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
came here and searched for "busy". good job :)

To the sibling comment about arbitrariness, we could use a hybrid where we trade off some bits from IEEE FP to introduce far reaches and also some precision there.. so like, keep 32 bits or 54 bits for IEEE compatibility, then switch to "extended" ranges for e.g. BB numbers, higher alephs, etc..

There was this one system for calculation with infinities that avoided the Hilbert Hotel problem.. can't find it but was called smth like Infinioid or some other play on the name. Would be neat to bolt those on too :)

Edit: "grossone" is the calculus for infinities.. love this work! https://www.theinfinitycomputer.com/

>>pmayrg+Tg6
> To the sibling comment about arbitrariness, we could use a hybrid where we trade off some bits from IEEE FP to introduce far reaches and also some precision there.. so like, keep 32 bits or 54 bits for IEEE compatibility, then switch to "extended" ranges for e.g. BB numbers, higher alephs, etc.

This sort of trick/hack is the reason why theorems in (algorithmic) information theory involve constant factors. For example, we can define an image compressor which outputs a single `1` when given the Lenna test image[0], and otherwise acts exactly like PNG except prefixing its output with a `0`. To decode: a `1` decodes to the Lenna image, and anything starting with `0` gets decoded as PNG without the leading `0`. This gives perfect compression with no loss of quality, when tested on that Lenna image ;)

[0] https://en.wikipedia.org/wiki/Lenna

>>chrisw+Jq6
Furthermore this method can be scaled up to cover more and more test images, with a modest increase of prefix bits