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For the formal underpinnings of the infinitesimal approach, the phrase to look for is "Nonstandard Analysis": https://en.wikipedia.org/wiki/Nonstandard_analysi

by makr17 4y ago

For the formal underpinnings of the infinitesimal approach, the phrase to look for is "Nonstandard Analysis": https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis Fun stuff.

drewrv 4y ago
I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis? I loved real analysis in college, for example I found cantors set kind of crazy. Does that "exist" in nonstandard analysis? Or are there other mind-bending implications?
- hansvm 4y ago
  Standard math is a subset of nonstandard, and you can prove the existence of NSA within ZFC, so they are exaxtly equivalent.
- blagie 4y ago
  > I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis? This is a bit of a philosophical question which hinges on what the word "equivalent" means. I disagree with the other comment. It is not totally equivalent, or even at all equivalent. I think about math in terms of set theory. You create sets, like the real numbers, the integers, or otherwise. You can add other elements to those sets. For example: * You can add a variety of infinities (countable, uncountable, positive, negative, etc.). You have a self-consistent system. * You can add imaginary numbers, think of the whole thing as a ball, with a zero on one end and infinity on the other. That gives you complex analysis, which is self-consistent as well. All of those are helpful, useful, and powerful, and generally lead to the same place where they overlap, they're not very compatible with each other, either formally or intuitively. I view the two formulations of calculus as similar. Even if they lead to the same results, that's not the same as saying they're equivalent. They're different formalisms.
zozbot234 4y ago
Edward Nelson's Internal Set Theory is a somewhat cleaner and more intuitive foundation that's practically equivalent to non-standard analysis.

by makr17 4y ago

drewrv 4y ago

I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis? I loved real analysis in college, for example I found cantors set kind of crazy. Does that "exist" in nonstandard analysis? Or are there other mind-bending implications?

hansvm 4y ago
Standard math is a subset of nonstandard, and you can prove the existence of NSA within ZFC, so they are exaxtly equivalent.
blagie 4y ago
> I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis? This is a bit of a philosophical question which hinges on what the word "equivalent" means. I disagree with the other comment. It is not totally equivalent, or even at all equivalent. I think about math in terms of set theory. You create sets, like the real numbers, the integers, or otherwise. You can add other elements to those sets. For example: * You can add a variety of infinities (countable, uncountable, positive, negative, etc.). You have a self-consistent system. * You can add imaginary numbers, think of the whole thing as a ball, with a zero on one end and infinity on the other. That gives you complex analysis, which is self-consistent as well. All of those are helpful, useful, and powerful, and generally lead to the same place where they overlap, they're not very compatible with each other, either formally or intuitively. I view the two formulations of calculus as similar. Even if they lead to the same results, that's not the same as saying they're equivalent. They're different formalisms.

zozbot234 4y ago

Edward Nelson's Internal Set Theory is a somewhat cleaner and more intuitive foundation that's practically equivalent to non-standard analysis.