# why correct for multiple comparisons in a correlation table?

I often read that in a correlation matrix the p-values for all correlations in a NxN matrix should be adjusted for multiple comparisons (for example: Multiple comparisons for correlation matrix?). Most of my co-authors also adamantly believe this but nobody is able to give me a source for this. My question is: do I have a fundamental misunderstanding of why we need to correct for multiple corrections to reduce type-1 error?

My understanding is that if we want to test a hypothesis with an alpha of 0.05, that within this hypothesis the nested set should have a confidence of 95%. So for example:

H0 = $\mu_{1} + \mu_{2} + \mu_{3} = 0$
H1 = $\mu_{1} + \mu_{2} + \mu_{3} \neq 0$

The full set should have a reliability of 95%. This is why when examining pairs causes type 1 error inflation, because the set of pairs:

$\mu_{1} = \mu_{2}$
$\mu_{1} = \mu_{3}$
$\mu_{2} = \mu_{3}$

All are subsets of the original nulhypothesis and the entire set should have reliability of 95%. To maintain this, a correction on the alpha (or p-values) for every pairwise comparison is needed to maintain 95% confidence.

Is this understanding correct? Because its implication would be that multiple comparison should occur within a nested set of comparison, but should not occur between non-nested comparisons? So if I have a new nulhypothesis about a different Y, than that nulhypothesis does not be corrected for the existance of the nulhypothesis test on the other dependent variable.

So, how are all bivariate correlations in a NxN matrix subject to multiple comparisons? Is this because the same variables are repeatly correlated with other variables and so form a nested set of their own? I’m really struggling with this, so if someone has a good (mathematical) paper for me to help me understand the fundamentals that would be awesome.