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If infinity is not a number, how can it be larger than any number? A number is a position on the number line. A larger number is the position further on the number line. Infinity is not on the number line. I understand the definition of the "extended real number system", but it doesn't really answer how infinity can be put in a relation to a number, such as "larger", other than completely arbitrary without sufficient logic. Finally the definition based on the Cauchy sequences is also questionable, as such sequences are seriously challenged by people like Norman Wildberger, a Canadian prof. of math. at the University of New South Wales, Australia.

So what is the consensus on this forum, is infinity larger than 1?

BCLC
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safesphere
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    Everything about infinity is questioned by Wildberger. Induction, infitude of natural or real numbers, etc. He is a finitist. – Eff Aug 08 '17 at 23:36
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    I feel sad when Wildberger confuses yet another person. First learn about the concepts he talks about en then review his comments on them. You will see that he is not to be taken seriously. What he wants to investigate is fine, but his claims that the rest is wrong/"logically weak" show that he doesn't have a clue what mathematics is about – Jens Renders Aug 08 '17 at 23:36
  • @Eff, I thought he was an ultrafinitist, since he also scoffs at prime factorisation. – mdave16 Aug 08 '17 at 23:44
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    @mdave16 what exactly is there to scoff at about prime factorization? – eyeballfrog Aug 08 '17 at 23:53
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    I see Mr. Wildberger is not a favorite on this forum. Or elsewhere, I assume. Was this the reason he had to move to Australia? :) – safesphere Aug 09 '17 at 00:10
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    some people are less welcoming than others, the work he does is fine and logically valid. People more have a problem with people who don't work on the normal problems or use different axioms. NJW is a finitist or an ultrafinitist, which is not the norm. It would be like a physicist who doesn't believe in gravity or something, since gravity can't be explained yet. – mdave16 Aug 09 '17 at 00:16
  • you wanted to know if there was a consensus, the consensus is about which axioms to use, and everything else just follows logically from it. – mdave16 Aug 09 '17 at 00:18
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    @mdave16 People gennerally do not have a problem with Wildbergers work or any kind of work that isn't following the norm. They do, however, have a problem with the fact that mister Wildberger has a problem with other peoples work. The work that is the norm. Please investigate other kinds of mathematics, but don't start calling the rest wrong. – Jens Renders Aug 09 '17 at 00:29
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    Jens has said better what I was trying to say, just pretend i said that in the comment above – mdave16 Aug 09 '17 at 00:30
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    @mdave16: I see your point and agree in principle. However, in this case the axiom itself seems flawed in its formulation. Specifically, infinity is defined as larger than any number. Well, "larger" normally means further on the number line. So nothing outside the number line can be defined as "larger" by the definition of "larger". So in this case the axioms contradict each other and therefore cannot produce a valid theory. However, I do see that the consensus here does seem to exist :) – safesphere Aug 09 '17 at 00:35
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    @safesphere read trough all the definitions in my answer and tell me where there is a contradiction. You won't find one (or I made a mistake ;) ). Maybe they contradict you pre existing ideas about it, but they aren't part of the definitions. This is a crucial concept in mathematics. All you know is what the axioms and definitions tell you, nothing more. I hope that clears things up – Jens Renders Aug 09 '17 at 00:50
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    @safesphere: Regarding "larger=further on the number line": That is true when one is working in the real numbers. However, there are other systems of numbers (for example: cardinal numbers and ordinal numbers) that need not adhere to the properties one usually ascribes to (real) numbers. *If* one assumes (as Wildberger has suggested) that it is ludicrous to imagine a number beyond that of all atoms in the known universe, then these numbers will be the same. Otherwise? Not necessarily. – Cameron Buie Aug 09 '17 at 00:57
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    See also Is infinity a number? – Theraot Aug 09 '17 at 09:37
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    @safesphere I find "A number is a position on the number line." to be a pretty narrow definition/concept of "number". The number line is a representation of a type of numbers (with inbuilt ordering), it is not the numbers themselves. Most answers accordingly point at your definition/concept of "number" being flawed or incomplete in one or the other way. – Chieron Aug 09 '17 at 11:22
  • Aren't there an infinite number of numbers that are between 0 and 1 ... so you could also say infinity is less than 1. – aslum Aug 09 '17 at 13:03
  • Note that differentiation (dx/dy/etc) only works when you can handle division by zero. Same with infinity - math tricks to handle something hard. – Thorbjørn Ravn Andersen Aug 09 '17 at 14:03
  • This wiki article on absolute infinity may be interesting (and hopefully relevant) to you. –  Aug 09 '17 at 14:44
  • @safesphere You might find the book Where Mathematics Comes From to be helpful. It discusses the conceptual underpinnings of mathematics in general, but it spends quite some time on the various mathematical and intuitive notions of "infinity" and how they relate. – zwol Aug 09 '17 at 14:49
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    A question for the OP, in order to try to clear up their misunderstanding Is 1.1 > 1? Running through your logic in nearly the exact manner you are with infinity and the reals. 1.1 is a rational number (11/10 is in Q), which is not a natural number (N). How can an element x in Q be greater than an element y in N if x is not in N? However we define it as greater, because 1.1 and 1 are both elements in Q and we define an ordering such that 1.1 > 1. Rejecting that infinity > any such z in the reals is equivalent to rejecting the prior claim that 1.1 > 1. – ndenarodev Aug 09 '17 at 14:51
  • @ndenarodev: Thanks for your comment and a willingness to help! I really appreciate it! To respond to your comment, we don't "define" 1.1>1. It is not a definition, but follows from the fact that these two numbers have a common property, such as value. This property can be visually represented by a position on the number line. Infinity does not have a specific value, has no position on the number line, and the way it is defined in the ERN theory cannot be in a "larger" relation to real numbers. – safesphere Aug 10 '17 at 17:22

