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The problem is that infinity is neither a real nor a complex number, nor an element of any algebraic field, and the proposition that "x * 0 = 0" only holds if x
by ryandv 2y ago
The problem is that infinity is neither a real nor a complex number, nor an element of any algebraic field, and the proposition that "x * 0 = 0" only holds if x is an element of some algebraic field. It is a theorem that depends on the field axioms.
The real numbers can be extended to include two special elements ∞ and -∞, but this extension does not constitute a field, and the range of expressions in which these symbols make sense is very strictly and narrowly defined (see Rudin's PMA, Definition 1.23):
(a) If x is real then
x + ∞ = +∞, x - ∞ = -∞, x / +∞ = x / -∞ = 0.
(b) If x > 0 then x * (+∞) = +∞, x * (-∞) = -∞.
(c) If x < 0 then x * (+∞) = -∞, x * (-∞) = +∞.
The extended real number system is most commonly used when dealing with limits of sequences, where you may also see such symbols appear:
3.15 Definition Let {sₙ} be a sequence of real numbers with the following property: For every real M there is an integer N such that n ≥ N implies sₙ ≥ M. We then write
sₙ ⟶ +∞.
In no other contexts do the symbols ∞ and -∞ make any sense. They only make sense according to the definitions given.
It's usually the case that when you see people discussing infinity that they are actually talking about sequences of numbers that are unbounded above (or below). The expression "sₙ ⟶ +∞" is meant to denote such a sequence, and the definitions that extend the real number line (as in Definition 1.23 above) are used to do some higher-level algebra on limits of sums and products of sequences (e.g. the limit of sₙ + tₙ as n becomes "very large" for two sequences {sₙ}, {tₙ}) to shortcut around the lower-level formalisms of epsilons and neighborhoods of limit points in some metric space, which is how the limits of sequences are rigorously defined.
In no case do the symbols ∞ and -∞ refer to actual numbers. They are used in expressions that refer to properties of certain sequences once you look far enough down the sequence, past its first, second, hundredth, umpteenth, "Nth" terms, and so on.
Thus when you see people informally and loosely use expressions such as "infinity times zero" they're not actually multiplying two numbers together, but rather talking about the behavior of the product of two sequences as you evaluate terms further down both sequences; one of which is unbounded, while the other can be brought arbitrarily close to (but not necessarily equal to) zero. You will notice that no conclusions can be drawn regarding the behavior of such a product in general, whether referencing the definitions comprising the extended real number system or the lower-level definitions in terms of epsilons and neighborhoods of limit points.
So much confusion today comes down to people confidently using words, symbols, and signs they don't understand the definitions nor meanings of. Sometimes I wonder if this is the real esoteric meaning of the ancient Tower of Babel mythos.
- yellowapple 2y agoInfinity doesn't need to be in some "algebraic field" for it to be patently true that an infinite amount of nothing is still nothing, and that adding zero to itself over and over again for an infinitely long time will never give you a result other than zero. It's only impossible to define if you overthink it, and/or maintain a needlessly narrow definition of what a "number" is. Or, if you really insist on speaking in mathematician-ese, an infinite series of zero is zero, and a zero-bounded summation is zero regardless of the summand: x · y ≡ Σ(y, i = 1) x = y times { x + x + … + x } ≡ Σ(x,i=1) y = x times { y + y + … + y } ∞ · 0 ≡ Σ(0, i = 1) ∞ = 0 times { ∞ + ∞ + … + ∞ } = 0 0 · ∞ ≡ Σ(∞, i = 1) 0 = ∞ times { 0 + 0 + … + 0 } = 0 Or, if you prefer programmer-ese: julia> i = 0 0 julia> while true println(i) global i += 0 end 0 0 0 0 0 0 0 0 0 ...and on and on until the heat death of the universe or you hit Ctrl-C. Either way, seems pretty straightforward to define if you have a clear definition of what multiplication is in the first place (and what either zero or infinite iterations of that definition will produce).
- ryandv 2y agoOk, let's assume you are correct and that ∞ · 0 = 0. Consider then the two sequences sₙ = n, tₙ = 1/n. By Definition 3.15 as provided in my last post, sₙ ⟶ +∞, and you will have to take it for granted that tₙ ⟶ 0 [0]. Intuitively we can see that the terms of {sₙ} are 1, 2, 3, ... tending to +∞; for {tₙ} we have 1, 1/2, 1/3, ... tending to zero, for progressively larger values of n. Now I ask what happens if we multiply the "infinite'th" terms of both sequences together. The first few terms of this product would be 1 · 1, 2 · 1/2, 3 · 1/3, and so on; I ask what the value x is in the limit sₙ · tₙ ⟶ x as we evaluate further and further "nth" terms of both sequences. You may have observed from the first three terms evaluated that sₙ · tₙ = n(1/n) = 1. Thus, as we continue to increase the value of n, it's always the case that sₙ · tₙ ⟶ 1 and the product tends to 1, because the product is constant and irrespective of n; we've "cancelled it out." The limit of the product is the product of the limits [1]; that is, sₙ · tₙ ⟶ +∞ · 0, as we first established that sₙ ⟶ +∞, tₙ ⟶ 0. If we thus take your supposition that +∞ · 0 = 0 for granted, we obtain sₙ · tₙ ⟶ 0, which contradicts our previous result that sₙ · tₙ ⟶ 1. Thus we can either dispense with the cited established theorems of analysis used to deduce that sₙ · tₙ ⟶ 1, or conclude that the supposition +∞ · 0 = 0 must be false. It might be the case that Σ(∞, i = 1) 0 = 0, but you can't extend this to conclude +∞ · 0 = 0 in general. Lots of intuitions from informal mathematics and even calculus start to break down once you examine the lower-level "machine code" of proof and analysis, especially once you start talking about concepts like infinity. [0] See Theorem 3.20 (a) in Rudin: https://archive.org/details/principles-of-mathematical-analysis-walter-rudin/page/56/mode/2up https://archive.org/details/principles-of-mathematical-analy... [1] See Theorem 3.3 (c) in Rudin: https://archive.org/details/principles-of-mathematical-analysis-walter-rudin/page/48/mode/2up https://archive.org/details/principles-of-mathematical-analy...