The theorem can be relativized to allow the sentence to use sets of natural numbers from " V " as parameters, in which case " L " must be replaced by the smallest submodel containing those parameters and all the ordinals.
22.
Even larger countable ordinals, called the " stable ordinals ", can be defined by indescribability conditions or as those \ alpha such that L _ \ alpha is a 1-elementary submodel of " L "; the existence of these ordinals can be proved in ZFC, and they are closely related to the nonprojectible ordinals.
23.
It states that if " X " is a transitive set and is an elementary submodel of some level of the constructible hierarchy L & alpha;, that is, ( X, \ in ) \ prec ( L _ \ alpha, \ in ), then in fact there is some ordinal \ beta \ leq \ alpha such that X = L _ \ beta.
24.
The paradox can be resolved by noting that countability is not absolute to submodels of a particular model of ZFC . It is possible that a set " X " is countable in a model of set theory but uncountable in a submodel containing " X ", because the submodel may contain no bijection between " X " and ?, while the definition of countability is the existence of such a bijection.
25.
The paradox can be resolved by noting that countability is not absolute to submodels of a particular model of ZFC . It is possible that a set " X " is countable in a model of set theory but uncountable in a submodel containing " X ", because the submodel may contain no bijection between " X " and ?, while the definition of countability is the existence of such a bijection.
26.
For example, imagine a large predictive model that is broken down into a series of submodels where the prediction of a given submodel is used as the input of another submodel, and that prediction is in turn used as the input into a third submodel, etc . If each submodel has 90 % accuracy in its predictions, and there are five submodels in series, then the overall model has only . 9 5 = 59 % accuracy.
27.
For example, imagine a large predictive model that is broken down into a series of submodels where the prediction of a given submodel is used as the input of another submodel, and that prediction is in turn used as the input into a third submodel, etc . If each submodel has 90 % accuracy in its predictions, and there are five submodels in series, then the overall model has only . 9 5 = 59 % accuracy.
28.
For example, imagine a large predictive model that is broken down into a series of submodels where the prediction of a given submodel is used as the input of another submodel, and that prediction is in turn used as the input into a third submodel, etc . If each submodel has 90 % accuracy in its predictions, and there are five submodels in series, then the overall model has only . 9 5 = 59 % accuracy.
29.
For example, imagine a large predictive model that is broken down into a series of submodels where the prediction of a given submodel is used as the input of another submodel, and that prediction is in turn used as the input into a third submodel, etc . If each submodel has 90 % accuracy in its predictions, and there are five submodels in series, then the overall model has only . 9 5 = 59 % accuracy.
30.
In particular, if " N " is any model of ZFC that has a ( parameter-free ) definable well-ordering of the universe, then " N " has definable Skolem functions, hence the set " M " of all definable elements in " N " is an elementary submodel of " N ", and therefore it is a model of ZFC with the required property.