8 Answers8

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Let me be very clear here, because there is a subtlety you have missed:

Infinity is not a real number.

Infinity is a number, in other contexts. For example, in the Extended Real Numbers, it is a number. This set is of a huge importance for subjects like measure theory and integration theory. In the Ordinals or in the Cardinals (used extensively in set theory), infinity isn't just a number, it is an entire range of numbers.

And yes, in all of these systems, infinity is greater than one.

Duncan Ramage
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    Thank you for beeing very clear! That first sentence already earns you an upvote. Too many people get told that "infinity is not a number" by their highschool teacher or youtube video or wathever... When they don't even have a definition for number. Go on beeing very clear! – Jens Renders Aug 08 '17 at 23:40
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    Perhaps we ought to give some indication of what we think a real number is? That said, this is the answer OP needs, and should read twice to be sure (s)he gets it. – Chris Aug 08 '17 at 23:48
  • For those who are interested in the set theory definitions of infinity, I find VSauce's How to Count Past Infinity is quite accessible to the layman... at least as accessible as any discussion of infinity can be. – Cort Ammon Aug 09 '17 at 00:23
  • @Cort Ammon: Thank you for the reference! I will check it out :) – safesphere Aug 09 '17 at 00:28
  • Thank you for your answer! Unfortunately I fail to see any logic in it :) To say, "infinity is a number in other contexts", is the same as to say, "infinity is an apple in other contexts". It makes no sense from the relation standpoint. "Larger" is a direction along the number line, not an arbitrary direction wherever we wish it to point. We can define a pack as a set of wolves plus a rainbow, but "defining" a rainbow as "also a wolf" that is more wolf than any wolf makes no logical sense. The "bigger" relation is not arbitrary for us to define, it has been there before we started counting. – safesphere Aug 09 '17 at 05:54
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    @safesphere To say, "infinity is a number in other contexts", is the same as to say, "infinity is an apple in other contexts". You do understand that we have different sets of numbers right? It is a number within the extended real numbers ($\overline{\mathbb R}$). It is not a number within the reals ($\mathbb{R}$). Pi ($\pi$) is not a number within the rational numbers ($\mathbb{Q}$), and $-1$ is not a number within the natural numbers ($\mathbb N$). Irrational numbers ($\mathbb R\setminus\mathbb Q$), e.g. $\sqrt{2}$, were once not considered numbers. – Eff Aug 09 '17 at 07:09
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    "infinity is an apple in other contexts" I think it's more accurate to say, in the reals there is no number that is infinite. But in the extended reals there is an number that is larger than all other number. we call that number $\infty$. Other than $\infty$ all other extended numbers are the reals. A better analogy than wolfs and rainbows would be a bee colony of workers and drones but no queen, then add a queen, then add a human bee keeper. It does make sense to so a beekeeper is also a bee and more bee then any other bee, depending on what you need, bee and more bee to mean. – fleablood Aug 09 '17 at 07:40
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    The real projective line includes infinity, in which $\infty "<" 1 "<" \infty$. – Frank Vel Aug 09 '17 at 14:08
  • Is there a solid reference for calling $\infty$ as an element of the extended real numbers a 'number'? I do not have my Rudin or Apostol with me right now, but I have always been thinking that $\infty$ is an element of $\overline{\mathbb{R}}$ that is not itself a number. – Tommy R. Jensen Aug 09 '19 at 12:20
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I'm going to try to clear up your confusions, as you are not the only one with them.

