Georg Cantor Theory of Correspondence vs. The Whole is Greater than the Part
Theory of Correspondence
Georg Cantor (1845-1918) establishes the theory of correspondence in order to compare the cardinal number of infinite sets. We can reasonably state whether on set is larger than another by correlating the members of each series. It would be impossible to count all the members in each infinite set to establish which one has more members but by creating a correspondence between members we can find out if one set has members that cannot be paired off with a member from the other set. Bertrand Russell (1872-1970) realized the axiom ‘the whole is greater than the part’ creates problems when applied to the infinite. He also knew we could not apply the axiom and the Cantorian principle of correspondence at the same time. In other words, with infinite collections the whole can be similar to the part which contradicts the axiom the whole is greater than the part.
If there are members that must be left over in either set, no matter what sort of correspondence between terms is proposed, then one set is indeed larger and therefore has a different cardinal number. In this correlation sense one infinite can be ‘bigger’ than another. A denumerable infinite is an infinite series that can be correlated to the series of integers; therefore all denumerable infinite collections are equivalent. Two sets are equivalent, as Bertrand Russell mentions, if “u is similar to a proper part of v, and v to a proper part of u (a case that can only arise when u and v are infinite), then u is similar to v; hence ‘u is greater than v’ is inconsistent with ‘v is greater than u’” (Russell p.122). Russell uses the term ‘similar’ to determine that u has the same cardinal number as a proper set of v and vice versa, thus u is equivalent to v. The cardinal number of a set is used to show how many members the set contains without referring to any particular order the members are in. One set has the same cardinal number as another set when all members of one can be put into one-to-one correspondence with all members of another. A set which has the same cardinal number as the set of integers is denumerable infinite i.e. its cardinal number is alpha-null (symbol unavailable).
Cantor proves that occasionally infinite sets can have different sizes. The set of real numbers and the set of natural numbers do have different sizes because some real numbers do not pair up with natural numbers. Therefore we can say the infinite set of real numbers is greater than the set of natural numbers. Henri Poincare states “Cantor’s proof could just as well be taken to establish merely that we could not devise a way of pairing off the natural numbers with real numbers.” (Moore p.121-122). While other philosophers thought the real numbers did not construct a genuine set at all, “presumably because real numbers were somehow too unwieldy to be grouped together in one determinate totality.” (Moore p.122).
The Whole is Equal to the Part
It is through the correlation between members of infinite sets that we are led to the denial of the axiom the whole is greater than the part. We find, for example, the set of whole numbers is equivalent to the sub-set of even whole numbers. It appears as though the set of even whole numbers would be smaller, because intuitively we might think it is half the size of all the whole numbers. However, when we correlate both sets there are no members that cannot be matched up and thus the sets are equivalent. For every term n in the set of whole numbers there is a term 2n found in the sub-set of even whole numbers. Given that both sets have no last member this is not really a surprising result, for they are clearly both denumerable infinite collections. It is only counter-intuitive if you think of them as finished in the finite sense of having both a first and a last term and conclude therefore the set of even whole numbers is half the set of whole numbers. As infinite collections do not have a finite number of members and do not commonly have two definite ends it is not surprising the axiom the whole is greater than the part does not apply.
So in infinite sets the entire set is equivalent to its infinite sub-set, or the whole is similar to the part. We do have a whole or completed series with the series of whole numbers (1,2,3...) which has the ordinal ω , because the series as all the whole numbers as its members. As A.W Moore states we can still claim the subset is smaller in the subset sense if not in the correlation sense.
William Lane Craig wonders if the Cantorian principle of correspondence also only applies to finite collections. He states that clearly the principle of correspondence and the axiom of the whole is greater than its part cannot both be true because “one asserts that the whole is greater than the part, while the other maintains that the whole is not greater than a part.” (Craig p.95). Using the Cantorian principle we say the set of even whole numbers is equivalent to the set of all whole numbers. Craig notes that an interesting feature of this correspondence is that “the number in the second series is always double the number in the first series.” (Craig p.73). It seems he might be alluding to the extensiveness of the set of whole numbers compared to the subset of even numbers. If we count to ten we have ten terms that belong to the set of whole numbers but only five terms that belong to the subset of even whole number. The set of whole numbers acquires more terms (twice as much) then the set of even whole numbers. With the principle of correspondence Cantor can assert that ω and ω +ω both have the cardinal alpha-null.. Despite the fact that the set of whole numbers increases its members at twice the rate of the even whole numbers the set can be correlated term for term with its subset. Craig claims if it is fact that ω and ω +ω have the same cardinal number it creates problems with infinite collections found in the real world. If an actual infinite collection were found in reality the set (1,2,3...) would be equivalent to the set with (1,2,3...1,2,3). Craig then concludes it is more reasonable to suggest that both principles are valid in reality and that actual infinite collections are impossible in reality. Therefore, for Craig, the principle of correspondence and the axiom the whole is greater than the part are both valid for finite collections found in reality, and infinite collections are reserved for the mathematical realm.
Russell states there is “no evidence for the axiom except supposed self-evidence, and its admission (to infinite collections) leads to perfectly precise contradictions.” (Russell p.360). Russell continues, “the similarity of the whole and part could be proved to be impossible got every finite whole, it becomes implausible to suppose that for infinite wholes, where the impossibility could not be proved, there was in fact no such impossibility.” (Russell p.360). Therefore although the axiom the whole is greater than the part is invalid for infinite collections we can still assert that the whole is similar to the part. Since the principle of correspondence entails that the whole and the part are similar it is valid for infinite collections whether they be in the mathematical realm or found in reality. Russell was attempting to discover whether the axiom the whole is greater than the part was valid or the principle of correspondence was valid for infinite collections. Russell concluded the axiom created paradoxes while the correspondence principle did not. While Craig’s motivation was to demonstrate the Cantorian principle also leads to absurdities when applied to an actual infinite.
Craig, William Lane. The Kalam Cosmological Argument. Macmillan Press, London 1979
Fraenkel, Abraham A. Abstract Set Theory. North-Holland Publishing, Amsterdam 1968.
Moore, A.W. The Infinite. Routledge New York 1990
Russell, Bertrand The Principles of Mathematics. W.W Norton & Company New York, 1943