If you "combine" Finitist Set Theory with Locally Finite Theories, what you get is a version of Axiom of Infinity with "processes" creating bigger and bigger sets with gaps in them.
@philipthrift On Thursday, May 23, 2019 at 11:34:21 AM UTC-5, Bruno Marchal wrote: > > 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] <javascript:>> > 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]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/e9647789-10b1-45e9-9f28-894971a5ab8a%40googlegroups.com.
