct (similar) 2 transformation. Hence 
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:
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) .
| 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. |
| [1] | Dedekind, R. (1888). Was sind und was sollen die Zahlen? (German). Braunschweig: Vieweg & Sohn.. |
Cite this web-page as:
Štefan Porubský: Infinite Sets.