# Does “5 < ∞“ make any sense?

Or is that like saying “5 < orange” since infinity is not a number?

Sure, it makes sense. It just means “5 is finite.” At least, that is my interpretation.

ETA what is the context in which you saw this?

Sure it makes sense. Every non-infinity number is < infinity.

But does it mean that? Given that some infinities are larger than others we could put the smaller infinity on the left side and it would still be infinite.

No context. Just a question.

I assumed you were talking about real numbers. All of those (e.g., 5) are finite.

If you want, let’s say, surreal numbers (these occur in game theory), then many are infinite, but you can still compare any two of them.

It makes sense in any context where one includes infinity as a number. These contexts tend to use the extended real numbers, which include both -\infty and +\infty, the latter of which can be written as just \infty.

A common context would be anything dealing with limits, such as calculus. There you can argue that a limit that gets bigger indefinitely is larger than a limit that settles on a finite number.

IANAMathematician but, to my understanding, there are a variety of infinities like the set of countable numbers, the set of non-countable numbers, the set of all ratios, etc.

And likewise, my understanding is that you can pick and choose your definitions for things depending on what you’re trying to accomplish.

If, for example, you have set A which is [1000, 2000] and set B which is [1, 2, 3], and you were to ask is A > B? You would first need to define whether, in this particular bit of math, you meant the > operator to mean the one with the greatest sum, the greatest average, the greatest number of elements, or whatever all else. There are different ways to think of the things as comparing to one another - the only question is which one you mean.

Math is open to anyone who can rigorously define what they mean with the symbols that they use and relay that to others in a way that is consistent. If you can accomplish that, then you can write any dang thing you want. If you can’t, then no.

So what?
The existence of other things bigger or smaller than infinity have nothing to do with the relationship between 5 and infinity.

27 is bigger than 5, but has no impact on the size of 5 relative to infinity.

But the fractions between 2 and 3 is an infinite set.

If , < means the sum, then it’s smaller than 5. If it means number of elements, then it’s bigger.

Some people are mixing up two different things. You can compare (less than, greater than, equal) any two real numbers. Similarly, if the axiom of choice is true, then you can compare the cardinality of any two (possibly infinite) sets.

NB there are many different infinite cardinal numbers or infinite surreal numbers, in which case better not merely write \infty

Just to say that “x < \infty” is a common mathematical idiom meaning that x is finite. One common place this shows up is when talking about the sizes of sets: If A is a set, then its size is often written as |A|, which could be a finite number or an infinite cardinal. In that context, |A|<\infty is a common way to write that A is a finite set.

It makes perfect sense to me, and here’s why:

In math, sets of real numbers are often specified in interval notation. For example, (3, 7) (“the open interval from 3 to 7”) denotes the set of all numbers strictly between 3 and 7 (3 < x < 7). 5, for example, is in this interval, because 5 > 3 and 5 < 7.

If we want all real numbers greater than 3 (i.e. we don’t want to stop at 7, or anywhere else), we can write this as (3, \infty) or (3, +\infty). This would be all real numbers strictly between 3 and infinity, which really just means all real numbers greater than 3. 5 is in this interval because 5 > 3 and 5 < \infty, even though the latter is true by default if we’re talking about real numbers. We use the infinity symbol so that we can write unbounded intervals in a way that is analogous to how we write bounded intervals.

Fair enough. I thought that you were saying , by definition, the less than sign means finite in addition to less than.

I think this is an irrelevant test. You need an expression, not necessarily a number. You don’t have to test whether infinity is a number.

Not true. One context in which infinity is considered a number is the projective reals. In this context, there’s no +\infty or -\infty; there’s just \infty = \frac1 0. Since projective infinity is neither positive nor negative, you can’t use it as an argument to a comparison operator.

Including infinity in a number system always breaks things. It’s just a matter of which things you want to break, and sometimes you have a choice of what to break, which leads to different number systems.

To be sure, yours is a good question. On one hand, notation should typically not be abused [here, in the sense that we should not compare two entities if they are not both members of some explicitly defined ordered set]. On the other hand, if everyone understands what is meant by “x<\infty” (see Post #11), then there is not really a problem in practice.

It’s just a shorthand way to say that 5 is finite.

Yes, there are many kinds of infinity, but this is just shorthand. If by \infty you mean ordinal infinity, for example, then it certainly makes sense. Although the first infinite ordinal is usually denoted \omega. (And the second is \omega+1).

Well I AM a mathematician and I just wanted to pipe in to say that this is exactly correct, although it even goes beyond that. As Chronos its sometime useful to define infinity so that there is only one infinity and plus and minus infinity are the same. In complex analysis it makes sense to define infinity as a single value that encompasses the infinite outer ring of the complex plane. These versions of infinity is different in flavor and basically incomparable to the various infinities related to cardinality (e.g. number of reals greater than number of integers) Mathematics all comes down to definitions, and with a nebulous concept like infinity there are lots of different ways that you can define it and its properties depending on what you want to use if for, and provided it doesn’t result in a contradiction that breaks your set of axioms.

As a bonus, here’s a simple example of how you can expand the number system to include “numbers” that are greater than any finite number: Extend the concept of “number” to include polynomials, and take comparisons to mean “in the limit of large x”. So, for instance, the “number” 2x is greater than the “number” 1000000, because for all sufficiently-large values of x, 2x > 1000000 (and likewise for any other finite number in place of a million). And you can actually retain most of the structure of arithmetic, this way: You can add, subtract, or multiple any two “numbers”, and get another “number”, and every “number” has a single well-defined additive inverse. But you can’t usually divide two of these “numbers”, or find multiplicative inverses, because the quotient of two polynomials usually isn’t a polynomial.

Or you could, instead of expanding to polynomials, expand to all functions (or maybe just analytic functions). But that has its own problems, because now you can do division, but you can’t always do comparisons: You can say, for instance, that sin(x) < 5, but you can’t say that sin(x) > 0 nor sin(x) < 0.

You can extend this to what are called formal Laurent series which are power series that are allowed to have a finite number of terms with negative exponents: \sum_{i=k}^{\infty} a_i x^i, where k is allowed to be any integer, positive, negative, or zero. Now you can freely add, subtract, multiply, or divide (of course, not divide by 0.) The inverse of a series whose first non-zero term is in degree k will start in degree -k.