1/3 is also a special kind of procedure. It's 1 divided by 3. Either you use the isomorphic 0.3 repeater procedure where you recursively add 3 digits until you get bored or you have enough, or you use division to generate the number sequence. The fun thing about fractions is that we've worked out some ways in which fractions can be multiplied with natural numbers and rational numbers to generate new fractions, and also some fractions conveniently are isomorphic with rational and natural numbers.
Important to note: When ggp asked for a recursive procedure to generate real numbers, they wanted that exactly same proceedure would generate all reals (not special procedure for each number)
If we have special procedure for each number, then procedure to generate 1/3 is just 1/3. …of course naively assuming notation of 1/3 is as valid as 0.33333…, and that base 10 is not the only possible base.
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.