What is a number?

Surprisingly, you can go through a full mathematics education and not once encounter a definition of "number". What you define is "Set" and "element of a set". These things are defined axiomatically by ZFC (there are alternatives though)

Some sets have common names, for example the natural numbers ($\mathbb{N}$), the real numbers ($\mathbb{R}$), the complex numbers ($\mathbb{C}$), hyperreal numbers ($*\mathbb{R}$), etc. Any element of such a set is commonly called a number. This is not a mathematical definition, just a common name.

However, the set of real numbers is well defined and thus so is the term "real number" (an element of that set). The same is true for the other examples I gave.

What is a relation?

Once we have sets, we put structures on them, extra information about the sets. Order relations are an example of such structure, and so is an operation like addition, or a concept of distance like a metric.

The general definition of a relation can be found here, and as you can see, the idea is the following. If I want to define a relation $R$ on a set $S$, I just have to say which elements of $S$ are in relation with each other, so for each pair $(a,b)$ I choose whether or not the are in relation with each other. If yes, we say $(a,b)\in R$, otherwise we say $(a,b)\notin R$. So in other words, a relation on $S$ is just a subset of $S \times S$

A special case of this concept is a partial order relation. Here we put extra demands on this relation. We demand 3 properties:

  • $\forall a \in $S$: (a,a) \in R$
  • $(a,b)\in R \text{ and } (b,a)\in R \implies a = b$
  • $(a,b)\in R \text{ and } (b,c)\in R \implies (a,c)\in R$

Not all relations have these properties, but some do and we call them partial order relations. A set along with a partial order relation on it is called a partially ordered set or poset. We can verify that $\mathbb{R}$ along with "$\leq$" is a poset. It even makes it a toset which we can intuitively think about as a line.

Now for infinity

There are many sets that contain an element that we call infinity, but I will look at just one example: the extended real numbers $\bar{\mathbb{R}}$. What is this thing?

Well we start with the set $\mathbb{R}$ and another set with 2 elements that aren't in $\mathbb{R}$. These elements have no special role yet, but we will call them $\infty$ and $-\infty$. Now we define the set $\bar{\mathbb{R}}$ to be: $$\bar{\mathbb{R}} = \mathbb{R}\cup \{\infty,-\infty\}$$

Now we put on this set a relation "$\leq^*$". We say that $(a,b)\in \bar{\mathbb{R}}\times \bar{\mathbb{R}}$ is in the relation "$\leq^*$" if and only if: $$(a,b\in\mathbb{R}\text{ and } a\leq b)\text{ or } a = -\infty \text{ or } b = \infty$$ We can again verify that this makes $\bar{\mathbb{R}}$ along with the relation "$\leq^*$" a poset. (again even a toset)

The answer to the question

$\infty$ is not a real number as $\infty \notin \mathbb{R}$, but we can call it a number because it is an element of the extended real numbers $\bar{\mathbb{R}}$.

We can't say it is bigger then any real number using "$\leq$", but we can say that it is bigger than any real number using "$\leq^*$".

So in the end it all boils down to definitions. You might object and say that the concept of infinity already existed before these definitions, and you are right. These definitions just form a mathematical model for it, so that we can be precise about it, so that we know we are all talking about the same thing, and so that we can answer questions about it with certainty.

