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The real breakthrough with Dedekind cuts (as far as I can see) is not that you can define irrational numbers this way, it’s that when you do, calculus works. I
by seanhunter 2y ago
The real breakthrough with Dedekind cuts (as far as I can see) is not that you can define irrational numbers this way, it’s that when you do, calculus works. In other words, they obey all the axioms you would want and in particular they have the Archimedean property and possess a least upper bound.
You could always say that Dedekind was “just” restating real numbers in modern analytic language, but the fact is that no-one had managed to do that up till then, and it wasn’t obvious how it could be done. This is what sparked the crisis in analysis with some people saying that calculus was more like a religion that you just had to believe in rather than being able to prove.
- seanhunter 2y agoBy the way, there is an accessible and (in my opinion) brilliantly clear explanation of the construction of numbers on the “Another Roof” youtube channel. “What is a number?” https://youtu.be/dKtsjQtigag?si=EPzNVgk47gXzNLra https://youtu.be/dKtsjQtigag?si=EPzNVgk47gXzNLra Explains what natural numbers are and why we should care about this set of definitions And “defining every number ever” goes from there up to the complex numbers https://youtu.be/dKtsjQtigag?si=DI39LQtGvBsOU00y https://youtu.be/dKtsjQtigag?si=DI39LQtGvBsOU00y
- seanhunter 2y agoOh and the reason I picked on the Archimedean property specifically is that it had been this that both Leibnitz’”infinitesimals” and Newton’s “fluxions” had handwaved around. They both relied no some sort of non-Archimedean numbers for limits and derivatives to work. The Archimedean property is that given two numbers x and y there exists an integer n such that nx>y. Infinitesimals and fluxions relied on the idea that they were so small this wouldn’t work and therefore they didn’t fit the definition of real numbers.