This seems to be a strengthening of elementary finite set theory, which is the theory of Zermelo minus the axiom of infinity.
The theory of Zermelo is ZF without the Replacement Axioms (needed to compare the well-ordering and the ordinals) and without the foundation axioms (when we reject set belonging to themselves). I would not say that set theory is used for the foundation of mathematics. It is mainly a theory on the infinities, lurking toward the inconsistent big unnameable one. Sort of vertical theological shortcut. Elementary finite set theory is Turing complete (Turing universal). It is a set theoretic version of something between RA and PA. It is a universal machinery with its universal machines, and all others. It is a what I call a universal number. Each one has its application and purpose “in life”. God loves them all (I guess) Bruno > On 22 May 2019, at 22:08, Philip Thrift <[email protected]> wrote: > > Finitist Set Theory > > https://en.wikipedia.org/wiki/Finitist_set_theory > > "The goal of an engineer who applies FST is to select axioms which yield a > model that is one-one correlated with a target domain that is to be modeled > by FST, such as a range of chemical compounds or social constructions that > are found in nature. ... An applied FST model is always the minimal model > which satisfies the applied axioms. This guarantees that those and only those > elements exist in the applied model which are explicitly constructed by the > selected axioms: only those urs [ https://en.wikipedia.org/wiki/Urelement ] > exist which are stated to exist by assigning their number, and only those > sets exist which are constructed by the selected axioms; no other elements > exist in addition to these." > > From: > Finitist set theory in ontological modeling > Avril Styrman & Aapo Halko, University of Helsinki > Applied Ontology (2018) > > Abstract > "This article introduces finitist set theory (FST) and shows how it can be > applied in modeling finite nested structures. Mereology is a straightforward > foundation for transitive chains of part-whole relations between individuals > but is incapable of modeling antitransitive chains. Traditional set theories > are capable of modeling transitive and antitransitive chains of relations, > but due to their function as foundations of mathematics they come with > features that make them unnecessarily difficult in modeling finite > structures. FST has been designed to function as a practical tool in modeling > transitive and antitransitive chains of relations without suffering from > difficulties of traditional set theories, and a major portion of the > functionality of discrete mereology can be incorporated in FST. This makes > FST a viable collection theory in ontological modeling." > > > Relation of finitist sets to processes: > > The term 'partition level' and the recursive definition of n-member are > adapted from: > - Seibt, J. (2015) Non-transitive parthood, leveled mereology, and the > representation of emergent parts of processes. > - Seibt, J. (2009). Forms of emergent interaction in general process theory. > > https://www.researchgate.net/publication/220607682_Forms_of_emergent_interaction_in_General_Process_Theory > > "General Process Theory (GPT) is a new (non-Whiteheadian) process ontology. > According to GPT the domains of scientific inquiry and everyday practice > consist of configurations of ‘goings-on’ or ‘dynamics’ that can be > technically defined as concrete, dynamic, non-particular individuals called > general processes. The paper offers a brief introduction to GPT in order to > provide ontological foundations for research programs such as interactivism > that centrally rely on the notions of ‘process,’ ‘interaction,’ and > ‘emergence.’ I begin with an analysis of our common sense concept of > activities, which plays a crucial heuristic role in the development of the > notion of a general process. General processes are not individuated in terms > of their location but in terms of ‘what they do,’ i.e., in terms of their > dynamic relationships in the basic sense of one process being part of > another. The formal framework of GPT is thus an extensional mereology, albeit > a non-classical theory with a non-transitive part-relation. After a brief > sketch of basic notions and strategies of the GPT-framework I show how the > latter may be applied to distinguish between causal, mechanistic, functional, > self-maintaining, and recursively self-maintaining interactions, all of which > involve ‘emergent phenomena’ in various senses of the term." > > cf. Locally Finite Theories > https://www.jstor.org/stable/2273942 > > @philipthrift > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/6e50b919-7aff-4811-a230-e1eb771e9cf6%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/6e50b919-7aff-4811-a230-e1eb771e9cf6%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/9D3E3717-413E-4035-AD00-59DACA52F46B%40ulb.ac.be.