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Jens Renders
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  • Now I undestand, thank you very much for your clarity. – Danilo Gregorin Afonso Aug 09 '17 at 00:46
  • A very nice answer, thank you so much for taking time to explain! I will study it to get a better understanding. However my initial impression is that a very simple logic is lost here in complex constructs. It is like me trying to convince myself that I need a new bike while my old bike rides just fine :) "Bigger" is a relation along the number line. Defining anything else as "even bigger" means that we are moving it infinitely further along this line. Therefore the definition becomes, "infinity is infinity", which is not a definition or recursive at best. – safesphere Aug 09 '17 at 05:35
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    +1 for touching on the concept of "what is a number anyway". – Asaf Karagila Aug 09 '17 at 05:52
  • @safesphere A better analogy would be getting a new bike with more gears, while the old one works fine. Sure you can get up that super steep hill with the old bike, but the new one makes it easier since you can use a higher(lower? never know which way it goes) gear. – DRF Aug 09 '17 at 07:32
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    A think a bike with a gear called infinity would be an instant transporter option... that works.... A trip of distance in 0 time is still a trip and it is something none of the gears can do but the gears can get arbitrarily faster and faster without limit or boundary. – fleablood Aug 09 '17 at 07:49
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    @safesphere , you statement: ""bigger" is a relation along the number line," isn't entirely correct. "Bigger" can be defined as needed for different contexts. If you're working within the context of real numbers, then your statement is correct. If you're working within the context of, say, ordinals, then it's not true at all. You can't take the definition from one context, and say it's the only one you're willing to entertain, at least, not if you want to be able to converse with anyone who works outside of that one context. – G Tony Jacobs Aug 09 '17 at 13:27
  • I've deleted my recent verbose comments. I hope you've had a chance to read them. Thanks again for your answer. – safesphere Sep 14 '19 at 23:31
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It's not clear to me what your objection is. The extended real number system is defined to be the real numbers plus the two symbols $+\infty$ and $-\infty$; the relation $<$ on this system is defined so that $-\infty < x$ and $x < \infty$ for real numbers $x$.

Robert Israel
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    Thank you for the answer, I appreciate it. My objection is that this definition is about as logical as "extended apples" defined as all apples plus an orange where the orange is tastier that any apple by definition. Sure, it is logical, but it makes no real life sense, because apples are not oranges.However, my question is if a consensus exists here on this issue and it looks like it does :) – safesphere Aug 08 '17 at 23:49
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    @safesphere I understand the point you're raising (I think) - it doesn't seem as if infinity can be counted as a number. In practice, we don't look at it as a number in the ordinary sense, but you can actually look at topics where it's convenient to be able to compare numbers to $\infty$ in a direct sense. In this context $\infty$ functions more as a symbol for "larger than any real". In other contexts, it may mean "more than any finite (integer) number," e.g. an "infinite set." – Chris Aug 08 '17 at 23:53
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    Mathematics is not real life. In real life we can't make arbitrary definitions, but in mathematics we can, and we do. – Robert Israel Aug 09 '17 at 00:20
  • @Chris: Thanks Chris! This actually makes more sense to me. You comment feels close to my understanding (or intuitive "definition") of infinity as a dynamic object that is a finite large number that is much larger than any other number in the context of any given problem or task. By "dynamic" here I mean that, if any such a number is insufficient for the task, then we just increase it as needed. To say "dynamic" is no different than to say "symbol". Perhaps not a strict definition, but I feel it makes sense. – safesphere Aug 09 '17 at 00:21
  • @Robert Israel: Yes, I understand and appreciate it. However, a large part of mathematics actually describes real life and this part uses infinity extensively. For example, a Dirac function in quantum mechanics is infinite and used everywhere. However, it does not correspond to any actual physical quantity being infinite in reality. Just for instance, a charge (or mass) density of an electron is conceptually infinite, because elementary particles have no size. However, the uncertainty principle jumps in to make sure that nothing is actually infinite there. – safesphere Aug 09 '17 at 00:27
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    You might enjoy The Unreasonable Effectiveness of Mathematics in the Natural Sciences – Robert Israel Aug 09 '17 at 00:46
  • @Robert Israel: Thank you Robert! I appreciate the link :) – safesphere Aug 09 '17 at 02:32
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    @safesphere: No part of mathematics describes real life. At all. Mathematics describes mathematical objects and their behavior and interactions with each other. It just happens that mathematics have roots in real life, and can be used by scientific disciplines to model real life. But it has absolutely nothing to do with real life, just like how chemistry and medicine have nothing to do with witch doctors. – Asaf Karagila Aug 09 '17 at 07:16
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    @AsafKaragila, it should be noted that that is a thoroughly modern idea—highly nontraditional. Many of the best mathematicians in history would strongly disagree with you. For MUCH longer than it has been a purely abstract pursuit built on arbitrarily chosen "axioms," mathematics has always been about the beauty and order of this, our physical universe. – Wildcard Aug 09 '17 at 07:25
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    @Wildcard: As would be non-procreational sex, premarital sex, the existence of human beings, the existence of the internet, computer-assisted proofs, programming languages, the equality symbol, electricity, equal rights to women, children rights, and so many other things. Yes, things are moving forward. Society is ever-changing, and mathematics changes with it. – Asaf Karagila Aug 09 '17 at 07:27
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    @AsafKaragila, you can have mathematics have nothing to do with real life or you can have it change along with the changing society. You can't have both. The illogic involved there is fascinating, actually. – Wildcard Aug 09 '17 at 07:29
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    @Wildcard: (1) That's a false dichotomy; (2) I don't see why you can't have that, of course you can have both; and (3) mathematics in modern time (say the last 70 years) is more of a social activity between mathematicians, than anything else, so as a social activity it is bound to change with the changes in society. – Asaf Karagila Aug 09 '17 at 07:34
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    @Asaf Karagila: "mathematics have roots in real life, and can be used by scientific disciplines to model real life". Thank you, it was exactly my point. – safesphere Aug 09 '17 at 07:35
  • @safesphere : Until you have experience with infinite things, you are not ready to say that a particular use of infinity in mathematics makes real life sense. (Hmm... That may seem harsher than intended.) This is the same problem students have with (both) relativities and quantum mechanics -- without experience with the very fast, the very heavy, and the very small, all of it fails to make real life sense. Until you have worked with (or thought very hard, with guidance from others who have this experience, to avoid foundational errors) these things, they cannot make real life sense. – Eric Towers Aug 09 '17 at 13:44
  • Almost every mathematical model of a "real life" situation is inexact, and involves some degree of idealization. Infinity is just one aspect of this. – Robert Israel Aug 10 '17 at 01:11
  • @Eric Towers: Hi Eric! Thanks for your guidance! I am afraid you misunderstood my intent. I am a physisist, not a mathematician, but I am very familiar with the use of infinities in QM, relativity, and such. The Dirac function is just one example. What I question here is not the use, but the definition, because the definition lacks a solid logic to say the least. So what you call "experience" is in fact getting used to the poor definition until you stop questioning it or simply forget it and use infinity intuitively like a very big real number. – safesphere Aug 10 '17 at 17:37
  • @safesphere : Not really. I'm a mathematician. I never treat infinity like a very big real number. In Calculus, I teach, "When we use "$\infty$" as the result of this limit, we are using it to represent the idea that the result is larger than any number that can be picked." I.e., I exactly teach the (formal) definition, without encumbering my students with quantifiers. – Eric Towers Aug 20 '17 at 01:03
  • @safesphere : Although now I am concerned about your phrase "the definition [of infinity]". All uses of infinity are bound to a context, so there are multiple definitions of infinity. All of them (abstractly) capture the idea of "bigger than anything from $X$" where $X$ is some context specific set. – Eric Towers Aug 20 '17 at 01:03
  • @Eric Towers: I was referring to the definition of extended real numbers. Not to start a debate with a 600-character limit, but just to express my opinion. You cannot logically define an object to be "bigger" than "anything" in a set with infinitely increasing values. This way your definition collapses to "infinity is infinity". In fact, you cannot define infinity based on a relation to an infinite set at all, because an infinite set cannot be defined before you define infinity. Are you seriously not seeing such an obvious logical flaw? Limits also do not define infinity for the same reason. – safesphere Aug 20 '17 at 05:58
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In the extended real numbers, one DEFINES infinity having the order relations $ -\infty < x < \infty $ $\forall x \in \mathbb{R}$

