[WallWextra]

If you define the reals axiomatically, you still need an existence proof. Which will involve addition and multiplication algorithms.

[yakubin]

Not algorithms. There will be infinite addition involved, and algorithms are finite.

Thinking of multiplication as repeated addition also won't explain anything about it. It's a separate operation. Deal with it. For similar reasons, you can't calculate x-th power of a number, when x is irrational, by decomposing it into exponentiation and roots.

This metaphor is just training wheels. At some point you should lose it.

Together with the notion that "multiplication is repeated addition" comes the notion that numbers are quantities. Only some of them are, and this isn't really what makes them numbers. Now what exactly gets repeated, when you don't have quantities?

[prionassembly]

Multiplication on the naturals is repeated addition.

You might see the natural numbers as training wheels for higher mathematics, but number theorists might disagree...

[eru]

> There will be infinite addition involved, and algorithms are finite.

People are just bit sloppy, and say algorithm when they mean something slightly different.

See https://stackoverflow.com/questions/28841260/what-is-the-dif... and https://en.wikipedia.org/wiki/Corecursion

Basically, you don't want an 'algorithm' here to produce the whole number.

All you need is some scheme that will produce the next digit in finite time (and the next one and the next one etc).

[titzer]

Algorithms can work on symbolic formulas, and symbols can represent anything; infinite objects, operations on infinite objects, infinite sets of operations on infinite objects, and so on.

[bopbeepboop]

Algorithms aren’t necessarily finite, eg “while True, print 1”.

Also, an irrational exponent is the product of component factors.

b = Prod(0,inf) a^[x_i * 10^(-i)] = a^x

So even with irrational numbers, operations can be decomposed - such as exponentiation into multiplication of integer exponents and roots.

Which makes sense, because in the sequence definition of reals you need a way to generate the resulting sequence from the two original sequences.

I think you’re trying to claim more than is true.

[gugagore]

Algorithms are not finite. E.g. Newton's method.

[qsort]

Well, no, not really. The standard definition of the reals is as the unique nontrivial totally-ordered, Dedekind-complete, Archimedean field up to isomorphism. So what you would really need is a uniqueness proof, with addition and multiplications "provided" by the hypothesis.

[howling]

And how do you prove that a totally-ordered, Dedekind-complete, Archimedean field does not lead to contradiction besides constructing it explicitly by bootstrapping from natural numbers?