Alphabeta MathConstruction of the Real Numbers via Cauchy Sequences
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 28 results · all verified · 21 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 7 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
2 · Logical flowchart
3 · Items
Definition
Field
Definition
Ordered field
Definition
Complete ordered field (least-upper-bound property)
Definition
The integers as equivalence classes of pairs of naturals
Lemma
The integer-defining relation is an equivalence relation
Definition
Arithmetic on the integers
Lemma
Integer addition and negation are well defined
Lemma
Integer multiplication is well defined
Theorem
The integers form a commutative ring
Definition
Order on the integers
Theorem
The integers form a totally ordered ring
Lemma
The integers have no zero divisors; multiplicative cancellation
Lemma
The naturals embed in the integers
Definition
The rationals as equivalence classes of pairs of integers
Lemma
The rational-defining relation is an equivalence relation
Definition
Arithmetic on the rationals
Lemma
Rational arithmetic is well defined
Lemma
The reciprocal on the rationals is well-defined
Theorem
The rationals form a field
Definition
Order on the rationals
Lemma
Every rational has a positive-denominator representative
Theorem
The rationals form a totally ordered field
Lemma
The integers embed in the rationals
Lemma
The rationals are Archimedean
Lemma
Absolute value and the triangle inequality
Definition
Cauchy sequence of rationals
Definition
Null sequence
Lemma
Every Cauchy sequence of rationals is bounded
Lemma
Null sequences are Cauchy
Theorem
Cauchy sequences form a commutative ring
Lemma
Null sequences form an ideal
Lemma
A non-null Cauchy sequence is eventually bounded away from zero, with constant sign
Lemma
The null ideal is maximal
Definition
The real numbers
Theorem
The reals form a field
Definition
Order on the reals
Theorem
The reals form a totally ordered field
Lemma
The rationals embed densely in the reals
Definition
Limits and Cauchy sequences of reals
Theorem
The reals are complete
Example
The sequence 1/n is null
False statement
FALSE: some rational number squares to 2
False statement
FALSE: the rationals are complete
Sources
- W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1
- M. Spivak, Calculus, 4th ed., Ch. 1
- Elias Zakon, Mathematical Analysis: Axioms and Basic Definitions
- University of Illinois Chicago notes: Ordered field axioms
- M. Spivak, Calculus, 4th ed., Ch. 8
- University of Wisconsin Math 521 notes: Real analysis
- T. Tao, Analysis I, 3rd ed., §4.1
- E. Landau, Foundations of Analysis
- Integer — construction from pairs of naturals (Wikipedia)
- L. S. Krapp, Constructions of the real numbers: a set theoretical approach (Oxford, 2014)
- T. Tao, Analysis I, 3rd ed., §4.2
- Rational number — formal construction (Wikipedia)
- Rational number (Wikipedia)
- Archimedean property (Wikipedia)
- T. Tao, Analysis I, 3rd ed., §4.3
- Absolute value — ordered rings (Wikipedia)
- T. Tao, Analysis I, 3rd ed., §5.1
- W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3
- Real number (Encyclopedia of Mathematics)
- T. Tao, Analysis I, 3rd ed., §5.2
- Cauchy sequence (Wikipedia)
- T. Tao, Analysis I, 3rd ed., §5.3
- T. Tao, Analysis I, 3rd ed., §5.4
- W. Aitken, Constructing the Real Numbers, Chapter 9, Section 6, Lemmas 17–18
- Dense set (Wikipedia)
- T. Tao, Analysis I, 3rd ed., §6.1
- T. Tao, Analysis I, 3rd ed., §6.4
- W. Aitken, Constructing the Real Numbers, Section 8, Lemma 30 and footnote 3
- T. Tao, Analysis I, 3rd ed., §4.4
- Square root of 2 — proofs of irrationality (Wikipedia)