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I disagree, only because infinity is not an element of the real or the complex numbers. You can make extended number systems (the hyperreals, for instance) but

I disagree, only because infinity is not an element of the real or the complex numbers. You can make extended number systems (the hyperreals, for instance) but it isn't commonly done. Infinity as a notation in the limit is not the same thing as infinity as an element of a set.

throwawaymath 8y ago
Can you clarify what you're disagreeing with? I don't think I stated anything controversial. No, ∞ is not an element of the real or complex fields, that's correct. But it is an element of the extended real and complex number systems, according to the following rules: for all x ∈ ℝ: 1) - ∞ < x < + ∞, 2) x + ∞ = + ∞, x - ∞ = - ∞, 3) (x / + ∞) = (x / - ∞) = 0, 4) x > 0 ==> x(+ ∞) = + ∞, x(- ∞) = - ∞, 5) x < 0 ==> x(+ ∞) = - ∞, x(- ∞) = + ∞ Infinity is not just an artifact of limit notation, because the algebraic rules defined here formalize the analytic theory underlying calculus. This set is distinct from the hyperreals for a number of reasons (most importantly, it's not a field). But just because it's not a field doesn't mean it's not a coherent algebra. Which really circles back to the original point: you can usefully define an algebra such that division by 0 is not undefined. In particular, note that by augmenting ℝ with +∞, -∞, we can derive (x / 0) = ∞ from the third rule. This behavior is only very slightly pathological because 1) it's (mostly) contained to the infinities, and 2) it still mostly preserves the order, completeness, addition and multiplication properties of the real numbers themselves. It's just not a field because we cannot do some propositional things that follow from Peano arithmetic (e.g. cancellation) and we can't e.g. take square roots of infinity. To make this point even more technical, we can choose to define numbers as cardinalities of sets under Zermelo-Fraenkel set theory instead of under Peano arithmetic. This allows us to neatly define arithmetic involving infinities via operations on infinite sets.
- fusiongyro 8y ago
  Upon rereading what you are saying, I realize I am not disagreeing with you. I am, however, easily triggered by this as an armchair student of math, because I think infinity forms a kind of a trap for amateurs like me. "Actual" infinity is not a member of the reals and it is not necessary to augment the reals to have a functioning real analysis. But you are right that it can be done, it can be added to the reals to form interesting algebraic structures and there are even books on nonstandard analysis that start from here (Elementary Calculus, which I have read part of but not finished).

throwawaymath 8y ago

Can you clarify what you're disagreeing with? I don't think I stated anything controversial. No, ∞ is not an element of the real or complex fields, that's correct. But it is an element of the extended real and complex number systems, according to the following rules: for all x ∈ ℝ: 1) - ∞ < x < + ∞, 2) x + ∞ = + ∞, x - ∞ = - ∞, 3) (x / + ∞) = (x / - ∞) = 0, 4) x > 0 ==> x(+ ∞) = + ∞, x(- ∞) = - ∞, 5) x < 0 ==> x(+ ∞) = - ∞, x(- ∞) = + ∞ Infinity is not just an artifact of limit notation, because the algebraic rules defined here formalize the analytic theory underlying calculus. This set is distinct from the hyperreals for a number of reasons (most importantly, it's not a field). But just because it's not a field doesn't mean it's not a coherent algebra. Which really circles back to the original point: you can usefully define an algebra such that division by 0 is not undefined. In particular, note that by augmenting ℝ with +∞, -∞, we can derive (x / 0) = ∞ from the third rule. This behavior is only very slightly pathological because 1) it's (mostly) contained to the infinities, and 2) it still mostly preserves the order, completeness, addition and multiplication properties of the real numbers themselves. It's just not a field because we cannot do some propositional things that follow from Peano arithmetic (e.g. cancellation) and we can't e.g. take square roots of infinity. To make this point even more technical, we can choose to define numbers as cardinalities of sets under Zermelo-Fraenkel set theory instead of under Peano arithmetic. This allows us to neatly define arithmetic involving infinities via operations on infinite sets.

fusiongyro 8y ago
Upon rereading what you are saying, I realize I am not disagreeing with you. I am, however, easily triggered by this as an armchair student of math, because I think infinity forms a kind of a trap for amateurs like me. "Actual" infinity is not a member of the reals and it is not necessary to augment the reals to have a functioning real analysis. But you are right that it can be done, it can be added to the reals to form interesting algebraic structures and there are even books on nonstandard analysis that start from here (Elementary Calculus, which I have read part of but not finished).