What is the biggest ordinal that exists metamathematically?

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The discussion centers on the concept of ordinals and their metamathematical implications, specifically addressing the notion that there is no largest ordinal. Participants highlight that for any given ordinal, one can always define a larger ordinal by adding one. Graham's number is mentioned as a notable finite number, but it does not represent the largest ordinal. The conversation emphasizes the distinction between sets and proper classes, asserting that while sets can be infinite, the concept of a maximum set is nonsensical.

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Garrulo
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What is the biggest ordinal that exists metamathematically??
 
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Whatever the biggest ordinal is, I can name it plus 1. Discuss.
 
Well, if there is not a maximum cardinal metamatemathically existing, ¿which is the colection of this matematical sets than exist metamathematically
 
But the Graham´s number is mathematically finite, isn´t it? I talk about ordinal numbers transfinite
 
Given any set - infinite or not - there is another set that is larger. If the set is infinite, then the other set is a "larger infinity" which means that it is so big that there is no way to ever match it up with the first.

It follows that there is no largest set and the idea of a set of everything makes no sense.
 
Not a set, but it exists the concept of propper class. A class that can´t be in another class
 

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