I’m by no means a mathematician so this is a layman’s confusion after watching Youtube videos.
I understand why the (new) real number couldn’t be at any position (i.e. if it were, its [integer index] digit would be different, so it contradicts the assumption). But how does this **prove** that the set of real numbers is larger? This number indeed couldn’t be at any position, but it can **still** be mapped to an integer, because there are infinite integers – neither set can be larger because neither is finite.
Or are just “larger” or there being “more real numbers than integers” wrong terms to describe cardinality?
Pardon my ignorance.
Thank you.
In: Mathematics
The description of one being larger than the other is misleading. The simplest explanation is that there are different types of infinities that behave in different ways mathematically.
One of these types of infinities is the “listable” or “countable” infinities. The total number of natural numbers (the set of {1,2,3,…} ) is “countably” infinite.
Cantor’s diagonal argument is used to show that the size of the set of all real numbers is not countably infinite, as you can never make an infinite list of all the real numbers.
I think the claim that one is “bigger” however is misleading. At first glance it might appear that there are more integers than even numbers, because even numbers is a subset of integers. But both of these sets are countably infinite and we can make a unique one-to-one correlation between the two sets.
Instead, in math I think it’s clearer to understand that there are different types of infinities which have different properties, and unless you go into high level mathematics, you will mostly encounter the countable infinities.
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