As he says, what was proved so far is that: a_0 < a_1 <= p <= t <= 2^(a_0) The first inequality is by definition (a_1 is "the smallest cardinal greater than a_0), the second one is a theorem by Hausdorff, the third one is supposedly easy, the last one is trivial. Also, if the first one is an equality, then the second one is, too.
The continuum hypothesis claims that they are all equal; that's been proved by Cohen to be independent of what you call "A" (and is generally called ZFC), by "Forcing" which is not easy, technical, and has won Cohen a Fields medal... And you're right in saying that what is proved in an axiom set is true in any larger axiom set.
However, the main point is that "we cannot have p<t, and hence p=t" is false; that is the whole point of indecidability. A statement can be "true", "false" or "not decidable". Many people apparently expected the p=t question to fall into this last group (with some enlarged axiom set "A+" permitting to prove the strict inequality).
And no, the statement "The continuum hypothesis is undecidable in ZFC" is proved within ZFC, otherwise it wouldn't have been proved at all... To give a trivial counter-example, as a corollary of Cohen's proof, the axiom choice "ZFC+CH" (ZFC + continuum hypothesis) is as valid an axiom set as ZFC is. Within it, you can certainly prove that "ZFC implies CH" since... CH is simply true.