In its proof theory.
Each variable is changed to a (dynamically-nested, indefinitely-sized) type variable, effectively a process where elements are added as needed. cf. https://poesophicalbits.blogspot.com/2012/04/persons-without-infinities.html Death to Platonism. @philipthrift On Friday, May 24, 2019 at 4:13:46 AM UTC-5, Bruno Marchal wrote: > > > On 23 May 2019, at 19:17, Philip Thrift <[email protected] <javascript:>> > 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]> 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/4b356c26-d513-42c0-b593-120ce8e99b16%40googlegroups.com.
