Is Rayo's number really that big? I was reading about large numbers, and came across Rayo's Number which is defined to be the smallest integer that is not nameable by any expression in the language of set theory that contains less than $10^{100}$ symbols.
Now, my question is: Is this number really that large?
If we pick some "random" number $2091580284...384901284021$ with $10^{100}$ digits, wouldn't it be non-nameable with less than a googol symbols? Wouldn't this number be bigger or equal to Rayo's Number?

No, a number like 10^(10^100) is much smaller than Rayo(10^100), because there are not a Googol digits, but a Googol symbols in first order set-theory, and it is pretty efficient to write down big numbers such as TREE(3), which can be expressed with MUCH LESS than a Googol symbols, you don't need anything like a Googol symbols to express TREE(3),and if we can say that :
              Rayo's Number > TREE(3)

We can certainly say that :
              Rayo's Number > Any number with 10^100 digits

So Rayo's Number is really that big.
