What is the biggest ordinal that exists metamathematically?

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The discussion centers on the concept of the largest ordinal in metamathematics, highlighting that any proposed largest ordinal can always be increased by adding one. It emphasizes that there is no maximum cardinality in metamathematics, as for any set, a larger set can always be defined. Graham's number is mentioned as a significant finite number, but it does not represent a largest ordinal. The conversation also touches on the idea that infinite sets can lead to "larger infinities," making the notion of a "set of everything" nonsensical. Ultimately, the concept of a proper class is introduced as a way to address collections that cannot belong to any other class.
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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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