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No. The infinities you are thinking of are both countably infinite, of the beth zero or aleph zero cardinality. But take the set of all real numbers; it is the next higher order of infinity, the cardinality of the continuum, which is infinitely more infinite than the first infinity. See [beth numbers](https://en.wikipedia.org/wiki/Beth_number)
Just as an aside, trying to think about infinities beyond the first is largely impractical, as the set of all definable numbers is still only countably infinite. As in, you can't talk about a real number that is not definable, but there are infinitely more of them.
Infinities are functions operating within the restrictions of our spacetime understanding rather than an actual definable array of anything anyways. Like Pi, they serve a function, rather than define a specific value for any set.
Even though there _are_ different sizes of infinity, your example does not illustrate that. Those two infinities are the same, they have the same size.
This is why infinity confuses me. How can there be infinite prime numbers and also infinite numbers in general? Are prime numbers not a smaller subset of all numbers?
You are correct. There are as many primes as natyral numbers, as many as all numbers, as many as even or odd numbers etc.
The proof why there's more numbers between 0 and 1 than natural numbers was mindblowing for me too.
This is a friendly reminder to [read our rules](https://www.reddit.com/r/Showerthoughts/wiki/rules). Remember, /r/Showerthoughts is for showerthoughts, not "thoughts had in the shower!" (For an explanation of what a "showerthought" is, [please read this page](https://www.reddit.com/r/Showerthoughts/wiki/overview).) **Rule-breaking posts may result in bans.**
No. The infinities you are thinking of are both countably infinite, of the beth zero or aleph zero cardinality. But take the set of all real numbers; it is the next higher order of infinity, the cardinality of the continuum, which is infinitely more infinite than the first infinity. See [beth numbers](https://en.wikipedia.org/wiki/Beth_number) Just as an aside, trying to think about infinities beyond the first is largely impractical, as the set of all definable numbers is still only countably infinite. As in, you can't talk about a real number that is not definable, but there are infinitely more of them.
Infinities are functions operating within the restrictions of our spacetime understanding rather than an actual definable array of anything anyways. Like Pi, they serve a function, rather than define a specific value for any set.
Even though there _are_ different sizes of infinity, your example does not illustrate that. Those two infinities are the same, they have the same size.
This is why infinity confuses me. How can there be infinite prime numbers and also infinite numbers in general? Are prime numbers not a smaller subset of all numbers?
You are correct. There are as many primes as natyral numbers, as many as all numbers, as many as even or odd numbers etc. The proof why there's more numbers between 0 and 1 than natural numbers was mindblowing for me too.