As you may have noticed, E. E. Escultura has shown up in the comments to this blog. In one comment, he made an interesting (but unsupported) claim, and I thought it was worth promoting up to a proper discussion of its own, rather than letting it rage in the comments of an unrelated post.
What he said was:
You really have no choice friends. The real number system is ill-defined, does not exist, because its field axioms are inconsistent!!!
This is a really bizarre claim. The field axioms are inconsistent?
I’ll run through a quick review, because I know that many/most people don’t have the field axioms memorized. But the field axioms are, basically, an extremely simple set of rules describing the behavior of an algebraic structure. The real numbers are the canonical example of a field, but you can define other fields; for example, the rational numbers form a field; if you allow the values to be a class rather than a set, the surreal numbers form a field.
So: a field is a collection of values F with two operations, “+” and “*”, such that:
- Closure: ∀ a, b ∈ F: a + b in F ∧ a * b ∈ f
- Associativity: ∀ a, b, c ∈ F: a + (b + c) = (a + b) + c ∧ a * (b * c) = (a * b) * c
- Commutativity: ∀ a, b ∈ F: a + b = b + a ∧ a * b = b * a
- Identity: there exist distinct elements 0 and 1 in F such that ∀ a ∈ F: a + 0 = a, ∀ b ∈ F: b*1=b
- Additive inverses: ∀ a ∈ F, there exists an additive inverse -a ∈ F such that a + -a = 0.
- Multiplicative Inverse: For all a ∈ F where a != 0, there a multiplicative inverse a^{-1} such that a * a^{-1} = 1.
- Distributivity: ∀ a, b, c ∈ F: a * (b+c) = (a*b) + (a*c)
So, our friend Professor Escultura claims that this set of axioms is inconsistent, and that therefore the real numbers are ill-defined. One of the things that makes the field axioms so beautiful is how simple they are. They’re a nice, minimal illustration of how we expect numbers to behave.
So, Professor Escultura: to claim that that the field axioms are inconsistent, what you’re saying is that this set of axioms leads to an inevitable contradiction. So, what exactly about the field axioms is inconsistent? Where’s the contradiction?