What is recorded here?
Dedekind constructed irrational numbers from gaps in the rational order and organized the resulting real-number system around cut completeness. He connected that continuity property with convergence of bounded increasing quantities. A rational order may contain partitions that no rational number realizes. Dedekind introduces a real number for such a gap, distinguishes order-equivalent cuts at rational endpoints and argues that the completed number system has a unique boundary for every proper cut. This gives analysis an arithmetic continuity principle instead of borrowing geometric intuition. His later construction of natural-number arithmetic is a different contribution.
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Scientific scope
The admitted target is the cut construction, order-completeness statement and its analytic connection. Cuts are proper nonempty order partitions; equivalent rational-endpoint cuts represent the same number. The cited source is a university transcription of the 1912 fourth edition, with formula/glyph defects, rather than the 1872 facsimile. It does not prove the continuity of physical space, certify every arithmetic-operation proof or assert a modern well-order of the reals.
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People and systems
Richard Dedekind
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English catalogue narrative
Dedekind: real numbers constructed through rational cuts
Dedekind constructed irrational numbers from gaps in the rational order and organized the resulting real-number system around cut completeness. He connected that continuity property with convergence of bounded increasing quantities.
A rational order may contain partitions that no rational number realizes. Dedekind introduces a real number for such a gap, distinguishes order-equivalent cuts at rational endpoints and argues that the completed number system has a unique boundary for every proper cut. This gives analysis an arithmetic continuity principle instead of borrowing geometric intuition. His later construction of natural-number arithmetic is a different contribution.
Scientific scope
The admitted target is the cut construction, order-completeness statement and its analytic connection. Cuts are proper nonempty order partitions; equivalent rational-endpoint cuts represent the same number. The cited source is a university transcription of the 1912 fourth edition, with formula/glyph defects, rather than the 1872 facsimile. It does not prove the continuity of physical space, certify every arithmetic-operation proof or assert a modern well-order of the reals.
People and systems
Richard Dedekind