Archimedean property of real numbers
E156205
The Archimedean property of real numbers is a fundamental axiom stating that for any real number, there exists a natural number larger than it, ensuring there are no infinitely large or infinitesimally small elements in the real number system.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Archimedean property | 1 |
| Archimedean property of real numbers canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1358743 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Archimedean property of real numbers Context triple: [Archimedes, knownFor, Archimedean property of real numbers]
-
A.
Peano existence theorem
The Peano existence theorem is a fundamental result in the theory of ordinary differential equations that guarantees the existence (but not necessarily uniqueness) of solutions under mild continuity conditions on the right-hand side.
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B.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
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C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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D.
axiom of choice
The axiom of choice is a fundamental principle in set theory asserting that one can select an element from each set in any collection of nonempty sets, with far-reaching consequences across mathematics.
-
E.
Banach fixed-point theorem
The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Archimedean property of real numbers Target entity description: The Archimedean property of real numbers is a fundamental axiom stating that for any real number, there exists a natural number larger than it, ensuring there are no infinitely large or infinitesimally small elements in the real number system.
-
A.
Peano existence theorem
The Peano existence theorem is a fundamental result in the theory of ordinary differential equations that guarantees the existence (but not necessarily uniqueness) of solutions under mild continuity conditions on the right-hand side.
-
B.
Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
-
C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
D.
axiom of choice
The axiom of choice is a fundamental principle in set theory asserting that one can select an element from each set in any collection of nonempty sets, with far-reaching consequences across mathematics.
-
E.
Banach fixed-point theorem
The Banach fixed-point theorem is a fundamental result in metric space theory that guarantees the existence and uniqueness of a fixed point for any contraction mapping and provides a method for finding it via iterative approximation.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
axiom of ordered fields
ⓘ
mathematical property ⓘ property of the real number system ⓘ |
| appearsIn | axiomatizations of the real numbers as a complete ordered field ⓘ |
| appliesTo |
ordered field of real numbers
ⓘ
real numbers ⓘ |
| category | order-theoretic property ⓘ |
| characterizes | Archimedean ordered fields ⓘ |
| contrastsWith | non-Archimedean property ⓘ |
| domain | ordered fields ⓘ |
| ensures |
compatibility between algebraic and order structures of the real numbers
ⓘ
no infinitely large real numbers relative to the natural numbers ⓘ no nonzero infinitesimal real numbers relative to the natural numbers ⓘ |
| equivalentTo |
For every real x > 0, there exists n in N such that n x > 1.
ⓘ
For every real x, there exists n in N such that |x| < n. ⓘ The set of natural numbers has no upper bound in the real numbers. ⓘ The set {1/n : n in N} has infimum 0 in the real numbers. ⓘ |
| expressedIn | first-order language of ordered fields ⓘ |
| formalStatement |
For every real number x > 0, there exists a natural number n such that 1/n < x.
ⓘ
For every real number x, there exists a natural number n such that n > x. ⓘ |
| historicallyNamedAfter |
Archimedes
ⓘ
surface form:
Archimedes of Syracuse
|
| implies |
every bounded increasing sequence of integers is eventually constant
ⓘ
every real number is finite with respect to the natural numbers ⓘ for any real x, there exists integer n with n-1 <= x < n ⓘ the integers are unbounded above and below in the real numbers ⓘ the natural numbers are unbounded above in the real numbers ⓘ the real line has no infinitely distant points ⓘ there are no infinitely large elements in the real numbers ⓘ there are no nonzero infinitesimal elements in the real numbers ⓘ |
| notTrueIn |
hyperreal numbers
ⓘ
non-Archimedean ordered fields ⓘ p-adic number fields ⓘ |
| relatedTo | Archimedean ordered field ⓘ |
| role | rules out infinitesimal and infinitely large elements in the real numbers ⓘ |
| trueIn |
field of rational numbers
ⓘ
field of real numbers ⓘ |
| usedIn |
calculus
ⓘ
construction of the real numbers from rationals ⓘ measure theory ⓘ number theory ⓘ real analysis ⓘ topology of the real line ⓘ |
| usedToProve |
basic inequalities in analysis
ⓘ
density of rational numbers in the real numbers ⓘ existence of floor and ceiling functions on real numbers ⓘ existence of integer parts of real numbers ⓘ limit properties involving sequences 1/n ⓘ that 1/n converges to 0 in the real numbers ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Archimedean property of real numbers Description of subject: The Archimedean property of real numbers is a fundamental axiom stating that for any real number, there exists a natural number larger than it, ensuring there are no infinitely large or infinitesimally small elements in the real number system.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Archimedean property