You can define a procedure that reaches 2 very trivially. e.g. you generate all possible 1 digit numbers, then 2 digit numbers, then 3 digit numbers, etc.
This is how natural numbers work in the first place. You're just adding a decimal point to all of the possible places it could go.
Well, you have to ask yourself what is PI really? You've been taught it is a number with infinite decimal places, however, practically speaking when you use PI you actually round. Practically speaking this doesn't matter because it's not actually possible to have a shape with infinite points in reality so you only need to approximate to whatever fidelity suits you.
How is this? This is because PI is not actually a number. It's a procedure that generates digits for approximating things about circles.
This is the same with sqrt(2). The sqrt procedure emits digits just as the procedure to find all real numbers does.
You can't "reach" PI for the same reason that a natural number can't reach "f(x) => x + 1". That is, natural numbers aren't procedures or functions.
What about 1/3, 1/7, …?
Previously outlines recursive procedure doesn’t generate those.
But yeah, if you deny existance of irrational numbers, and redefine Real:=Rational, then you can generate these “real” numbers recursively and it does follow that all infinities have same cardinality here.
Btw. what is the diagonal of a unit square formed by 4 objects at the corners? I assume it is a rational number.
Btw2. If you take that answer and multiply by itself, what do you get?
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.
It seems to me that this comes at the price that now in your geometry the diagonal of a unit square does not have a defined length. Only an approximate one of 1.41421, but what does this approximate?
What price? In an algebraic setting you are never going to convert sqrt(2) into a real number, and in any practical setting you're going to have to round because nothing real actually has infinite precision.
It would appear that sqrt(2) does not have a meaning now, as sqrt(-1) in R. So it does not seem to matter if you do not convert it in an algebraic setting. There is no such number that you can reach with the enumerative approach, which will only give you the rational numbers. There are lots of proofs that sqrt(2) is not rational.
But in all probability we are discussing the wrong thing here. Our difference it's likely at a deeper conceptual level than this.
This is how natural numbers work in the first place. You're just adding a decimal point to all of the possible places it could go.