Is it possible to generate an infinite number of collisions given an infinite amount of strings? Let us assume that we have a string $w$ such that it belongs the set $C$, where every member of $C$ is not null, and contains every string of characters that can be permuted (numbers, letters etc.). For a secure crypto-hash function such as SHA-512, is it possible to generate an infinite amount of hash collisions given an unbounded amount of time and computation for every possible string $w$?
Yes and no.

SHA-512 is by definition limited to strings of at most $2^{128}$ bits long.  So there is only a finite set of possible inputs, not an infinite set.  But you're guaranteed to have many collisions even in that finite set, because there are only $2^{512}$ holes to put $2^{2^{128}}$ pigeons into.

SHA3-512,* in contrast, is not limited to any finite input space.  So there are collisions of infinitely many strings, in principle.

But you're not likely to find any.  Unless you know something about SHA-512 or SHA3-512 that I don't, you would have to try approximately $2^{256}$ before you would be likely to find a single collision.  I assure you most decidedly that ain't nobody got time for that.



* It's also silly to use SHA3-512 for any purpose.  Its design was selected for political, not technical, reasons after certain well-known news in 2013.  There's no substantive reason to use it instead of SHAKE256-512.