[mFixman]

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.

[tomtomtom1]

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

[mFixman]

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

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

[tomtomtom1]

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