[falserum]

> 1/3 is also a special kind of procedure.

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.

[ludston]

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.

[falserum]

Is 1/10 (also known as 0.1) a valid number in base 10?

What about same number expressed in base 3? (I think in base 3 that would be written as 1/31)

And what about number 0.1 in base 3? (Which is equivalent to 1/3 in base 10)