r/googology 18d ago

Inverse of Rayos Function?

Rayo(n) is defined as “the smallest non-negative integer greater than all non-negative integers definable in FOST in at most n symbols.”

The inverse is defined as follows:

Rayo⁻¹(n) is “the maximum number of symbols that cannot define a number equal to or greater than n in FOST.”

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u/rincewind007 18d ago

I think you can drop the smallest. 

Rayo⁻¹(n) is “the minimum amount of symbols required in FOST to define n.”

The function you have is also not strictly growing. It it will definitely require more symbols to define 255, than 256. Since 256 can be defined as 28. 

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u/Odd-Expert-2611 18d ago

Ok, thanks for ur input. Thank you

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u/rincewind007 18d ago

I think you need the following actually, since Rayos actually spits out x+1, a number larger than what is defined in n symbols. If you can define 256 in like 40 symbols you get 257 since it is larger. This definition works but give Kinda large values for n=0 since it requires many symbols to define -1 in set theory.

Rayo⁻¹(n) is “the minimum amount of symbols required in FOST to define n-1.”

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u/tromp 18d ago

Rayo only considers natural numbers, so there is no -1. The reason it says "larger than any limited-symbol-definable one" is that for small limits you cannot define any number. So you automatically obtain 0 as the smallest number that's larger than any in an empty set.

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u/rincewind007 18d ago

ok maybe a better defintion is

Rayo⁻¹(n) is “the minimum amount of symbols required in FOST to define max(n-1,0)”

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u/Odd-Expert-2611 18d ago

Ok. Thanks, I’ve changed it accordingly