To explain some of the original post by organic.
The link to some external website (michael cleverly) is something that just states that there exist distinct numbers p and q such that eventually after repeating the collatz opertation then they become the same sequence. In particular, if we start with 25 or 18, then rapidly they reduce to the same case (and eventually become 1 thank goodness, or Collatz is trivially false).
There are trivially infinitely many numbers which are of the form 3n+1 and m/2 simultaneously. This does not say anything about the decidability of the Collatz problem.
As you've said the there are simultaneous solutions to something, it cannot be that you mean for there to be an integer n with 3n+1=n/2, so the rest of the posts are not relevant.
If |K|=|N| then of course nothing can distinguish between |K| and |N| because they are equal.
If you meant can anything distinguish between N and K, then of course the answer is equally trivially yes, no element of K is a multiple of 3.