Suppose that $${\displaystyle \kappa }$$ is a cardinal number. Zermelo–Fraenkel set theory with Choice (ZFC) implies that the $${\displaystyle \kappa }$$th level of the See more

There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal μ, there is an … See more

• Worldly cardinal, a weaker notion
• Mahlo cardinal, a stronger notion
• Club set
• Inner model
• Von Neumann universe See more

• Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics, vol. 76, Elsevier Science, ISBN 0-444-10535-2
• Hausdorff, Felix (1908), "Grundzüge einer Theorie der … See more

The term "α-inaccessible cardinal" is ambiguous and different authors use inequivalent definitions. One definition is that a cardinal κ is … See more

Firstly, a cardinal κ is inaccessible if and only if κ has the following reflection property: for all subsets $${\displaystyle U\subset V_{\kappa }}$$, there exists $${\displaystyle \alpha <\kappa }$$ such that $${\displaystyle (V_{\alpha },\in ,U\cap V_{\alpha })}$$ See more