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> no computational model can represent all real numbers (by a trivial cardinality argument). I agree with your points about this problem being difficult. But t

> no computational model can represent all real numbers (by a trivial cardinality argument). I agree with your points about this problem being difficult. But the cardinality argument is neither here nor there. The cardinality argument is based on classical logic, and has no relevance to computation. In constructive mathematics it is perfectly possible that all real numbers can be represented by finite algorithms. And even that all functions defined on all the reals are continous: a finite constraint of the output can be computed from a finite constraint on the input. But, starting from a definition of the reals as a Cauchy sequence we have a long way to go to get something which can be practically computed on. I think RealPCF [0] was a step in the right direction. [0]: https://www.cs.bham.ac.uk/~mhe/papers/realpcf.pdf https://www.cs.bham.ac.uk/~mhe/papers/realpcf.pdf

qsort 6y ago
Again, I don't disagree with any of this. The argument advanced by the OP is that floating point numbers are not intuitive to use for the general public because they don't work like the unique complete ordered field up to isomorphism. All I'm saying with the cardinality argument is that because real numbers are fundamentally unlike integers, obvious methods of doing things won't work, and we are stuck (at least for the time being) with leaky abstractions. I make no claim about the existence of better/more intuitive solutions. My only counterpoint is that while the ideas outlined in the paper are certainly valuable, I don't think the framework is mature enough to replace the default representation for real numbers in a programming language like Java, at least not without further work and/or proof that it can be adapted to a more general setting without disastrous performance decay. > But, starting from a definition of the reals as a Cauchy sequence... I trust that what you're saying is true, but I'm not familiar with that topic, I'm sorry. Thanks for sharing the paper.

qsort 6y ago

Again, I don't disagree with any of this. The argument advanced by the OP is that floating point numbers are not intuitive to use for the general public because they don't work like the unique complete ordered field up to isomorphism. All I'm saying with the cardinality argument is that because real numbers are fundamentally unlike integers, obvious methods of doing things won't work, and we are stuck (at least for the time being) with leaky abstractions. I make no claim about the existence of better/more intuitive solutions. My only counterpoint is that while the ideas outlined in the paper are certainly valuable, I don't think the framework is mature enough to replace the default representation for real numbers in a programming language like Java, at least not without further work and/or proof that it can be adapted to a more general setting without disastrous performance decay. > But, starting from a definition of the reals as a Cauchy sequence... I trust that what you're saying is true, but I'm not familiar with that topic, I'm sorry. Thanks for sharing the paper.