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564:. If the axiom of choice holds, then trichotomy holds between arbitrary cardinal numbers (because they are
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of numbers usually expresses that some tacitly given ordering relation on
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is a trichotomous one. An example is the law "For arbitrary real numbers
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337:{\displaystyle \forall x\in X\,\forall y\in X\,(\,\lor \,\,\lor \,)\,.}
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If a trichotomous relation is also transitive, then it is a
426:) } is transitive and trichotomous, and hence a strict
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contains an early formulation of the law of trichotomy
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461:) } is trichotomous, but not transitive; it is even
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Law (all real numbers are positive, negative, or 0)
518:to be zero, relying on the real number's additive
353:A relation is trichotomous if, and only if, it is
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665:An Introduction to Mathematical Analysis
560:of well-orderable sets even without the
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42:is either positive, negative, or zero.
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640:& Michael J. Hoffman (1993)
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642:Elementary Classical Analysis
550:Zermelo–Fraenkel set theory
522:structure. The latter is a
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646:W. H. Freeman and Company
529:In classical logic, this
591:Law of noncontradiction
691:. Dover Publications.
687:Bernays, Paul (1991).
596:Law of excluded middle
520:linearly ordered group
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470:Trichotomy on numbers
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18:Trichotomous relation
689:Axiomatic Set Theory
601:Three-way comparison
543:intuitionistic logic
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531:axiom of trichotomy
638:Jerrold E. Marsden
566:all well-orderable
554:Bernays set theory
370:strict weak order
366:strict total order
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45:More generally, a
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16:(Redirected from
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719:Order theory
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667:, page 11,
644:, page 27,
428:total order
382:On the set
63:if for all
40:real number
32:mathematics
729:3 (number)
713:Categories
607:References
355:asymmetric
348:Properties
627:MathWorld
586:Dichotomy
359:connected
312:∧
293:¬
289:∧
270:¬
263:∨
241:¬
237:∧
223:∧
204:¬
197:∨
175:¬
171:∧
152:¬
148:∧
125:∈
119:∀
112:∈
106:∀
572:See also
535:integers
377:Examples
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34:, the
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