The infinite in mathematics has two manifestations. Its occurrence in analysis has been satisfactorily formalized and demystified by the
-
ÎŽ method of Bolzano, Cauchy and Weierstrass. It is of course the âset-theoretic infiniteâ that concerns me here. Once the existence of an infinite set is accepted, the axioms of set theory imply the existence of a transfinite hierarchy of larger and larger orders of infinity. I shall review some well-known facts about the influence of these axioms of infinity to the everyday mathematical practice and point out to some, as of yet not understood, phenomena at the level of the third-order arithmetic. Technical details from both set theory and operator algebras are kept at the bare minimum. In the Appendix, I include definitions of arithmetical and analytical hierarchies in order to make this paper more accessible to non-logicians. In this paper I am taking a position intermediate between pluralism and non-pluralism (as defined by P. Koellner in the entry on large cardinals and determinacy of the Stanford Encyclopaedia of Philosophy) with an eye for applications outside of set theory.
Let me recall von Neumannâs definition of the cumulative hierarchy. We define sets
Vα for ordinals
α by recursion, so that
and
Vα+1 is the power-set of
Vα for every
α. If
ÎŽ is a limit ordinal, we let
Vα =
Vα. The Power-Set Axiom asserts the existence of
Vα+1, granted
Vα exists. The existence of
VÎŽ for a limit ordinal
ÎŽ follows from the Replacement Axiom. We therefore have an increasing collection of sets, indexed by all ordinals, that provides framework for all of mathematics as we know it. All of number theory is formalized within
VÏ. Every countable set, such as
, or the free group with two generators, has an isomorphic copy inside
VÏ (where
Ï is the least infinite ordinal). These sets, as well as all real numbers (defined via Dedekind cuts) belong to
VÏ+1. The set of real numbers therefore belongs to
VÏ+2. If
A is a separable metric structure, such as
2, Tsirelsonâs Banach space, or Cuntz algebra
2, then it has a countable dense subset
A0 that can be identified (equipped with all of its metric, algebraic, and relational structure) with an element of
VÏ+1. Therefore,
A itself, identified with the equivalence classes of Cauchy sequences in
A0, belongs to
VÏ+3. This also applies to objects that are separable in at least one natural metric, e.g., II
1 factors (
2 metric) or multipliers of separable C*-algebras (strict metric). Quotients, coronas, ultrapowers, or double duals of separable objects, as well as their automorphism groups, all belong to
VÏ+n for a relatively small natural number
n. Therefore
VÏ+Ï already provides framework for most of non-set-theoretic mathematics.
a Nevertheless,
VÏ+Ï is not a model of ZFC since it fails the Replacement Axiom.
Accepting the existence of the empty set and the assertion that every set has the power-set has as a consequence the existence of Vn for all natural numbers n. However, the cardinality of V6 is roughly 1019,738. Current estimates take number of fundamental particles in the observable universe to be less than 1085. While these estimates are based on our current understanding of physics and are therefore subject to change, this shows that we have no concrete model of V6. Can we nevertheless assert that V6 exists? Can we claim that the power-set axiom is true in the physical world? For example, consider the set X of all electrons contained in this sheet of paper at this very moment. Does the power-set of X exist? (The problem may be in the comprehension axiom, or rather the question whether X is a set?)
The fact that most physical laws are only approximately true does not diminish their usefulness in concrete applications. (As von Neumann pointed out in
[54], the truth is much too complicated to allow anything but approximations.) Regardless of whether the set of all reals
(or any other infinite set) exists or not, its formal acceptance provides us with remarkable mathematical tools. Accepting the transfinite hierarchy, together with some substantial large cardinal axioms, may be comparably illuminating. Present lack of arguments pro this view is fortunately counterbalanced by the complete absence of arguments against it.
2. Independence
By Gödelâs Incompleteness Theorems, every consistent theory
T that includes Peano Arithmetic and has a recursive set of axioms is incomplete. Moreover, the sentence constructed by Gödel is a
arithmetical sentence (see the Appendix). This means that it asserts that every natural number
n has a certain property that can can be verified by computation. Such a sentence can be independent f...
