(February 1, 2012 at 1:28 am)Pendragon Wrote: I don't care. Can a computer AI (artificial Intelligence) find a structure it can make a decision with, your goldbach conjecture?
For this kind of programming, one needs constructionist tools. Infinity does not work on finite systems. You have to deal with reality sometimes, not hallucinations of god, and infinity.
"Infinity does not work on finite systems"...huh? It's quite easy to define transfinite arithmetic consistent with Cantor measures.
If you just disregarded infinity altogether, you'd never get anything like Wolfram Alpha, which can solve very sophisticated algebra and complex analysis problems that require a notion of infinity--in the case of complex analysis, it occasionally requires an action "point at infinity" to be described.
If you got rid of infinity, you'd get rid of most applications of computers to solving systems of differential equations; finding convergence or divergence requires a notion of what it means to "diverge to infinity".
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