It might be a stupid question, but I am having a look at independence and this question came to my mind :
Let say you have a proposition P1 independent of ZFC. If you find, in this same axiomatic system, a proposition :
P2 that imply P1
P3 that imply (not P1)
P4 equivalent to P1
Does it mean that P2 is false, that P2 is undecidable, or just nothing ?
Does it mean that P3 is false, that P3 is undecidable, or just nothing ?
Does it mean that P4 is false, that P4 is undecidable, or just nothing ?
If you can advise me good reading about independence, and undecidability it would be nice. If you want to change the tags or the title it could be nice too... I don't know what to write.
Thanks in advance