Danilo Gregorin Afonso
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    Isn't it more like this: one adds infinity to the set, en then DEFINES the order relation like that. Small diffrence tough – Jens Renders Aug 08 '17 at 23:41
  • I understand the difference between a definition and conjecture, but not every definition makes sense. For example, would it be OK for me to define my cat as smarter than my neighbor's dog? Would it really make sense? As I explained in my question, "larger" normally means further on the number line, so a definition of anything outside of the number line being "larger" than anything on the number line is meaningless in this sense. – safesphere Aug 08 '17 at 23:56
  • @JensRenders sorry, but I don't see the difference. Could you explain better? – Danilo Gregorin Afonso Aug 09 '17 at 00:37
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    @DaniloGregorin Read my answer, I hope that clears things up. As you will see there, $\infty$ is just some element, nothing more. The definition of the order is where you define that it is bigger than anything else – Jens Renders Aug 09 '17 at 00:39
  • @safesphere You know what intelligence is, so you can in some sense measure wherther your cat's is grater than the do's (for example, who takes less time to learn to go get a ball -- of course, such a meaure is very imprecise, but you got my point). If you don't like that, you can DEFINE some "intelligence" property by choosing something, comparing your cat's to the dog's and saying then that the cat's is greater. – Danilo Gregorin Afonso Aug 09 '17 at 00:39
  • @JensRenders , sorry, I didn't see at first. Clicking on the otification took me directly to my answer. Gonna read it now. Thanks. – Danilo Gregorin Afonso Aug 09 '17 at 00:42
  • We can put infinity on a number line if the number line isn't linear. Let the real number $x$ be put at the distance $\frac x{|x|+1}$ on the number line then if $x < y$ then $y$ will be further on the number line. Put $\infty$ at $1$. then $\infty$ is further on the number line than any other number. – fleablood Aug 09 '17 at 07:57
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    @safesphere Sure, you can define your cat to be smarter than your dog. By doing so, you're defining part of the concept of smartness: perhaps your definition is, more fully, that "X is smarter than Y if X is my cat and Y is my dog, and, otherwise, if X is smarter than Y by the dictionary's definition of 'smart'". However, dog-lovers will argue that your definition isn't very good, and you may have introduced an inconsistency with the dictionary definition, which would allow you to prove anything you want with this definition (so mathematicians would say it's a bad definition). – David Richerby Aug 09 '17 at 12:46
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In the context of ultrafinitism, not only doesn't infinity exist, infinite sets such as the set of natural numbers $\mathbb{N}$ don't exist either. Wildberger has emphasized his support for this vision in many of his video lectures. This view is far more radical than finitism. It's not a good idea to on the one hand adopt Wilderger's views and on the basis of that question the way conventional mathematicians work with infinity, as in Wildberger's framework there is no such thing as infinity in the first place. You have to work within a well defined framework, so you have to either accept the conventional view or work within a well defined ultrafinitistic framework. In the former case, there is no problem as is pointed out in the other answers.

