Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one.

Ludwig Wittgenstein,

Philosophical Grammar

Welcome

This website is for mathematicians, philosophers and everyone interested in concepts about infinity. We present a dynamic, relative notion of infinity instead of the usual static and absolute one. The static notion considers infinity as a size, whereas a dynamic notion regards it as a process with indefinitely large states. This is a relative notion, similar as a child may call a (finite) number infinite, if it is larger than it can imagine. A more mathematical formulation is that the number is indefinitely larger than the numbers in the current context of investigation. But at any time the context may change and thus an indefinitely large, but finite number has to be replaced by a larger (still finite) one. The basis for establishing these ideas is the concept of indefinitely extensibility, which is for us the basic property of the potential infinite.

This dynamic treatment of infinity avoids the paradoxes that arise with the usual static concepts. We use model theory and type theory as the framework for these concepts. More precisely, for a classical first-order logic we develop a variant of the common Tarskian model theory with a dynamic interpretation of the carrier (also called domain or universe). As a consequence, the universal quantifier has to be interpreted in a new, no longer tautological way. These models use the structure of indefinitely extensible sets with its indefinitely large stages.

The Background

Around 1900, in the foundational crisis of mathematics, modern logic and set theory emerges. The relation between logic, with its main parts proof theory and model theory, and set theory is twofold:

Set theory is a first order theory, like many other theories. Model theory provides a model for it (or several ones).
In its foundational role however, set theory provides the basis for model theory. In particular, the carrier of a model is seen as a set.

In order to play its foundational role, the concept of an actual infinite set is crucial. This is mainly due to the interpretation of the universal quantifier: A formula $\forall \mathsf{x}$ … is interpreted as "for all elements in the domain …". In order to use this phrase "for all …" naively, the totality of elements has to exist completely (i.e., as an actual infinite set in case of an infinite carrier).

In contrast, if infinity is seen dynamically (i.e., as a potential infinite), then the only reference to a totality of objects is by reference to some state with finitely many elements. No naive reference to "all elements" is possible anymore. There was no model theory available that provided this kind of interpretation of the universal quantifier. The main opponent of set theory as a foundation of mathematics at that time was intuitionism. Notwithstanding the fact that intuitionism relies on the concept of a potential infinite, it does not make explicit that the potential infinite is an increasing finite. So intuitionism only uses the dynamic character of a potential infinite in subsequent constructions and arguments, but does not explain it.

The Project

Contemporary mathematics uses both conceptions of infinity, the potential infinite and the actual infinite, for instance the potential infinite for analysis and actual infinity in set theory. A potentialist considers infinite sets and series as unbounded, never ending processes, having finitely many elements at each stage. In contrast, actual infinite sets and series are completed totalities of objects. We claim that it is possible to develop mathematics with a notion of a potential infinite alone, without any restrictions on the logical axioms and rules. Instead of potential infinite we often use the term indefinitely extensible, since the possibility to always increase a set of objects in any circumstances whatsoever is the basic property of a potential infinite. Even if we claim to have completed the task of collecting "all" elements, this is a relative and dependent claim, whereas extensibility is absolute and unconditional.

Set theory plays a special role in mathematics since it may be considered as a foundation for it. This is due to the fact that all mathematical concepts can be reduced to sets. We will consider sets as indefinitely extensible, which allow indefinitely large stages as a substitute for actual infinite sets. The difference of both conceptions, sets as indefinitely extensible vs. completed, is best seen at the following example. Let us temporarily call a set $M$ normal if it is not a member of itself, i.e. $M \not\in M$ . With the notion of an actual infinite set it is not possible to form the "set of all normal sets" (this is known as Russell's paradox). If we regard sets as indefinitely extensible processes, then the step of building "all normal sets" leads to the set of "all normal sets that exist at the moment". But due to this operation we created a new set and referring now to "all normal sets" leads to a larger set. Note that the usual set theoretic axioms are still valid, only their interpretation changes. In particular, the axiom of infinity postulates a potential infinite set.

The constructive and temporal language is only for the sake of motivation. The concepts that we develop are abstract, allowing a non-constructive reading and a notion of dependency and context instead of time. We want to emphasize that all concepts that we develop do not require any specific philosophical view on mathematics. What is essential here is a dynamic view, not necessarily a constructive one. Even the finitistic perspective is not required: Instead of a finite versus infinite distinction we may consider a small versus large distinction as well, e.g. small means "size of a set" and large means "size of a proper class". But the locution "large" should be understood as being extensible whereas "small" should be understood as being fixed.