Ontology:Q800
- set theory (top-level category) 1-1-1
Characteristics in draft[edit]
Properties[edit]
- item type
- Z (wiki feature; pronounced C) 1-1-1
- label (en)
- set theory (top-level category) 1-1-1
- alias (en)
- analysis of how kinds of numbers compare with each other
- analysis of collections of unique objects
- external identifier
- sub-case of [Item]
- case of
- field of mathematics
Wavebuilder combinations[edit]
- pronounced [P] pronounced Wavebuilder: forms result [Item]
- ZFC set theory 1-1-1
- along with [Item]
- [S2] axiom of choice
- pronounced [P] pronounced Wavebuilder: forms result [Item]
- non-well-formed set theory 1-1-1
- along with [Item]
- [S0] anti-foundation axiom
Usage notes[edit]
ZFC set theory is the form of set theory most commonly in use, in contrast with non-well-formed set theories. One factor in this is that set theory is most often used to characterize kinds of numbers, such as how real numbers compare with rational numbers or what numbers are prime. For this use, ZFC set theory axioms are usually sufficient. However, if somebody wishes to try to use set theory for the purpose of categorizing real-world phenomena either in terms of conceptual, metaphysical labels or basic descriptions of the movements of individual objects, well-formed set theory ceases to make as much sense. Taking the classic grade-school-teacher thought experiment of apples in a basket, if we assume every apple had a produce sticker on it giving it a number, it would be hard to say that the set of apples is a separate thing from the physical apples that compose it, and tempting to say that the set definitionally contains itself. If the set of apples is not the basket, and the basket can only contain what is strictly the set of apples, there is no set of three apples that does not contain three apples, and there is usually no collection of three apples which is not a set of three apples. This is to say that every time three apples are put together, a particular back-and-forth recursion occurs: three apples generate a set of three apples which contains three apples which are a set of three apples which entails that the set of three apples contains a set of three apples which is the same set as itself.
Within this wiki, Categories and Items that represent conceptual categories are presumed to operate on non-well-formed set theory. Z and S Items (Z1 and S1) are loosely analogous to the numbered apples, while level-0 and level-2 Items are analogous to non-well-formed sets of apples that sit in the baskets. The categories that divide things are never separable from the things themselves, and there is always some particular reason that things are grouped together, even if it is purely historical or stochastic.