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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Natural numbers $\mathbb N$

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Summary

The standard Peano system, realized here on the carrier $\omega$ with zero, successor, addition, multiplication, and order.

Record metadata

Canonical name: Natural numbers $\mathbb N$
Concept ID: natural_numbers
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Number theory
Domains: Number theory, Algebra, Order theory, Set theory
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Number system

Carrier(s)

set $\omega$

Data

$0$
$S(n)=n\cup\{n\}$
addition and multiplication by recursion
usual well-order

Axioms / constraints

Peano axioms

Canonically induces

commutative semiring structure

Notes

The atlas distinguishes the abstract Peano structure from its selected von Neumann carrier $\omega$ .

Concept sources

nLab — natural number — natural number · mathematical reference · source ID nlab-natural-number
Wikipedia — Natural number — Natural number · encyclopedic reference · source ID wp-natural-number
Wikipedia — Peano axioms — Peano axioms · expository · source ID wp-peano

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

First infinite ordinal $\omega$ Natural numbers $\mathbb N$

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

Relation ID: omega_to_natural_numbers
Relation type: Representation / equivalence representation-equivalence
Direction: source → target
Endpoint roles: source: Representation / equivalence with; target: Representation / equivalence with
Authored annotation: standard carrier of $\mathbb N$

Authored explanation

Equip $\omega$ with zero, successor, recursively defined arithmetic, and its membership order to obtain the standard natural-number system.

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

nLab — natural number — natural number · mathematical reference · source ID nlab-natural-number
Wikipedia — Natural number — Natural number · encyclopedic reference · source ID wp-natural-number

Peano system Natural numbers $\mathbb N$

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

Relation ID: peano_system_to_natural_numbers
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: standard Peano system

Authored explanation

$\mathbb N$ on carrier $\omega$ is the standard second-order Peano system.

How to interpret this relation type

The target is a member or subtype of the broader source class.

Relation sources

Wikipedia — Peano axioms — Peano axioms · expository · source ID wp-peano
nLab — natural number — natural number · mathematical reference · source ID nlab-natural-number

Semiring Natural numbers $\mathbb N$

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

Relation ID: semiring_to_natural_numbers
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: initial arithmetic semiring

Authored explanation

With its usual operations, $\mathbb N$ is a commutative semiring and is initial among semirings in the standard unital convention.

How to interpret this relation type

The target is a member or subtype of the broader source class.

Relation sources

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.

Natural numbers $\mathbb N$ Countable infinity $\aleph_0$

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

Relation ID: natural_numbers_to_aleph_zero
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: cardinality $\aleph_0$

Authored explanation

The carrier of $\mathbb N$ is countably infinite and has cardinality $\aleph_0$ .

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Wikipedia — Aleph number — Aleph number · encyclopedic reference · source ID wp-aleph-number
nLab — natural number — natural number · mathematical reference · source ID nlab-natural-number

Natural numbers $\mathbb N$ Gödel numbering

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

Relation ID: story_natural_numbers_to_godel_numbering_add_data
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: choose an effective coding of syntax

Authored explanation

Use natural numbers as codes and choose a computable encoding of symbols, formulas, and finite proofs; Gödel numberings are not unique.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Gödel — Über formal unentscheidbare Sätze — Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I · original research paper · source ID godel-incompleteness-paper
Open Logic Project — Incompleteness — Incompleteness · open textbook · source ID openlogic-incompleteness

Natural numbers $\mathbb N$ Difference pairs of natural numbers

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

Relation ID: natural_numbers_to_integer_pairs
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: form formal differences

Authored explanation

Use $\mathbb N\times\mathbb N$ to represent differences before quotienting.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Wikipedia — Integer: Construction — Integer · encyclopedic reference · source ID wp-construction-integers
nLab — integer — integer · mathematical reference · source ID nlab-integer

Natural numbers $\mathbb N$ Integers $\mathbb Z$

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

Relation ID: natural_numbers_into_integers
Relation type: Canonical embedding canonical-embedding
Direction: source → target
Endpoint roles: source: Embeds canonically into; target: Contains a canonical copy of
Authored annotation: $n\mapsto[(n,0)]$

Authored explanation

The map $n\mapsto[(n,0)]$ is an injective semiring homomorphism into the nonnegative integers.

How to interpret this relation type

Map one structure injectively into another by the standard structure-preserving inclusion or representation. This records a canonical copy, not necessarily literal set containment.

Relation sources

Wikipedia — Integer: Construction — Integer · encyclopedic reference · source ID wp-construction-integers
nLab — integer — integer · mathematical reference · source ID nlab-integer

Natural numbers $\mathbb N$ consistent effective theory + arithmetic coding(construction junction)

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

Relation ID: j_first_incompleteness__natural_numbers
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: sufficient arithmetic

Authored explanation

Require the theory to represent enough elementary arithmetic of the natural numbers.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Open Logic Project — Incompleteness — Incompleteness · open textbook · source ID openlogic-incompleteness
Gödel — Über formal unentscheidbare Sätze — Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I · original research paper · source ID godel-incompleteness-paper

Natural numbers $\mathbb N$ consistent effective theory + internal provability(construction junction)

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

Relation ID: j_second_incompleteness__natural_numbers
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: sufficient arithmetic

Authored explanation

Require the theory to represent enough elementary arithmetic of the natural numbers.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Open Logic Project — Incompleteness — Incompleteness · open textbook · source ID openlogic-incompleteness
Gödel — Über formal unentscheidbare Sätze — Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I · original research paper · source ID godel-incompleteness-paper