What transcends the absolute infinite?

Answer

Hi Jeroen,

The infinite is itself endless. Cantor has demonstrated a method that makes increasingly strictly larger objects, the so-called powerset. The powerset of a set A is the set of all subsets of A. Cantor showed that the powerset of A, denoted as P(A), contains strictly more elements than the set A itself.

This concept allowed to show that not everything is a collection (burali and Forti’s paradox). Before Cantor’s time, it was thought that every collection of similar objects is a collection, but that is incorrect. Let us suppose that everything is a set, then we define a set A that contains all the sets, this is well defined from the original assumption that everything is a set. Cantor’s reasoning then says,

A⊂P(A)⊆A

The first inclusion is strict, Cantor has shown, but P(A) is again a set because according to our assumption everything is a set and consequently this is a part of A that includes all sets. This is of course a contradiction because now it says that A is a strict part of itself.

This not only proves that some things are not a collection, but that larger objects must exist. It never stops, you have sets then something even bigger, is a class than a universe etc… So you have to talk about the class of all sets and for example the class of nonlinear systems, etc. Roughly a class is a collection of objects that have the same property in common where a collection is a collection of objects for which the property is explicit. Nonlinear systems is not an explicit property, it is a negation of an explicit property. The linear systems do form a collection.

All these objects can be defined axiomatically, in 1903 the axioms of a collection were laid down in the work of Zermelo and Fraenklin.

Your question was directly about numbers. One of the controversial things Cantor introduced was the notion of transfinite number, which is a number that is not strictly (absolutely) infinite but still greater than any finite number.

He introduced the concept of ordinal and cardinal number. For example, ω is the first non-finite ordinal number or the smallest transfinite number. Cantor’s construction also allowed you to make an ever larger ordinal. The difference between cardinals and ordinals is that ordinals respect a rank and a cardinal does not. Two different infinite sets can have the same cardinality (same magnitude) but with a different ordinality. The set of natural numbers is ω₀ and the set of real numbers is ω₁.

Cantor was a believer and a mathematical realist. He believed that mathematics was discovered and consequently he thought that the construction of the transfinite numbers was endless. He equated the infinite of the transfinite numbers with god. On August 3, 1899, Cantor announced in a letter to Dedekind that he was dropping the concept of “absolute infinity” because it conflicted with the Burali-Forti paradox (Burali-Forti, Cesare (1897), “Una questione sui numeri transfiniti” , Rendiconti del Circolo Matematico di Palermo 11:154-164). He concluded in that letter that the system of all ordinals is an inconsistent absolute infinity.