This, and the comments that Snowolf made, are (IMO anyway) the biggest problem with creationist arguments. Creationists tend to take a reductionist view of arguments, i.e. everything's a two-horse race and it's all me or thee. It's evolution or creationism, it's christianity or atheism, it's carbon-14 or nothing, etc. The problem is, it's almost never so simple. For instance, the circular argument posted in that vid near the beginning of this thread (bible is infallible because it's the word of god, it's the word of god because it says so, and you have to accept that because it's infallible [loop] ). The problem with this argument is that I can prove the Truth of Islam, Hinduism, and any other religion that has a "holy book". You could even make a pretty good argument for religions with only verbal traditions using the same "inspired by [insert deity here]" argument. So now the xians have the additional problem of proving that THEIR circular argument is right while the other religions' circular arguments are nonsense.
With "proofs" like carbon-14, they forget that there are many different pieces of evidence that not only point to an old universe, but also corroborate each other. Various radioactive dating techniques corroborate each other, and agree with dendrochronology, ice core samples from the arctic, sediment samples from the bottom of the ocean, DNA ancestry tracing, language evolution, magnetic fossilization in lava flows, etc, probably many more than I can think of while sitting here recovering from work.
Then there's the historical problems: No record whatsoever of the Jews in Egypt, or of all the firstborn male Egyptians dying; no evidence of ten thousand-odd hebrews wandering the desert for 40 years; no evidence of the Flood; Archeological evidence that indicates that the Hebrews didn't defeat the Canaanites, but are the Canaanites, and so on.
Anyway, the point is, it's one thing to take it on faith (which I believe is the whole point according to JC) and another to try to present it as a rigorously defensible theory.
Yes, there are many different forms of clock (the DNA mutation one is particularly interesting) that all have a high degree of accuracy over a particular timescale and furthermore corroborate each other where they overlap.
On the other hand religion has "but the bible says, but the bible says"
On the point about the flood though, it's interesting that this particular legend is an undeniably widespread one List of flood myths - Wikipedia, the free encyclopedia
Does that mean that the bible is the accurate version? Preposterously unlikely (not to mention arrogant of Christians to think so). Far from being proof the other versions serve to dilute the biblical version. Much more likely that some much earlier large scale flood event passed down and retold by many, many generations (Chinese Whispers anyone) until separate myths become the canonical versions and the origin lost.
But that's old bible stuff, take the new testament... compiled (by the church - PEOPLE with a vested interest in power and control) from selected documents written years after Jesus' life, then versioned and (mis)translated.
Yet today it's taken as indisputable fact...
Even if the writings had a basis in truth, a child would be able to see the flaws in being able to assert the accuracy and authenticity with any real conviction. Let alone the layer of personal interpretation that is inevitably ladled on top by the reader...