TY - GEN

T1 - Generating Custom Set Theories with Non-set Structured Objects

AU - Dunne, Ciarán

AU - Wells, Joseph Brian

AU - Kamareddine, Fairouz

N1 - Publisher Copyright:
© 2021, Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - Set theory has long been viewed as a foundation of mathematics, is pervasive in mathematical culture, and is explicitly used by much written mathematics. Because arrangements of sets can represent a vast multitude of mathematical objects, in most set theories every object is a set. This causes confusion and adds difficulties to formalising mathematics in set theory. We wish to have set theory’s features while also having many mathematical objects not be sets. A generalized set theory (GST) is a theory that has pure sets and may also have non-sets that can have internal structure and impure sets that mix sets and non-sets. This paper provides a GST-building framework. We show example GSTs that have sets and also (1) non-set ordered pairs, (2) non-set natural numbers, (3) a non-set exception object that can not be inside another object, and (4) modular combinations of these features. We show how to axiomatize GSTs and how to build models for GSTs in other GSTs.

AB - Set theory has long been viewed as a foundation of mathematics, is pervasive in mathematical culture, and is explicitly used by much written mathematics. Because arrangements of sets can represent a vast multitude of mathematical objects, in most set theories every object is a set. This causes confusion and adds difficulties to formalising mathematics in set theory. We wish to have set theory’s features while also having many mathematical objects not be sets. A generalized set theory (GST) is a theory that has pure sets and may also have non-sets that can have internal structure and impure sets that mix sets and non-sets. This paper provides a GST-building framework. We show example GSTs that have sets and also (1) non-set ordered pairs, (2) non-set natural numbers, (3) a non-set exception object that can not be inside another object, and (4) modular combinations of these features. We show how to axiomatize GSTs and how to build models for GSTs in other GSTs.

UR - http://www.scopus.com/inward/record.url?scp=85113492175&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-81097-9_19

DO - 10.1007/978-3-030-81097-9_19

M3 - Conference contribution

SN - 9783030810962

T3 - Lecture Notes in Computer Science

SP - 228

EP - 244

BT - Intelligent Computer Mathematics. CICM 2021

A2 - Kamareddine, Fairouz

A2 - Sacerdoti Coen, Claudio

PB - Springer

ER -