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Between $2$ and $3$ there are infinite numbers and between $2$ and $4$ there are infinite numbers.

So which "infinity" is greater?

J.-E. Pin
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user304327
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  • my answer: infinity is undefined it is not comparable. Both the infinities are same – Bhaskara-III Jan 11 '16 at 16:02
  • the infinity between two integers is greater than the infinity which comprises $1, 2, 3 \dots$ if that answers your question – Rob Jan 11 '16 at 16:03
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    I wonder why there are so many down votes. This is a question by a non mathematician and it makes sense if you interpret it as a question on cardinals. – J.-E. Pin Jan 11 '16 at 16:07
  • I agree with J E Pin. The first time someone told me that two sets like this are the same size I thought they were wrong, only once someone explained to me how we are measuring "size" in this sense did I understand what they were saying. It is an important fundamental question. – Sean English Jan 11 '16 at 16:09
  • I agree with @J.-E.Pin. Maybe it was a stampede effect. In any case... Dear user, you don't have to be embarrassed of asking any question to the point of starting it by "I am not a mathematician". We are always glad that the questions that we treat sparkle the curiosity of people from other fields. – Rodrigo Jan 11 '16 at 16:11
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    It's mean to downvote this. If you are not familiar with cardinalities, you could assume that a larger interval has a "greater" infinity than a smaller. This question shows actual curiosity, it's better than many others posted here which just ask for solutions! – adjan Jan 11 '16 at 16:19
  • In the duplicate question the intervals are $(0,1)$ and $(0,2)$, but the proof is essentially the same. – Asaf Karagila Jan 11 '16 at 16:57

2 Answers2

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They are the same size infinity, specifically they both are the same size as the whole real line. To see that they are the same, notice we can map the interval $[2,3]$ onto the interval $[2,4]$ using the function $f(x)=2x-2$ this hits everything in $[2,4]$ since any $c$ in $[2,4]$ is hit by $(c+2)/2$which is in $[2,3]$ and only one thing hits $c$. (In this case we say that f is one-to-one and onto, which means f is a bijection. This is how we define"size" of infinite sets, if there exists a bijection between them, they are the same size.

Sean English
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Since you are not a mathematician, I will try to explain in simple terms why mathematicians consider that these two "infinities" are equal. Imagine you have a classroom and you want to know whether you have more students than chairs (or the opposite). Then you ask the students to seat down and the result is clear: if some students are still standing, there are more students than chairs, if some chair is empty, there are more chairs than students. And if no student is standing and all chairs are occupied, there as as many chairs as students. Mathematically speaking, you have constructed a bijection between the set of students and the set of chairs: this just means that one associates with each student the chair on which he/she is sitting.

Now, when you have infinite sets, like the intervals [2, 3] and [2, 4] you are considering, mathematicians use the same idea to compare them: they try to establish a bijection between them. In this case, as explained in SE318's answer, the function $x \to 2x - 2$ gives a bijection between [2, 3] and [2, 4], that is, a correspondence that sends every element of [2, 3] to an element of [2, 4] such that (1) every element of [2, 4] is reached (see "all seats are occupied") and two distinct elements of [2, 3] are never sent to the same element of [2, 4] (see "a chair is occupied by only one student"). In this case, mathematicians say that the sets [2, 3] and [2, 4] have the same cardinality, which is a kind of measure to compare infinities.

By the way, Georg Cantor was probably the first mathematician to give a rigorous answer to your question.

J.-E. Pin
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