The meaning of the symbol $\infty$ in Spivak's calculus book

Question

Spivak in "Calculus" writes

... symbols of $\infty$ and $- \infty$ are purely suggestive: there is no number $``\infty"$ which satisfies $\infty \geq a$ for all numbers $a$.

What is the meaning of being purely suggestive?

Source:

Answer 1

$\infty$ and $-\infty$ are not real numbers. So when ever we use them in real analysis we need to define their use carefully, as he has done in the extract you quote.

For instance the interval $(a,b)$ is defined as $\{ x \in \mathbb R \ : \ a < x < b \}$ whenever $a, b$ are both real numbers.

This definition is in trouble if the symbols $a$ or $b$ are $-\infty$ or $\infty$. In those cases, we need a different definition, as Spivak gives. This is to say, up to that point of Calculus, we know what $$x < y$$ means for real numbers $x$ and $y$. We have not yet defined what for instance $-\infty < x < b$ would mean.

This might seem pedantic, especially in the example of interval notation, but mathematical results can often turn on carefully delineated definitions and expressions. So it's important to get even these basic definitions right in one's early training in rigorous mathematics, which is what Spivak's Calculus is all about.

In addition to the use of $\pm\infty$ in interval notation, Spivak will later give careful definitions of expressions such as

$$\lim_{x\to\infty} f(x),$$ $$\lim_{n\to\infty} a_n,$$

$$ \lim_{x\to a} f(x) = \infty \quad\text{ where $a$ is a real number,}$$

and

$$\int_a^\infty f(x) \ dx \quad\text{ where $a$ is a real number}.$$

Those definitions will not treat $\infty$ as a real number. Instead, the definitions are expressed using real numbers in such a way that the suggestion of $\infty$ is formalized.

Answer 2

You may have heard the adage, "infinity is not a number, it's a concept" (or at least you have now). This is in fact true, since $\infty$ is not a real number (if it were, terrible things would happen).

Infinity $-$ $\infty$ $-$ is the object such that no number is greater than it: $$ \forall x\in \mathbb R : |\infty| > |x|$$

We use this object to talk about the "extreme" ends of the real numbers (e.g., when talking about taking limits), because it is easier to say "as $x \to \infty$, foobar is true" than it is to say "for any [possibly very large] number $M \in \mathbb R$ there exists a number $N>M$ such that foobar is still true".

Answer 3

In short, the symbol has separate but analogous meanings in different contexts. While we often use it in place of a real number in notation, careful usage will not treat it as a real number---consider, for example, the notation $\lim_{x \to \infty} f(x)$.

To expand orthogonally to Simon S' excellent answer (and perhaps besides the cursory paragraph above, this is really more of a comment), let me point out that one can define the so-called extended real numbers, which is the set $\{-\infty\} \cup \Bbb R \cup \{\infty\}$ in particular endowed with an order $\leq$, and in this setting, we have $-\infty < a < \infty$ for all $a \in \Bbb R$. It also comes endowed partially defined operations $+, -, \cdot, /$ that extend those defined on $\Bbb R$ and are otherwise still reasonably well-behaved. This construction also dovetails nicely with some existing conventions. For example, for real $a, b$, $a < b$, we define the open interval $(a, b) := \{x : a < x < b\}$, and extending this formally to the extended reals gives us the usual definition, for example, of $(a, \infty)$ for $a$ real or $-\infty$. Note that partially defined here means that some operations are not defined for all combinations of inputs. For example, $\infty + (-\infty)$ is not defined.

Even though the extended real numbers, by design, have a good deal in common with the real numbers, the usual definitions of limit, integral, etc. are by default given in terms of real numbers, and so cannot be applied to the extended reals without appropriate extension.

Answer 4

You could put it this way: If we write $$\begin{align}\lim_{x\to a}f_1(x)&=b\\ \lim_{x\to c}f_2(x)&=\infty\\ \lim_{x\to \infty}f_3(x)&=d\\ \end{align}$$ then there is a qualitative difference between these three lines. That is, the second is not just the same as the first with $c$ in place of $a$ and $f_2$ in place of $f_1$ and $\infty$ in place of $b$. Indeed, by definition the first means $$\let\epsilon\varepsilon\forall \epsilon>0\colon\exists\delta>0\colon\forall x\colon |x-a|<\delta\to |f_1(x)-b|<\epsilon $$ and the second means $$\forall M\in\mathbb R\colon \exists\delta>0\colon |x-c|<\delta\to f_2(x)>M$$ which looks significantly different (whatever $\infty$ could mean, it gets eliminated in this definition and there certainly is no "$\epsilon$-neighbourhood of $\infty$"). Similar for the third line. Thus, in principle, we ought to have introduced three distinct notations, e.g., $$\begin{align}\lim_{x\to a}f_1(x)&=b\\ \operatorname{divergesupbeyondall}\limits_{x\to c}f_2(x)&\\ \operatorname{limforawfullylarge}\limits_{x}f_3(x)&=d\\ \end{align}$$ (Note that one of the notations isn't even an equation any more). For simplicity, we abuse the notation for the finite case and introduce the symbol $\infty$ in a place of that notation to suggestively express what happens.