Count Iblis
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    Thank you for your insight! Although I respectfully disagree that the choice is limited to the "conventional view" or ultrafinitism. The concept of infinity is successfully used in physics and math. So the question s not if infinity must be removed, but how it is defined and what exactly it is. None of the answers here explained how not a real number could be "larger" than a real number. – safesphere Aug 09 '17 at 05:01
  • @safesphere Sure, there are other options. But it is an important point that Wildberger has chosen to work in a system where there is no such thing as infinity. – David Richerby Aug 09 '17 at 12:50
  • @David Richerby: Hi David! I cannot speak for NW or judge if he is right or wrong, I merely referred to him to point out that other competent opinions exist. I am sure he's not alone in his opinion, plus other opinions also likely exist. My view is that in real life, such as physics, we cannot avoid infinity as a theoretical instrument, but never a result. So using it is fine with me. What I question here is the fact that it's definition as ERN looks like a parched workaround without a solid logic and so we need a better one, which potentially can open new directions in mathematics. – safesphere Aug 10 '17 at 18:05
  • @safesphere As pointed out here on page 12: "Often, authors forget to mention the first, very important, step in this logical procedure: replace the classical field theory one wishes to quantize by a strictly finite theory." Now, when mathematicians like Cantor ended up with the continuum they had classical physics concepts in minds, but according to QFT set up as in 't Hooft's book, that continuum actually arises out of an infinite scaling limit of a finite structure. – Count Iblis Aug 10 '17 at 21:57
  • @Count Iblis: A nice reference, thank you! A nice description that what we think is infinite is really not while infinity as an important mathematical tool to get results is not even strictly defined. My favorite part from that page, "Since, today, the answers to our questions are so well known, it is often forgotten how these answers can be derived rigorously and why they take the form they have. What is the strictly logical sequence of arguments?" It is shocking to me that very smart people here don't see the lack of logic in the ERN "definition" :) – safesphere Aug 10 '17 at 23:41
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The "teens" are a set or list of numbers: 13, 14, 15, 16, 17, 18, 19. This set is not a number itself. Furthermore the range of numbers 13-19 (including the non-whole numbers) is not a number. And yet it is entirely natural and correct to say that the teens are greater than 1.

In the same way, Infinity is not a number, but it is greater than 1.

curiousdannii
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    I think it more precise to say "each of the teens is greater than one", as the greater than relation is on elements of the set, not elements and subsets or ranges. One might also extrapolate that to "each of the infinities is greater than one" though that wording may raise eyebrows. Still +1 for an accessible analogy. – bishop Aug 09 '17 at 03:27
  • -1 because IMO it's very important to distinguish between numbers, which do have the ordering relation, and sets of numbers (not to speak of “vectors of numbers”... I'm looking at you, Matlab), which don't. – leftaroundabout Aug 09 '17 at 14:33
  • +1, because infinity is most generally seen as a set, and it can only be considered as a number when the distinction between its members is moot. While this technically holds for all "numbers", e.g. $1$, most typical "numbers" are essentially degenerate in most common contexts, whereas $\infty$ is far more prone to having distinctions even in common contexts. For example, ${\infty}-{\infty}$ is a very simplistic expression that can't be resolved without considering the specifics of the two $\infty$'s more precisely, while $1-1$ is usually sufficiently well-defined for most uses. – Nat Aug 11 '17 at 01:07
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The other responses seem to have drawn out the main elements here:

  • Infinity is not a real number: $\infty \notin \!R$
  • +/- infinity are part of the extended reals system $\pm\infty \in \bar{\!R}$

However, we haven't really discussed the motivation for the construction of the extended reals system. I don't think this was put together to enable the arithmetic handling of infinity - in fact $\bar{\!R}$ is very limited in its algebraic properties when compared to $\!R$ - but to help handling certain limits and topologies, so I'm not sure it really helps with the use of infinity under discussion.

Going back to the OP, I would hesitate to talk about the infinity (concept) as being greater than 1, preferring instead only to use $\infty$ as part of the formal language of limits and the like.

  • Surely infinity is larger than 1 or else all math and physics would crash and burn. The problem is that according to the extended real numbers "definition", infinity is not really larger than 1, but is "defined" as "larger" than 1. Can I "define" myself as "richer" than you? "Larger" is a relation based on the property of value. What is the value of infinity? Infinite, right? Great, so we just defined infinity as "infinity". Saying that infinity is "larger" than "any" of the infinitely increasing real numbers is the same as saying that infinity is "infinity". This is hardly a definition. – safesphere Aug 11 '17 at 16:53
  • I do not understand these arguments. We add a new element $\infty$ to $\mathbb{R}$ and we add a limited amount of new rules which tells us the meaning of something like $x + \infty$ and $\infty x,$ etc., and we also have to add a rule about how to extend the ordering of $\mathbb{R}$ to fit $\infty$ into it at any place of your choosing. E.g. we would be entitled to declare that $\infty$ is larger than $0$, and smaller than any positive real number. There is nothing magical about the designation of $\infty$ by the name 'infinity' to say that it has to be larger than all real numbers. – Tommy R. Jensen Aug 09 '19 at 12:37
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Is infinity larger than 1?

