Topics Mathematics

Question Can we have a POSITIVE understanding of such concepts as infinity? What I mean is that, whilst I am sure that we can well grasp the concept of finiteness, can we do more than negate it (which would yield not-finiteness), can we understand infinity from the inside instead of by negating everything that lies outside of it? Thanks, Andrea Jasson

Accepted:
November 26, 2005

Comments

There are two kinds of definitions of infinity, one which is"negative" in your sense and one which is "positive". The former is,historically, the older.

There are many equivalent definitions offinitude. My own favorite is due to Gottlob Frege: A set is finite ifits members can be ordered as what he called a "simple" series that hasan end, but it would take some time to explain what Frege meant by a"simple" series. What's nice about Frege's definition, though, is thatit amounts to a formal elaboration of the idea that a set is finite ifit can be counted. A definition close in spirit to Frege's isdue to Ernst Zermelo: A set is S finite if it can be "doublywell-ordered", that is, if there is a relation < on S with thefollowing properties:

< is a linear order: for all x, y, and z in S, either x<y or x=y or y<x
Everynon-empty subset of S has a <-minimal member; that is, if T ⊆ S andT is non-empty, then for some x∈T, for every other y∈T, x<y.
Everynon-empty subset of S has a <-maximal member; that is, if T ⊆ S andT is non-empty, then for some x∈T, for every other y∈T, y<x.

Givensuch a notion of finitude, we can then say that S is infinite if it isnot finite.

The other way of proceeding is due to Richard Dedekind: We say that S is Dedekind infiniteif there is some proper subset T of S that can be put in 1-1correspondence with S; that is, if there is some proper subset T of Sand a 1-1 function f that maps S onto T. The set of natural numbers{0,1,2,3,...}, for example, is Dedekind infinite: The function f(x) =x+1 maps the set of natural numbers one-one onto the set of positivenatural numbers, {1,2,3,...}. We then say that S is Dedekind finite if it is not Dedekind infinite.

Theformer, negative conception is usually just called "infinity", whereasthe latter, positive one is called "Dedekind infinity".

What is the relation between these two conceptions of infinity?

It is fairly easy to prove that if a set is Dedekind infinite, then it is infinite. Dedekind proved, in Was Sind und Was Sollen Die Zahlen?, where he introduced the notion of Dedekind infinity, that everyinfinite set is Dedekind infinite. Most people accepted Dedekind'sproof, but it was clear from the outset that there was somethingstrange about it. It turns out that the sense of oddity is due to thefact that, at a crucial point in the proof, Dedekind tacitly appeals towhat is called the "axiom of choice". The axiom of choice had not been isolated at that time: Was Sind? was published in 1888, but written some years earlier; Zermelo identifies the axiom of choice only about 1905. The way Dedekind gets the effect of choice is rather peculiar and involves what philosophers would call a use-mention confusion. But one can patch Dedekind's proof using choice, and most mathematicians nowadays accept the axiom of choice and so regardthe two conceptions as ultimately equivalent. But it can be proventhat, without the axiom of choice (more precisely,without countable choice), one cannot prove thatevery infinite set is Dedekind infinite: If the axiom of (countable) choice is false, then there may be a Dedekind finite set that is not finite. So there is an importantdifference between the two conceptions nonetheless, and the nature of this difference canitself be studied. (It has to do with the kinds of logical resourcesthat are deployed in each definition.)