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> While with infinite sets, the point is that they aren't (to the best of my knowledge) either 1) directly useful or 2) particularly interrelated with anything

> While with infinite sets, the point is that they aren't (to the best of my knowledge) either 1) directly useful or 2) particularly interrelated with anything else. Formal treatments of calculus are based around infinitesimals. Infinitesimals are the same set of ideas about infinite sets, but applied to the spaces between arbitrarily close points. Specifically formal treatments of calculus work because between any two points there are an infinite number of points and the size of that infinity is the same size as the entire set of real numbers. At least this is my undergrad level of comprehension. I'm certain there are more exotic treatments, but this is the one that you have to do all the proofs for.

porb121 5y ago
> Formal treatments of calculus are based around infinitesimals formal treatments of calculus don't use infinitesimals, and analysis using infinitesimals is explicitly separate from traditional analysis
- jyounker 5y ago
  You're absolutely right. I don't know what I was thinking when I wrote this last night. I meant to say that formal treatments are based around being able to find arbitrarily small neighborhoods around a point, and that this only works when you have something the size of the reals, for which you can find a real-sized space between any two points. It's this infinite self-similarity at any arbitrarily small scale that is at the heart of calculus.
adenadel 5y ago
This is not true. The usual calculus that people learn (and usually learn with rigor in a real analysis course) does not touch infinitesimals. The approach via infinitesimals is even called "nonstandard analysis". https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis
- jyounker 5y ago
  You're absolutely right. I was going for a different concept. See explanation above. I should stop writing after 02:00.

porb121 5y ago

> Formal treatments of calculus are based around infinitesimals formal treatments of calculus don't use infinitesimals, and analysis using infinitesimals is explicitly separate from traditional analysis

jyounker 5y ago
You're absolutely right. I don't know what I was thinking when I wrote this last night. I meant to say that formal treatments are based around being able to find arbitrarily small neighborhoods around a point, and that this only works when you have something the size of the reals, for which you can find a real-sized space between any two points. It's this infinite self-similarity at any arbitrarily small scale that is at the heart of calculus.

adenadel 5y ago

This is not true. The usual calculus that people learn (and usually learn with rigor in a real analysis course) does not touch infinitesimals. The approach via infinitesimals is even called "nonstandard analysis". https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis

jyounker 5y ago
You're absolutely right. I was going for a different concept. See explanation above. I should stop writing after 02:00.