> On 23 May 2019, at 19:17, Philip Thrift <[email protected]> wrote: > > > 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.
I guess you mean we get this in the meta-theory? If not explain me how you get omega, the first infinite ordinal, *in* the theory, without some infinity axiom. Bruno > > @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 >> <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 >> <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 >> >> <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 <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/e9647789-10b1-45e9-9f28-894971a5ab8a%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/e9647789-10b1-45e9-9f28-894971a5ab8a%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]. 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