Pseudo Rayo's number is the smallest positive integer that cannot be named by an expression in the language of first order set theory (FOST) with a googol symbol or less. Upper bound of the pseudo Rayo's number is \((10^{100})^{10^{100}} = 10^{10^{102}}\), according to the discussion in the comment of this blog post. It is not a very big number in googology, but it is very difficult to express its exact value without using this expression.

How can we estimate the lower bound of pseudo Rayo's number? In Rayo's number, it is shown that number n can be expressed in (9n2+43n)/2 symbols, by using "0" and "+1" operations. If we can express "*2" operation, we can express all numbers below 2n in 0, n operations of "+1" and n operations of "*2". If we assume that numbers of repetition of such operations can be expressed in some form of quadratic equation of n, the lower bound of the pseudo Rayo's number is in the order of \(10^{10^{50}}\). As it is the lower bound, I make it \(10^{10^{45}}\).

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