Ontology:Q819

1. ZFC set theory 1:(-)1:(-)1

Characteristics in draft

Properties

item type

Z (wiki feature; C (pronounced C)) 1:(-)1:(-)1

label (en)

ZFC set theory 1:(-)1:(-)1

alias (en)

Zermelo-Fraenkel Choice-axiom set theory (ZFC)

ZFC (set theory)

external identifier

Zermelo-Fraenkel set theory

sub-case of [Item]

set theory (top-level category) 1:(-)1:(-)1

case of

field of mathematics

prototype notes

set theory where sets are "computational" and pointers into the set cause a kind of infinite loop bug in the logic

Components

model combines claims

[S2] Sets are collections of unique objects / Set cardinality measures a set's number of elements which are unique and partly defines the set

model combines claims

[Z2] Sets are equal if they contain all of the same unique elements / axiom of extensionality

model combines claims

[S2] Sets can contain other sets

model combines claims

[S2] Sets cannot contain themselves / Every non-empty set contains a member which makes it not equivalent to itself / axiom of regularity

model combines claims

[Z2] Every element in a set is a set / No element in a set is an atom (atomic data structure)

model combines claims

[Z2] Any two sets can join into a greater set / For any two sets there can be a set containing them / axiom of pairing

model combines claims

[S2] Any element in a set can represent the whole set / When constructing a new set out of one element from each set it does not matter which one is used / AC (axiom of choice)

Wavebuilder combinations

[P] (pronounced [P]) Wavebuilder (pronounced Wavebuilder): forms result [Item]

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along with [Item]

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forming from [Item]

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Usage notes

Set theory axioms have much more wide-ranging implications than anybody would immediately think of. For instance, the axiom of choice: what happens if we apply the axiom of choice to people? We get something that looks roughly like Liberal-republicanism. But centuries of history have shown there to be problems with this concept. For one, if a group of people is relatively stable, it is still possible for somebody to be sent as a representative that a lot of people feel doesn't represent them. For another, if the boundaries of groups of people are rapidly changing due to changing ideologies or any other reason, there will be arguments over where voting districts should be drawn — whether people who are sorting into different groups should be in the same district or different districts, how many parties votes should be potentially nominating, how many parties should be represented in parliament, how many parties should be competing for president if applicable. Any debate about gerrymandering ultimately comes down to an argument about what sets accurately contain what people and what sets of people can safely apply the axiom of choice in order to treat broad groups of people as one person.

Studying the axioms of ZFC set theory and which axioms truly make sense in the physical world has very important implications for reforming or replacing Liberal-republicanism.