I’m not entirely sure I understand your comment, but the fact that there are more real numbers than natural numbers can be readily shown using something called cantor’s diagonal argument. It goes something like this:
Suppose the set of real numbers and natural numbers had the same size. Then we could write down an infinite list, where each line represents some real number written in it’s decimal representation. So something like
1: 3.141592652: 1.414213563: 0.24242424
...
This list goes on forever. We will now construct a new real number r as follows: The first number after the decimal point of r shall be different from the first number after the decimal point of the first number in our list, the second shall be different from the second decimal of the second number on the list, and so on (the name diagonal argument comes from this, we consider the entries on the diagonal from top left towards the bottom right).
The key point now is that this constructed real number is different from every single number on the list: After all, we have made sure it differs from each number on the list in at least one place. Therefore, it is impossible to write down the real numbers in such a way that each real number gets its own natural number: There are simply too few natural numbers for this. In particular, there are at least two different “sizes” of infinities.
The set, that has a measurable starting point, is a finite object. If infinity can be measured, then you have created a finite space, one that can be measured, within an infinite space, infinity itself, which cannot be measured. Infinity remains untouched and undivided. The sets that represent infinity are finite objects, that represent an infinite space, a representation which they can never truly achieve.
Maybe we should talk about what “infinite” means. I’d like to propose the following idea: a sequence of things is infinite, if there is always “one more” object to consider. We could also say that for any number of finite steps, there is always another object of the series we haven’t looked at yet.
As an example, the sequence of natural numbers would satisfy this: if I start considering the sequence 1, 2, 3 and so on, if I ever stop after finite time (say 1729 steps), I can always compute +1 to find another element of the sequence I haven’t seen yet.
Also consider the following: the set of all numbers between 0 and 1 is in some sense bounded. However, I can find an infinite sequence of numbers in this set: consider
1/1, 1/2, 1/3, 1/4, …
These numbers are always between 0 and 1, and are infinitely decreasing.
Perhaps the confusion comes from you talking about infinity as in a number which is larger than any real or natural number, while I’m talking about sizes of sets of infinite size. As I had demonstrated earlier, we can show the existence of uncountable infinite vs countably infinite sets, while such distinctions don’t really come up in limit theory and calculus.
I’m not entirely sure I understand your comment, but the fact that there are more real numbers than natural numbers can be readily shown using something called cantor’s diagonal argument. It goes something like this:
Suppose the set of real numbers and natural numbers had the same size. Then we could write down an infinite list, where each line represents some real number written in it’s decimal representation. So something like
1: 3.14159265 2: 1.41421356 3: 0.24242424 ...This list goes on forever. We will now construct a new real number r as follows: The first number after the decimal point of r shall be different from the first number after the decimal point of the first number in our list, the second shall be different from the second decimal of the second number on the list, and so on (the name diagonal argument comes from this, we consider the entries on the diagonal from top left towards the bottom right).
The key point now is that this constructed real number is different from every single number on the list: After all, we have made sure it differs from each number on the list in at least one place. Therefore, it is impossible to write down the real numbers in such a way that each real number gets its own natural number: There are simply too few natural numbers for this. In particular, there are at least two different “sizes” of infinities.
The set, that has a measurable starting point, is a finite object. If infinity can be measured, then you have created a finite space, one that can be measured, within an infinite space, infinity itself, which cannot be measured. Infinity remains untouched and undivided. The sets that represent infinity are finite objects, that represent an infinite space, a representation which they can never truly achieve.
Maybe we should talk about what “infinite” means. I’d like to propose the following idea: a sequence of things is infinite, if there is always “one more” object to consider. We could also say that for any number of finite steps, there is always another object of the series we haven’t looked at yet.
As an example, the sequence of natural numbers would satisfy this: if I start considering the sequence 1, 2, 3 and so on, if I ever stop after finite time (say 1729 steps), I can always compute +1 to find another element of the sequence I haven’t seen yet.
Also consider the following: the set of all numbers between 0 and 1 is in some sense bounded. However, I can find an infinite sequence of numbers in this set: consider 1/1, 1/2, 1/3, 1/4, …
These numbers are always between 0 and 1, and are infinitely decreasing.
Perhaps the confusion comes from you talking about infinity as in a number which is larger than any real or natural number, while I’m talking about sizes of sets of infinite size. As I had demonstrated earlier, we can show the existence of uncountable infinite vs countably infinite sets, while such distinctions don’t really come up in limit theory and calculus.