Ao [ 's theorem does not apply to the Archimedean property, because the Archimedean property cannot be stated in first-order logic.
12.
Ao [ 's theorem does not apply to the Archimedean property, because the Archimedean property cannot be stated in first-order logic.
13.
The reason is that these numbers are known ( by a theorem ) to be irrational, and the real number system satisfies the Archimedean property.
14.
In fact, the Archimedean property is false for the hyperreals, as shown by the construction of the hyperreal number " ? " above.
15.
:: : Unfortunately, you have to click through to read about the Archimedean property . talk ) 17 : 19, 27 December 2013 ( UTC)
16.
The last step, that ?! 0 as " n " ?! ", is often justified by the Archimedean property of the real numbers.
17.
If we allow more parameters, for instance all of the reals, we can specify a type \ { 0 that is realized by an infinitesimal hyperreal that violates the Archimedean property.
18.
A crucially important property of the real numbers is that it is an Archimedean field, meaning it has the Archimedean property that for any real number, there is an integer larger than it in absolute value.
19.
There are two senses in which the term may be used, referring to geometries over fields which violate one of the two senses of the Archimedean property ( i . e . with respect to order or magnitude ).
20.
Super Dedekind completeness implies Dedekind completeness; Dedekind completeness implies both Dedekind-completeness and the projection property; Both Dedekind ?-completeness and the projection property separately imply the principal projection property; and the principal projection property implies the Archimedean property.
