[madhadron]

It's a tradeoff. You can have excluded middle in your logic or infinitestimals in your extended reals. For mathematicians dealing with all the wild stuff coming out of studying infinities in the calculus, getting rid of excluded middle was a non-starter, so the system based on limits was created. If non-constructible proofs via contradiction aren't useful to you, as in physics, then you can certainly use infinitesimals.

[farrelle25]

That's interesting - I read something similar in Bell's 'A primer of infinitesimal analysis' where he said the price for 'Smooth World' infinitestimals is giving up the Law of Excluded Middle (LEM).

Don't really understand why (he said something about unconstrained use of LEM allows discontinuous functions...)

Is there any link to Brouwer's Intuitionism where LEM is rejected too (?!)

Ah it's all an interesting can of worms...

[Ericson2314]

https://ncatlab.org/nlab/show/real+numbers+object you can definitely have real numbers without the infinitesimals in constructive math, however.