One can also glue together different linearly ordered sets to form a circularly ordered set.
2.
The set of all Dedekind cuts is itself a linearly ordered set ( of sets ).
3.
There are in fact many ways to construct such a linearly ordered set of numbers, but fundamentally, there are two different approaches:
4.
For example, given two linearly ordered sets and, one may form a circle by joining them together at positive and negative infinity.
5.
Beyond the problem of defining set representatives for ordinal numbers, Scott's trick can be used to obtain representatives for cardinal numbers and more generally for linearly ordered sets ( Jech 2003 : 65 ).
6.
:" Law of Finite Chains Between Events in the Combined Ordering " : There are no infinite chains ( " i . e . ", linearly ordered sets ) of events between two events in the combined ordering ?!.
7.
Examples of transitive relations include the equality relation on any set, the " less than or equal " relation on any linearly ordered set, and the relation " " x " was born before " y " " on the set of all people.
8.
Thus, constructing the set of Dedekind cuts serves the purpose of embedding the original ordered set " S ", which might not have had the least-upper-bound property, within a ( usually larger ) linearly ordered set that does have this useful property.
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