Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-02
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The reals are the quotient of rational Cauchy sequences by the maximal ideal of null sequences

Example

The reals are the quotient of rational Cauchy sequences by the maximal ideal of null sequences.

Facts & Assumptions

Given: The Cauchy-sequence ring C and the null ideal N.

[L1]

The real numbers are defined as the quotient of rational Cauchy sequences by null sequences (The real numbers).

[L2]

Rational Cauchy sequences form the ring C (Cauchy sequences form a commutative ring).

[L3]

N is an ideal of C (Null sequences form an ideal).

[L4]

N is maximal in C (The null ideal is maximal).

[L5]

Quotienting a commutative ring by a maximal ideal gives a field (R/M is a field if and only if M is a maximal ideal).

[L6]

The constructed real numbers form a field (The reals form a field).

Verification

technique · direct
1.1

By [L1], the underlying set and operations of R are the quotient C/N.

L1L2L3L4L5L6given
2.1

Since [L3] and [L4] make N a maximal ideal, [L5] also identifies C/N as a field.

step 1.1L1L2L3L4L5L6given
3.1

This is the stated quotient realisation.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources