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If you look at the integers between say Graham's number (https://en.wikipedia.org/wiki/Graham%27s_number) and TREE3(https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem) you can observe that practically all of these integers, while "computable", cannot be defined within the known constraints of this universe.

Which raises an interesting question: In what meaningful sense do these numbers exist? They are just out of reach as the non-definable real numbers...

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You're mixing up "defined" with "defined via a decimal numeral" we can define these numbers without much difficulty via finite formula that compute them. This is a completely valid definition, it is just not a decimal numeral.

An interesting idea might be "useful integers" which requires whatever definition we have to allow approximation of any finite subsequence with error converging to zero given more computational power.

GP did not mix up anything. Some of those finite formulas also will be too long to be written within the constraints of this universe. The pigeonhole principle applies just as much to finite formulas as it does to finite strings of decimal digits.

In what meaningful sense do any numbers exist? This comes up with my kids sometimes ... are numbers real?

I liked this Numberphile video on that topic: https://www.youtube.com/watch?v=1EGDCh75SpQ

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