# why is there an infinite amount of numbers between 1 and 2, but none of them include 3?

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and why does it have a beginning and an end and a middle if it’s infinite

In: 0 3 is not between 1 and 2 it’s after. Decimals go on for forever so you can have 1,1.01,1.001……2 the same is true for numbers between 2 and 3 or any other two numbers you choose Because 3 is bigger than 2 so isn’t part of “between 1 and 2”. But there are 3s in decimal places, such as 1.3, 1.03, etc.

It’s infinite because just as there’s a middle (1.5), there’s a middle of that on each side (1.25, 1.75). And those have middles. And those and those, and those, on for infinity.

You can think of it as 1 + 1/x where x is any number from 1 to infinity. So infinite. (If you have x < 1 then you’re out of your 1 to 2 range.) >why is there an infinite amount of numbers between 1 and 2

Because any number you can find in between 1 and 2, you can always find a new number in between that and either 1 or 2. So you can always find a new number.

>but none of them include 3?

Because 3 is greater than 2, therefore it isn’t between 1 and 2.

>and why does it have a beginning and an end and a middle if it’s infinite

Because of how “infinite” is being used here. It is not infinite in its size (with size meaning the numerical distance between 1 and 2, that is a *finite* distance of 1). In this context when we are talking about infinity we are talking about how many numbers are crammed into that finite space. In which case, there are an infinite amount of numbers because no matter how many numbers you have, there will always be more. It kinda breaks your brain but it’s true: infinity isn’t a number. *You can always add more to infinity.* Which means there can be larger and smaller infinite sets.

Let’s say you’re the Maitre d’ at a restaurant with an infinite number of tables. And as luck would have it, an infinite number of guests have shown up and all the tables are full. Now Jennifer Aniston shows up wanting a table. What do you do?

Simple! Scootch everyone down by renumbering the tables. Table 1 is now table 2, and table 2 is table 3, and table N is now table N+1. This works into infinity; for every N there must be an N+1 so the entire infinite number of tables can be reseated. You’re worried about running into a problem at the last table? There is no last table! There’s always one more. So every N moves to N+1… the whole infinite number of them. Jen can be seated at Table 1 which is now empty.

Ok now an infinite number of Hollywood elites show up needing tables. No prob. Table N becomes Table 2N. For every N there must be a 2N, right? So now half your tables are empty: all the odd numbered tables from 1 to infinity, an infinite number of new tables to accommodate an infinite number of Hollywood stars. Now you’re making money!

The takeaway:

1. Infinity isn’t a number
2. Infinity isn’t a maximum. You can always add to it. [removed] When we say that, the meaning is that “infinity” doesn’t necessarily mean “everything”. There can be an infinite set (all the number between 2 and 3) but this set doesn’t include ALL the possible numbers in existence (for instance the number 3).

In other words, just because there can be infinite possibilities in a set, this set doesn’t have to include every possibility. As a couple people have pointed out infinite doesn’t mean everything.

Don’t think of it like a span that never ends, Think of it as a hole or a pool with infinite depth. Even if we put nicely defined borders around it so that the start and end are framed very clearly, the depth of it is still infinite. 1.1

1.11

1.11111

….

1.1111111111111111111111111111111111111111

You see where this is going?

You can add infinite amount of digit into there and never go above 2.