If we say that infinitesimals exist, it still happens that 1 = 0.999…. It just happens that 0.999 ≠ 1 - 𝛚.

0.999… = 1 is a property of the way we write some rational numbers, not of the number system itself.

wouldn't 0.999.. be equal to 1 - 10^w since it's only a countably finite series of nines.

0.999...9 with a countable finite amount of nines is clearly less than 1.

0.999... with infinite nines is equal to 1.

sorry I typoed I meant, countably infinite. by w I meant the ordinal.

I have never seen a number system with infinitesimals where the addition wasn't updated to ignore smaller classes if they come with larger ones.

That is, for any number system I've seen, 1 = 1 + dx, and infinity = infinity + 100.

The obvious one seems terrible enough, that division is no longer the inverse of multiplication: (1/3)*3 != 1

Nonstandard analysis exists (with infinitesimal and infinite numbers) , but 1/3 and 9/9 is the same there. The problem is that the numbers 0.333... and 0.999... don't really exist.

Completeness is one of the most important properties of real numbers. Basically, you will have to completely throw away real analysis.

1/3 does equal 0.3.. though.