Extending the Non-extendible: Shades of Infinity in Large Cardinals and Forcing Theories

Abstract

This is an article whose intended scope is to deal with the question of infinity in formal mathematics, mainly in the context of the theory of large cardinals as it has developed over time since Cantor’s introduction of the theory of transfinite numbers in the late nineteenth century. A special focus has been given to this theory’s interrelation with the forcing theory, introduced by P. Cohen in his lectures of 1963 and further extended and deepened since then, which leads to a development and further refinement of the theory of large cardinals ultimately touching, especially in view of the discussion in the last section, on the metatheoretical nature of infinity. The whole undertaking, which takes into account major stages of the research in large cardinals theory, tries to present a defensible argumentation against an ontology of infinity actually rooted in the notion of subjectivity within the world. This means that rather than talking of a general ontology of infinity in the ideal platonic or in the aristotelian sense of potentiality, even in the alternative sense of an ontology of the event in A. Badiou’s sense, one can argue from a subjective point of view about the impossibility of defining cardinalities greater than the first uncountable one $\aleph _{1}$ that would correspond to a distinct existence in real world terms or would be supported by a mathematical intuition in terms of reciprocity with experience. The argumentation from the particular standpoint includes also certain comments on the delimitative character of Gödel’s constructive universe L and the influence of the constructive approach in narrowing the breadth of an ‘ontology’ of infinity.

Appendix I: The Notion of Actual Infinity as a Precondition for the Separability of Space Points

To strengthen my arguments about formal infinity taken as the abstract form of a subjectively founded continuous unity, I present the example (which may be skipped by the mathematically uninterested reader) of a special topology in which the uncountability and separability of the points of the corresponding topological space is not by essence associated with infinity in objective real world terms. To formally ground my claim I will be based on a well-known mathematical space proved to be homeomorphic to the set of irrational numbers, this latter set provided with the subspace topology inherited from the standard Euclidean topology of the real line. The space in question, the Baire space $N^{\omega }$, is taken to be the set of maximal elements of the cpo²⁴ of finite and infinite sequences of natural numbers equipped with the subspace topology of the Scott topology.²⁵ This topology has as a base the collection of sets of the form $\uparrow s\cap N^{\omega }$, where by $\uparrow s$ is denoted the upper set of the finite sequence of digits s, that is, $\uparrow s= \{\xi ;\; s\le \xi \}$ where the order relation $\le$ is defined by: $\xi \le \xi ^{'}$ iff $\xi ^{'}$ extends $\xi$ (or $\xi$ is a prefix of $\xi ^{'}$); that is, $\xi ^{'}$ starts with exactly the same digits as $\xi$ and possibly has infinitely more. Usually this partial order is called in the mathematical literature a prefix ordering and is the key to the definition of several order-induced topologies of computational interest.

The interesting thing about the space $N^{\omega }$ as a Scott space is that it can be turned into a Stone space by taking as basic open sets except for the basic Scott opens of the form $\uparrow s$ (s finite) also their complements $N^{\omega }\setminus \uparrow s$. Generally it is proved that in this refined topology the base $\mathcal {B}$ of the clopen sets as above is a sub-Boolean algebra of the power-set of the original space $N^{\omega }$ and that every ultrafilter of $\mathcal {B}$ is a $\mathcal {B}$-neighborhood filter $\mathcal {N}_{B}(x)$ for a unique point x of $N^{\omega }$, where $\mathcal {N}_{B}(x)$ is the set $\{U\in \mathcal {B};\;x\in U\}$ (Smyth 1992, pp. 734–735). Moreover, by Stone’s representation theorem one has that the Stone space $N^{\omega }$ is a Hausdorff, compact, zero-dimensional and totally disconnected space (Porter and Woods 1988, pp. 171–172). What is of a special importance here, given my intention, is the Hausdorff property of $N^{\omega }$, namely the fact that for any two distinct points x, y of $N^{\omega }$ (which are actually infinite sequences of natural numbers) it can be proved that there are neighborhoods $U_{x}$ and $Y_{y}$ belonging respectively to the $\mathcal {B}$-neighborhood filters $\mathcal {N}_{B}(x)$ and $\mathcal {N}_{B}(y)$ such that: $U_{x}\cap Y_{y}=\emptyset$.

This means that the way to separate any two distinct points x, y of $N^{\omega }$ (which as homeomorphic to the space of irrational reals is a space of uncountable cardinality), is by generating at least two disjoint open neighborhoods of them. Further and to the extent that the points of $N^{\omega }$ can be thought of as corresponding to infinite decimal expansions of irrationals, it follows that however close any two distinct points can be in terms of proximity of their decimal expansion they can still be separated by disjoint open sets. Put more intuitively, the possibility of discerning between any two points of $N^{\omega }$ regardless of their proximity ad infinitum is conditioned on the formation of two disjoint neighborhoods to which they respectively belong, these latter ones meant as non-finite (by virtue of their topological openness) yet definite wholes in presentational immediacy. Essentially, the possibility for distinguishing space points in the form of infinitely extending sequences of natural numbers (or of binary digits) ordered by prefixing is ultimately conditioned on the possibility of forming at once their disjoint neighborhoods as definite continuous wholes in the present now of consciousness. In turn, completed wholes as actual ‘infinities’ freely generated in reflection independently of any spatio-temporal and causal concerns point to a concept of infinity constituted as the objective form of the continuous unity of each one’s temporal consciousness.

Consequently there is a case to be made here for the fact that, in shifting the focus of discussion to a fundamental constitutional level, the notion of continuity as such is not necessarily connected with the mathematical notion of uncountability. In fact, conventional uncountability is underdetermined with regard to its subjective origin in all classical proofs (e.g., think of Cantor’s diagonal proof of the uncountability of real numbers) to the extent that it is circularly conditioned on a pre-existing notion of ‘infinity’ in the sense of a subjectively generated and completed totality non-eliminable within a formal logical-mathematical universe.