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Garrulo
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Why in ZC (ZFC-reemplacement+separation) can´t exist Von Neumann Universet not even till \omega2. Sorry, I am new in the forum and I dont´know use Latex
Seems this question is answered here: http://math.stackexchange.com/questions/1402271/von-neumann-universe-in-zcGarrulo said:Why in ZC (ZFC-reemplacement+separation) can´t exist Von Neumann Universet not even till \omega2. Sorry, I am new in the forum and I dont´know use Latex
The Von Neumann Universe in ZC is a mathematical concept proposed by mathematician John von Neumann in the early 20th century. It is a theoretical construction of the collection of all sets in the Zermelo-Fraenkel set theory with Choice (ZC), which is a commonly used axiomatic system in mathematics.
The Von Neumann Universe is constructed by starting with the empty set, and then taking the power set (the set of all subsets) of each successive set in the hierarchy. This process continues indefinitely, creating an infinite hierarchy of sets.
The Von Neumann Universe is significant in understanding the properties and structure of sets in ZC. It provides a framework for studying the nature of infinity and the hierarchy of sets, and it is also used in the proof of the well-ordering theorem.
While the Von Neumann Universe is a theoretical construct and cannot be physically visualized, it can be represented visually through a diagram called the Von Neumann hierarchy. This diagram shows the sets in the hierarchy as nodes, with arrows connecting them to their respective power sets.
There are certain limitations to the Von Neumann Universe in ZC, such as the fact that it only contains sets that can be constructed within the ZC system. It also does not include sets that are too large to be contained in the hierarchy, such as the set of all real numbers. Additionally, the Von Neumann Universe does not address the existence of sets that cannot be defined within the ZC system, known as proper classes.