submodel sentence in Hindi
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- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- A variation on the method of forcing can also be used to demonstrate the consistency and unprovability of the axiom of choice, i . e ., that the axiom of choice is independent of ZF . The consistency of choice can be ( relatively ) easily verified by proving that the inner model L satisfies choice . ( Thus every model of ZF contains a submodel of ZFC, so that Con ( ZF ) implies Con ( ZFC ) . ) Since forcing preserves choice, we cannot directly produce a model contradicting choice from a model satisfying choice.
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