Can one non-cardinal infinity be greater than other non-cardinal infinity?

Question

As far as I know, there are two different notions to the word "infinity" in Mathematics.

First notion of infinity has to do with the cardinality of a set: if a set contains infinite number of elements, the set is said to be an "infinite set"; this notion of infinity deals with "how many".

Second notion of infinity deals with "how large" rather than "how many" - here, infinity is simply a "infinitely large number", that is, one that is greater than any numbers.

I have heard that some mathematician proved that one infinity can be larger than other infinity, but I have also heard that the proof only has to do with the first notion of infinity.

What about in terms of the second notion of infinity - say you have two infinitely large numbers (again, this is different than having a set with infinite number of elements). Can one infinitely large number be greater than other infinitely large number?

Answer 1

Your second question needs clarification (what exactly do you mean by "number"?), but here's one interpretation:

An ordered field is a set $S$ together with operations $+$ and $\times$, and a binary relation $<$, such that

• $(S, +, \times)$ is a field

• $<$ is a total order on $S$, and

• $<$ is "compatible" with the field structure (e.g. $a<b$ implies $a+c<b+c$, etc.)

An ordered field is Archimedean if for each $x>0$ there is some natural number $n$ such that $1+1+...$ ($n$-many times) is $>x$. For example, $\mathbb{R}$ is an ordered field. Perhaps surprisingly, there are lots of non-Archimedean ordered fields! In such fields, an element $x$ is often called "infinite" if $x>1+1+...$ ($n$ times) for every natural number $n$. Now infinite elements can be distinguished - e.g. if $x$ is infinite, so is $x+1$, and $x<x+1$.

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Note that this is not what is meant when people talk about limits at infinity - there, if we're even treating "$\infty$" as a real thing and not just a convenient shorthand, we're considering a number system (usually called the extended real line) $\mathbb{R}\cup\{\infty, -\infty\}$. Among other things, this system satisfies $\infty+1=\infty$. It's easy to check (and a good exercise) that this system is not a field.

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Tl;dr: it depends what you mean by "number." Note that, by contrast, cardinality of a set is precisely defined.

Answer 2

I think you're confused about your "second type of infinity". In talking about limits of or at infinity, infinity is not regarded as a number, rather having an infinite limit as, say, $x \to 0$ or $x \to +\infty$ is a way a function c\inftyan behave when $x$ is close to $0$ or very large.

In that regard, there is a sense in which one "infinity" is greater than another, although it isn't a comparison of two "infinite numbers", rather comparison of the way two functions of $x$ behave in the limit. You might look up Big O notation.