Measurable cardinal to get inner model, also in cumulative hierarchy

In summary, If κ is a(n inaccessible) measurable cardinal, then there exists an elementary embedding j:(V,∈)→(M,∈), with critical point κ, whereby (M,∈) is an inner model of ZFC and the construction of j can follow through taking a κ-complete, non-principal ultrafilter U and constructing κV/U. There is a direct relationship between M and Vκ, where j maps elements of Vκ to elements of M and preserves the structure of Vκ.
  • #1
nomadreid
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If M is the universe of the inner model which a measurable cardinal helps collapse V into, is there any connection between M and the cumulative hierarchy "universe" indexed by the measurable cardinal?
If κ is a(n inaccessible) measurable cardinal, then there exists an elementary embedding j:(V,∈)→(M,∈), with critical point κ, whereby (M,∈) is an inner model of ZFC and the construction of j can follow through taking a κ-complete, non-principal ultrafilter U and constructing κV/U.

In the von Neumann cumulative hierarchy, (Vκ, ∈) is a model of a number of sentences.

Is there any direct relationship between M and Vκ?
 
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Yes, there is a direct relationship between M and Vκ. The elementary embedding j:(V,∈)→(M,∈) maps elements of V to elements of M, and so some elements of Vκ will be mapped to elements of M. Furthermore, since j is an elementary embedding, it must preserve the structure of V and thus also preserve the structure of Vκ. Thus, while each element of Vκ may not necessarily map to an element of M, any relationships between elements of Vκ that existed in V (i.e. the relationships that form the structure of Vκ) will be preserved in M.
 

1. What is a measurable cardinal?

A measurable cardinal is a type of large cardinal number in set theory. It is defined as a cardinal number that can be equipped with a non-trivial measure, which is a mathematical tool used to assign a size to subsets of the cardinal. Measurable cardinals have important implications in the study of set theory and the foundations of mathematics.

2. What is an inner model?

An inner model is a mathematical structure that is used to represent a certain set of axioms or principles. In the context of set theory, inner models are used to represent the axioms of set theory in a more manageable and understandable way. They are often used to prove consistency results and to study the properties of large cardinal numbers.

3. How does a measurable cardinal relate to inner models?

A measurable cardinal is important in the study of inner models because it can be used to construct inner models that satisfy certain desirable properties. In particular, measurable cardinals are used to construct inner models that satisfy the axiom of choice and the generalized continuum hypothesis, which are important principles in set theory.

4. What is the cumulative hierarchy?

The cumulative hierarchy is a mathematical concept that is used to organize sets into a hierarchy based on their complexity. It starts with the empty set at the bottom and then builds up to more complex sets by adding elements. The cumulative hierarchy is an important tool in set theory and is used to define and study concepts such as cardinal numbers and ordinals.

5. How does the cumulative hierarchy relate to inner models?

The cumulative hierarchy is closely related to inner models because inner models are often constructed using elements from the cumulative hierarchy. In particular, the construction of inner models using measurable cardinals relies on the use of the cumulative hierarchy to define and organize the sets that make up the inner model.

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