I am a little confused. I was working with a definition of continuity mentioned here https://mathworld.wolfram.com/RationalNumber.html : “It is always possible to find another rational number between any two members of the set of rationals. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable.”
I understand that Rationals aren’t complete, and my question is why this is important for scientific inference. In other words, are we using the reals only because it makes the math easier, or is there a concrete example of inference which completeness helps with?
Specifically, since the context Jaynes is interested in is designing a (hypothetical) robot’s brain, and in order to achieve that we need to associate degrees of plausibility with a physical state, I don’t see why that entails the property of completeness which you mentioned? In fact, we mostly use digital and not analog computers, which use rational approximations for the reals. What does this system of reasoning lack?
The word used for the property referred to in the Wolfram article really should be dense, not continuous. The set of rationals is dense, but incomplete and totally disconnected.
The main property lacking is exactly what I stated earlier: for some perfectly reasonable questions, rationals only allow you to work with approximations that you can prove are always wrong. That’s mathematically very undesirable. It’s much better to have a theory in which you can prove that there exists a correct result, and then if you only care about rational approximations you can just find a nearby rational and accept the error.
Thank you.
I am a little confused. I was working with a definition of continuity mentioned here https://mathworld.wolfram.com/RationalNumber.html : “It is always possible to find another rational number between any two members of the set of rationals. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable.”
I understand that Rationals aren’t complete, and my question is why this is important for scientific inference. In other words, are we using the reals only because it makes the math easier, or is there a concrete example of inference which completeness helps with?
Specifically, since the context Jaynes is interested in is designing a (hypothetical) robot’s brain, and in order to achieve that we need to associate degrees of plausibility with a physical state, I don’t see why that entails the property of completeness which you mentioned? In fact, we mostly use digital and not analog computers, which use rational approximations for the reals. What does this system of reasoning lack?
The word used for the property referred to in the Wolfram article really should be dense, not continuous. The set of rationals is dense, but incomplete and totally disconnected.
The main property lacking is exactly what I stated earlier: for some perfectly reasonable questions, rationals only allow you to work with approximations that you can prove are always wrong. That’s mathematically very undesirable. It’s much better to have a theory in which you can prove that there exists a correct result, and then if you only care about rational approximations you can just find a nearby rational and accept the error.