4 ms·
In fact, BB(k) is roughly equivalent to asking "what's the biggest non-infinity namable number in k characters".
by shouyatf 7y ago
In fact, BB(k) is roughly equivalent to asking "what's the biggest non-infinity namable number in k characters".
- baddox 7y agoThat doesn’t sound right, because “BB(k) plus 1” is larger than BB(k) and has fewer than k characters for sufficiently large k.
- shouyatf 7y agoThat's why I say 'roughly'. Natural language is very vague and easy to be used to create illogical self-referential statements.
- dandanua 7y agoWhat you say is a Berry paradox. To be consistent we fix a programming language before any such definitions. So you have to count all characters in the BB(k) subprogram. Hence the total program for “BB(k) plus 1” definitely will have more than k characters.
- baddox 7y agoI don't think my example (for example, "BB(11111) + 1") demonstrates the Berry paradox. The original comment "the biggest non-infinity namable number in k characters" is the Berry paradox, which is what I was pointing out.
- yters 7y agoBut then you can have an even more powerful BB that has access to a halting oracle. Then you can have the oracle hierarchy. And then maybe there is another meta direction on top of that. It seems there is no limit to metaness, and that unlimited nature cannot be captured by notation, otherwise one can go meta on whatever notation is used. It is like the corollary to "there is no highest number," there is also no fastest growing function. Whatever function one names, it is always possible to use that function to define an even faster growing function.