zlacker

>>tromp+(OP)
The article discusses different encodings of numbers to give non-dense representations of numbers exceeding (2^64)-1. (These representations are inherently non-dense by the pigeonhole principle.) But I feel like this is missing a key point. A 64-bit string that we agree represents only numbers can represent 18446744073709551616 different numbers. The choice of what numbers are represented is completely up to us. If we want certain properties (dense integers) we end up with the highest being 18446744073709551615. If we want other properties (nearly logarithmic distribution, signedness, and good mapping to hardware for arithmetic) we might end up with FP64 with a maximum value around 10^308. And if we want no interesting property constraints except being dual to a program on a Turing machine, we end up with a busy beaver number. But... remember, we can choose any 18446744073709551616 values we want to be representable. There's no restriction on the interpretation of these strings; or, equivalently, the amount of external information required to explain the interpretation of these strings is unbounded. As a result, we can choose any computable number to be encoded by the string 64'b1, or by any other string, and exceed any desired bounds.

>>addaon+ab6
> we can choose any computable number to be encoded by the string 64'b1

First of all, every finite number is computable by definition.

And second, your encodings will, unlike those in the lambda calculus, be completely arbitrary.

PS: in my self-delimiting encoding of the lambda calculus, there are only 1058720968043859 < 2^50 closed lambda terms of size up to 64 [1].

[1] https://oeis.org/A114852

>>tromp+Ne6
It's interesting that you are counting how many lambda calculus terms can fit into 64 bits given a particular encoding, but you're not making room in the 64 bits for an explanation of how your encoding works (let alone how lambda calculus works!). Of course, any encoding you used to include an "explanation" in the 64 bits would itself require some explanation. I guess what I'm getting at is that I'm not sure that your approach is any less arbitrary than any other approach.