As other answers have said, yes, because infinity isn't a real number. Jens Renders's answer describes this well.

However:

Infinity isn't a number, just a description of a number

Saying that a number is infinite describes it, not defines it, much like saying that it's positive, negative, prime, composite, irrational, etc.. While all numeric assertions are technically just descriptions, descriptions like "$1$" are precise whereas descriptions like "prime" or "infinite" refer to more than just a single value.

In general, $\infty$ should be read as "a number that is infinite", which can concisely stated as "infinity" so long as we're careful to remember that that's shorthand for a class of numbers. It's probably more precise to say that $a{\in}{\infty}$ rather than $a={\infty}$, where the set $\infty$ is usually implicitly defined as the set of all numbers arbitrarily larger than all all non-infinite numbers described in the same context.

When less ambiguity is desired, systems like the hyperreal system provide a framework for more rigorous descriptions. However, in general you can invent your own systems for dealing with infinite values as long as they're consistent. The only real requirement is that all infinite values are, by definition, larger in magnitude than all real values, which are in turn larger in magnitude than all infinitesimal values.

When we're not being that explicit, statements about $\infty$ are limited by the amount of information that they contain. For example, ${\infty}-{\infty}$ is undefined because it basically just says "the difference between two infinitely large values"; and since the difference between two infinitely large values could literally be anything, depending on what the infinitely large values are, then their difference is undefined.

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Nat
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  • Thank you for your answer. Could you please clarify the following? When you say, "infinity [is a] shorthand for a class of numbers", are they real finite numbers or infinite numbers? If infinite, then this definition becomes recursive. Otherwise they are not infinite. And if they are different from real numbers, such as belong to a different number line, then how can you say they are "larger"? I also see your point on description, but how does it help with the definition? So this "number" is "infinite", fine. Now, what does this mean exactly? :) – safesphere Aug 09 '17 at 03:34
  • @safesphere "Infinity" is a shorthand description for the set of all values that're larger-in-magnitude than the set of all finite in value. As for recursion, would you consider "$x=1$" to recursively rely on itself? Infinity's property of being greater than all reals is similar; it's an assertion rather than a conclusion drawn from itself. – Nat Aug 09 '17 at 03:37
  • @safesphere If it helps, you seem to be questioning infinity and trying to understand it from the perspective of a constructionist. You'd probably like the hyperreal system; its virtue is that it clearly and rigorously defines what "infinite" means in a manner that's consistent with other mathematics. – Nat Aug 09 '17 at 03:47
  • Thank you very much for the references. I appreciate it. What I am questioning is not infinity, but the way it is defined. To me it appears that the concept is there and this is the reason it works so well in math and physics, despite the fact that its definition may not be logically consistent. For example, say, I have a physics equation with the Dirac function and it gives me some results. Well, would it then matter how we define infinity on this forum? No. So it is there, it works, but its definition is logically weak, although without affecting the results. – safesphere Aug 09 '17 at 04:18
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    @safesphere Definitely, and you're right to question it there. Classical math had a lot of handwaving, leading people to get kooky about stuff like infinity; for example, Cantor went kinda screwy near the end there, thinking that infinity was God and God was infinity. Rigorously defining stuff seems to be the solution. – Nat Aug 09 '17 at 04:22
  • @Nat, or, of course, there's always the possibility that he was right. ;) But never mind; carry on with the "rigor." – Wildcard Aug 09 '17 at 07:28
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    @Wildcard: Of course he was right :) However, there's a principal difference between the mathematical infinity (or simply infinity) that we are talking about here and absolute infinity (or simply absolute) that he was talking about. Absolut is a result of division by zero. It is not a real number, has no value relation to real numbers, and therefore cannot be larger or smaller or equal to one. In contrast, infinity is indeed larger than one, but is not rigorously defined, because the ERN "definition" confuses two different objects and is logically flawed. We simply need a better definition. – safesphere Aug 11 '17 at 00:03