Various Intuitions and Conceptualizations of Measurable Cardinals.

Thread starter biffus22
Start date Feb 14, 2024
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Set theory

Feb 14, 2024

biffus22

The concept of a "measurable cardinal" is rather difficult for many students of "Intermediate" Set Theory to grasp in terms of more basic set theoretic concepts -- as opposed say to concepts dealing with the relations among various "universes" or "models" etc. In fact, much of the problem may derive from the difficulty of imagining a "non-principal ultrafilter" within the P(X), the Power set of X. Whatever the difficulties involved, the concept is, it seems, also very difficult to teach. I am thus interested in how other people here understand, or better, "grasp," what a measurable cardinal is and is not, and how they attempt to teach their students the concept using more basic concepts such students already comprehend.

Last edited: Feb 14, 2024

Related to Various Intuitions and Conceptualizations of Measurable Cardinals.

1. What are measurable cardinals?

Measurable cardinals are a type of large cardinal in set theory. They are defined as cardinals with certain properties that allow for the existence of non-trivial elementary embeddings from the universe of sets into itself.

2. What are various intuitions behind measurable cardinals?

Various intuitions behind measurable cardinals include the idea of having large cardinalities that cannot be reached by smaller cardinalities through set functions, as well as the existence of non-trivial elementary embeddings that preserve the structure of the universe of sets.

3. How are measurable cardinals used in set theory?

Measurable cardinals are used in set theory to study the structure of the universe of sets, as well as to prove consistency results and establish the existence of certain mathematical objects that cannot be constructed in Zermelo-Fraenkel set theory alone.

4. What are some properties of measurable cardinals?

Some properties of measurable cardinals include being inaccessible cardinals, having non-trivial elementary embeddings, and satisfying certain combinatorial properties that allow for the construction of certain mathematical structures.

5. How do measurable cardinals relate to other large cardinals?

Measurable cardinals are a specific type of large cardinal, and they are related to other large cardinals through the hierarchy of large cardinal properties and the consistency strength of various large cardinal axioms in set theory.