Once you drop infinity as an axiom you have to think about fractions and repeating decimals differently.
1/3 isn't a real, it's a fraction. Fractions can be used to generate reals, and they can be used in algebra along with reals.
0.(3) is also not a real. It's also just representative of a procedure that can generate reals.
Both 1/3 and 0.(3) can still be used in algebra in the same way as before. You don't lose any capability because you can't practically expand 0.(3) to infinite decimal places in the first place.
You can have as many different symbols as you like, so long as you can actually write them on paper. As soon as you tell me that one of those symbols is a number with infinite digits I will disagree with you, given that you are unable to tell me what those digits are before the universe ends.
I can tell you that the number is 0.1 in base 3, which defines the number precisely. Or I can tell you that it in base 10, it has an infinite representation, and all the digits are 3. There, I told you what they all are, and the universe has not ended yet.
> Or I can tell you that it in base 10, it has an infinite representation,
Does it really have an infinite representation? I can't imagine an infinite representation fitting on a page. I'm pretty sure you're representing it as 1/3 or 0.(3). Neither of those representations are infinite. They're only a few characters really.
1/3 isn't a real, it's a fraction. Fractions can be used to generate reals, and they can be used in algebra along with reals.
0.(3) is also not a real. It's also just representative of a procedure that can generate reals.
Both 1/3 and 0.(3) can still be used in algebra in the same way as before. You don't lose any capability because you can't practically expand 0.(3) to infinite decimal places in the first place.