I am aware that in an infinite consecutive set of all positive integers, in theory there should be infinite twin prime numbers, but let's imagine an infinite set of only all the prime numbers.
Here are some examples of consecutive twin prime numbers in a consecutive set of only all prime numbers:
$(101, 103) (107, 109)$
$(137, 139) (149, 151)$
$(179, 181) (191, 193) (197, 199)$
$(419, 421) (431, 433)$
$(809, 811) (821, 823)$
$(1019, 1021) (1031, 1033)$
...
$(4217, 4219) (4229, 4231) (4241, 4243)$
...
I know that there are no proofs yet, but theoretically:
A)Is there a limit to how many consecutive twin primes can occur in such a set?
B)Are we expecting to have these consecutive twin primes occur infinitely in such a set?