Using ∞ as an algebraic term

Question

Would it ever be possible to use ∞ as an algebraic term, putting it in equations with non infinite terms, incorporating a coefficient? Would a new type of infinity need to be defined or could aleph null work?

Answer 1

There's a lot different approaches, depending on what you want to do. In analysis for example, the notion of the extended reals is useful, for example in the context of measure theory when you want a meaningful way to say "this set $A \subset \mathbb R^n$ has infinite measure."

Formally, the extended reals $\overline{\mathbb{R}} = \mathbb{R} \cup \{+\infty,-\infty\},$ equipped with partially defined arithmetic operations $+,-,\times,\div.$ In particular we have the following additional operations, \begin{align*} a \pm \infty &= \pm\infty, \\ \infty + \infty &= \infty \\ b \times \infty &= \infty, \\ (-1) \times \infty &= - \infty, \\ a \div \infty &= 0 \\ b \div 0 &= \infty \\ \end{align*} for $a,b \in \mathbb R$ with $b>0.$ Note that some expressions are simply left undefined, such as $\infty - \infty.$

Informally speaking, the idea behind analysis is to study infinitesimal changes by thinking of them as limits of finite operations. For example the idea that $\lim_{x \rightarrow 0} \sin x / x = 1$ is that for very small values of $x,$ we have $\sin x / x$ is very close to $1.$ One way to motivate the above definitions is to observe that if we replace each occurrence of $\infty$ by $x$ and each occurrence of $0$ by $\varepsilon$ in the above, taking limits $x \rightarrow \infty$ and $\varepsilon \rightarrow 0$ we reach those expressions. So from this limit perspective, the above definitions are the "correct" rules arithmetic with $\infty$ should satisfy.

However for example the limit of $a - b$ when both $a$ and $b$ are sent to infinity is ambiguous, depending on how quickly $a$ and $b$ grows in relation to each other. This in a way is why we leave it undefined.

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As the other comments and answers show, there are a lot of other ways of giving meaning to $\infty$ and ways of doing arithmetic with them. I would like to stress that there is no single correct way to define it, and that different approaches are useful for different things. For a very different (and very interesting) approach for example, see ordinal arithmetic.

Answer 2

In order: yes, yes, yes, no, no.

$\infty$ is an algebraic term which can occur in equations, it just isn't a very interesting one. For example: $2\infty=\infty$, $\infty+1=\infty$, and $\infty/2=\infty$ are all perfectly viable, they just don't mean anything.

Outside of this, $\infty$ can be used as an algebraic term via points at infinity. If you want, you can also define $\infty$ in such a way as to treat it like a number, but this is not advisable. $\aleph_0$ is the cardinal number defined as the cardinality of the set of all natural numbers. I would not use it as an algebraic term.