@CoronaCoreanici @starkatt epsilon is more properly 1/ω, right?
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@CoronaCoreanici @starkatt @Felthry Honestly, squaring to get -1 is a lot /easier/ for us, with the 2D thing and multiplying-by-i-as-rotation. :3

@CoronaCoreanici @starkatt @Felthry "You can't square something to get -1!" *plops a second axis on the number line*

@CoronaCoreanici @starkatt @Felthry Wait, it does?

@IceWolf @CoronaCoreanici @starkatt aaaa i can't keep up with this right now
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@CoronaCoreanici @IceWolf @starkatt also I feel like 0.9999... would still be equal to 1 even if infinitesimals exist? Because you can't write an infinitesimal out like that, 1-ε is not the same as 0.9999..., right?
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@Felthry @CoronaCoreanici @starkatt Well, wouldn't 0.999999... be an infinitely small amount smaller than 1, in that case? It's like a limit.

...whoo boy, this would totally break calculus wouldn't it...

@CoronaCoreanici @IceWolf @starkatt honestly limits always have felt really kludgey to us, like they're not a very elegant way of going about this kind of stuff

which is probably part of why we prefer to think of derivatives as literal ratios of infinitesimals
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@CoronaCoreanici @IceWolf @starkatt I guess all of this is an engineer's intuition more than a mathematician's rigor but I feel like our intuition could be made rigorous in a way that allows one to think of dx and dt and stuff as real infinitesimals that you can do math to
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@CoronaCoreanici @starkatt So now that I have the time to come back to this--what makes the archimedean property fundamental? It doesn't sound like some underpinning of mathematics, it sounds more like an interesting property.
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@CoronaCoreanici @starkatt we have a vague idea of what a dedekind cut is; it's considering a number as an ordered pair of ({the set of all numbers less than it}, {the set of all numbers greater than it}), right?
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@CoronaCoreanici @starkatt it feels kind of Weird that you can do that and get ℵ₁ reals, when you only have ℵ₀ rationals. It feels like there could only be ℵ₀ ways to divide them up in a consistent manner
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@CoronaCoreanici @starkatt I thought the cardinality of the reals was ℵ₁? huh
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@CoronaCoreanici @starkatt and with your saying that it's not fundamental, I have to wonder if there's a consistent formulation of mathematics that lets all our intuitive understanding be completely valid
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@CoronaCoreanici @starkatt Our intuitive understanding of stuff has so far worked pretty well for engineering work

I feel like we wouldn't be suited for pure mathematics, though
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@CoronaCoreanici @starkatt @Felthry So reading Wikipedia, to put it another way, if you keep adding copies of s it'll eventually get past r, for any s you can pick?