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> So, when we assume ∛7 "exists" in a formula, we're assuming that this infinite, never-ending decimal is somehow a complete object.

> This is why, Prof. Wildberger says he "doesn't believe in irrational numbers."

OK. I’m with you.

> His new method to solve polynomials also avoids radicals and irrational numbers, relying instead on special extensions of polynomials called "power series," which can have an infinite number of terms with the powers of x.

So, he believes this other infinite, never ending thing exists? Isn’t that a tad inconsistent?

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It could be the reporting?

Wildberger believes you can always get more precise calculations with more terms, but there is never a “completed” infinite object.

He had extensive videos on YouTube but something happened. Not sure if he ever got them all back.

> Irrational numbers, he says, rely on an imprecise concept of infinity and lead to logical problems in mathematics.

(Emphasis added)

As a mathetician, he doesn’t have a problem with the concept of infinity, it’s about the structure of the definition and how you are able to reason about it. Power series are heavily studied and understood.

> Power series are heavily studied and understood.

The different sizes of infinity are also heavily studied and understood.

> Power series are heavily studied and understood

And cube roots aren’t? Worse, the Taylor series are power series, and the cube root function can be described as a power series.

Personally, I find it hard to track how mathematicians reason about infinities.

They seem to make some very nuanced distinctions that don't make sense to me, but which satisfy them.

A lot of it comes down to set theory and the idea of mappings between sets. So countably infinite sets are those for which there exists a bijective (one-to-one and onto) mapping f:ℤ→X which is an extension of the idea of counting a finite set by showing that there’s a mapping of the set {1, 2, …, n}→X for a set X of size n. From this, we’re able to find larger infinities like |ℝ| where there is no bijective map from ℤ to ℝ. David Foster Wallace (yes, that David Foster Wallace) wrote a reasonably readable book about infinity that could be a good starting point in exploring the subject.

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