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One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up s
by moyix 13y ago
One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up something akin to standard analysis (computable analysis [2]).
[1] http://en.wikipedia.org/wiki/Computable_number http://en.wikipedia.org/wiki/Computable_number
[2] http://en.wikipedia.org/wiki/Computable_analysis http://en.wikipedia.org/wiki/Computable_analysis
- gizmo686 13y agoUnfourtuantly, there exist numbers which are definable but not computable.
- moyix 13y agoSure. Chaitin's Omega is a good example. The question is whether such numbers occur in the real world.
- deleted 13y ago[deleted]
- jfarmer 13y agoThe trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)
- moyix 13y agoThis reminds me of the self-defeating property of an "uninteresting" number -- a reasonable definition might be "any number that does not have any property of human interest", but then of course there is a smallest such number, and so that has the interesting property of being the first uninteresting number, a contradiction! You're right though that a more precise definition of "real-world numbers" is needed, but I confess that my attempts to think of one in the past few minutes have been essentially circular (coming down to "the ones we know how to compute")!
- jfarmer 13y agoWell, we can and have made the idea of a computable number precise: http://en.wikipedia.org/wiki/Computable_number http://en.wikipedia.org/wiki/Computable_number It's not clear whether the universe is computable, however, in the sense that we only find computable numbers in nature. This is kind of an epistemological catch-22, though. How would we know whether this were the case or not?