The same happens when we switch from the sum $$\sum_{n=1}^N a_n$$ to the series $$\sum_{n=1}^\infty a_n$$ which is an object of totally different kind (e.g., the commutative law may not hold any more; or just closure: the sum of rationals is always rational, but a series with rational summands may converge to an irrational number or not at all).

Answer 5

In the sense that Spivak discusses (i.e. the way it is generally used in calculus), "$\infty$" is really a sort of abbreviation to allow you to write certain limits and sets in a way consistent with other sets. Here are a couple of examples:

1. The interval $(a,\infty)$ is the set of real numbers that are larger than $a$. This is by analogy with the interval $(a,b)$, but this interval has an endpoint on the right, above which there are numbers not in the set. $\infty$ acts as a placeholder in the notation here: in set notation, we could write $$ (a,b) = \{ x \in \mathbb{R}: a < x \} \cap \{ x \in \mathbb{R} : x < b \} \\ (a,\infty) = \{ x \in \mathbb{R}: a < x \}, $$ so the $\infty$ in the second can be interpreted as "the thing which makes $\{ x \in \mathbb{R} : x < \infty \} = \mathbb{R}$". Since there is no real number with this property, we can just define "$\infty$" as having this property: for real analysis, this is basically all we need. The rest can be obtained from careful definition. (notice, for example, that the ordering on $\mathbb{R}$ ensures that $-\infty<x$ for every real number $x$.
2. In limits, "$n \to \infty$" means that for any given number $K$, there are eventually $n$ larger than $K$. (At its most basic, this is the archimedean property of the natural numbers.)

Answer 6

Start with the set $\mathbb{R}$. And adjoin two new elements to it, $\mathbb{R}\cup \{ +\infty,-\infty\}$. Denote this resulting set as $\overline{ \mathbb{R}}$. On this new set we do not define any algebra, i.e. we do not define what it means to add/multiply across infinities (because it is not possible to preserve all the rules we would like). However, we do define an ordering i.e. a reflexive, anti-symmetric, transitive relation; by declaring $\infty$ to be the final object and $-\infty$ to be the initial object.

With this extension we can now define the sup/inf of any set. Furthermore, the topology on $\overline{\mathbb{R}}$ induced by ordering is now compact.

If you do not know what a "topology" is yet, then ignore what I said. But the rest of what I said is why it is useful to make such an extension.

Answer 7

Your blurb is pointing out the difference between $\infty$ and practically every other symbol used in math to denote a number, such as in this case with the variable $a$. However large you define the value of $a$ to be, it is understood to be a finite, unique number. It follows that for any $a$, there is a finite, unique number $a+1$. This is true in the general case.

Infinity by definition is not a finite, unique value. That means that $\infty$ is not a number. This also means that other symbols and concepts that apply to finite numbers, no matter how "arbitrarily large" (a term used specifically to avoid using "infinity" when talking about extending a trend beyond what is practical to calculate empirically), do not apply to $\infty$. For instance, there is no such thing as $\infty + 1$, and $\infty$ can never "equal" anything.

Infinity therefore represents a journey, not a destination.

The water gets muddied when the concept of "trans-infinite numbers" enters the picture. This is a concept in elementary set theory. What is the cardinality of the set of natural numbers? Well, there are an infinite number of them, so the cardinality of the set of natural numbers is itself infinite. What is the cardinality of the set of real numbers? Well there are an infinite number of those as well so the cardinality of the set of real numbers is also infinite.

Here's the kicker: are there more real numbers than there are natural numbers? Classical thought says no; infinity is infinity and there's no such concept as something being "more infinite" than another.

However, intuition leads to an alternate line of reasoning. For instance, there is only one natural number "1", but there are an infinite number of real numbers that can be constructed by starting with the integer 1 and concatenating a decimal point plus an arbitrary number of other integers (for the simplest such construction, append the irrational Champernowne's Constant to any integer). There are, therefore, as many numbers between 1 and 2 as there are unique combinations of integers. That means there are more real numbers than integers even though there are an "infinite" number of both.