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Canonical static concept record

Von Neumann zero $0$

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Summary

The chosen realization $0:=\varnothing$ .

Record metadata

Canonical name: Von Neumann zero $0$
Concept ID: von_neumann_zero
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Set theory
Domains: Set theory, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Set-theoretic numeral

Carrier(s)

set $\varnothing$

Canonically induces

first finite ordinal and finite cardinal

Notes

This is the standard von Neumann encoding; abstract zero is not intrinsically identical with the empty set.

Concept sources

Wikipedia — Von Neumann ordinal — Von Neumann ordinal · encyclopedic reference · source ID wp-von-neumann-ordinal
nLab — natural number — natural number · mathematical reference · source ID nlab-natural-number
Open Logic Project — Set Theory — Open Logic Project: Set Theory · open textbook source · source ID openlogic-set-theory

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Empty set Von Neumann zero $0$

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: empty_set_to_von_neumann_zero
Relation type: Representation / equivalence representation-equivalence
Direction: source → target
Endpoint roles: source: Representation / equivalence with; target: Representation / equivalence with
Authored annotation: chosen as zero

Authored explanation

The von Neumann convention uses the empty set as the set-theoretic representative of the abstract numeral $0$ .

How to interpret this relation type

Reinterpret an object, pass to an equivalent presentation, or relate canonically corresponding structures; the carrier may change.

Relation sources

Wikipedia — Von Neumann ordinal — Von Neumann ordinal · encyclopedic reference · source ID wp-von-neumann-ordinal
nLab — natural number — natural number · mathematical reference · source ID nlab-natural-number

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.