[Question] What are ranks? In your opinion.

[Daniel]

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@daniel
can you expand on your point 2)? This seems an interesting philosophical position, I'd like to have more details.

OK, more stuff. I understand renormalization to be the process in physics based on moving up to the successor hyperoperator. I suspect the result of this is that the hierarchy of physics, chemistry and biology mapping at some level to the hyperoperators. I'm sure physics has at least the complexity of tetration. Assuming tetration manifest in physics, it's complexity would be that of tetration at least. But it seems like the Universe would be well served by creating a computer to produce higher order hyperoperators and their physical manifestation. That would be us! So humans and other equivalent alien beings create the higher hyperoperators. But the mind is amazing, it was able to create Gödel's theorems. In the same way, I hope that the hyperoperators as a whole can be understood.

About answer 3) I agree that a civilization able to master rank 4, 5, 6... would need incredible computational power available and/or a major conceptual and theoretical advance in how to manage functions of high complexity. Anyways, do you see, Daniel, the problem of fractional or even complex ranks to pose an even bigger obstacle, of do you see both big integers and non-integer ranks as having comparable difficulties for such an advanced civilization?

While I have given thought to non-integer and complex hyperoperators, I have nothing to report. My difficulty is I have no expectation of what would be found in such systems, no tests to validate any possible solutions. I do think the integer hyperoperators including \[x\uparrow^\infty \infty\] would need to be understood, before complex hyperoperators because a complex hyperoperator expression would likely be created from the entire set of hyperoperators.