Re: What definitive proof is there for the age of the earth?

There is seldom "definitive proof" of anything in science. But what you 
often do get is a large number of strands of separate pieces of evidence all 
pointing in the same direction. A picture is built up that is totally 
compelling because it would require a huge co-incidence, a cosmic 
conspiracy, to provide any explanation that is significantly different.

Much, but not all of the best evidence for the age of the earth comes from 
the decay of radioactive elements, in one way or another. Here are pointers 
to a couple of other Mad Scientist answers that deal with some of this 
evidence:

David Barlow 

Andrew Karam (including good references)

Here is some simple evidence that I will put before you: 
There are 81 stable elements, with, between them, just under 300 stable 
isotopes.
Every one of these stable isotopes has been found on earth.
Several radioactive isotopes are also found on earth. They fall into three 
categories:
(1) Very long-lived isotopes, for example Uranium-235, Uranium-238, Thorium-
232, Potassium-40.
(2) Short-lived isotopes that are produced via the decay of these long-lived 
isotopes, for example Uranium-234, Radium-226, Radon-222.
(3) Short-lived isotopes of light elements that are produced in nuclear 
reactions between high energy particles from the sun and the atmosphere, for 
example Carbon-14, Hydrogen-3, Beryllium-10.

There is no ongoing source of heavier elements, either stable or 
radioactive. It is currently believed that they were produced in a 
supernova, which also scattered them as the dust which formed part of the 
early solar system. So any of the isotopes of category (1) that are present 
on earth today are remnant fractions of what was present when the earth 
formed.

Now any radioactive isotope decays with a fixed half-life. If it has a half 
life of 1 day, then at this time tomorrow only half of it will be left, only 
one quarter the next day, and so on. After 10 days there will only be one 
thousandth of the original amount, one millionth after 20 days, one 
billionth after 30 days, and after 50 days, you can forget it!

Here are some category 1 isotopes that are abundantly present on earth:

uranium-238 has a half-life of 4.51 billion years
uranium-235 has a half-life of 0.71 billion years
thorium-232 has a half-life of 14.1 billion years
potassium-40 has a half-life of 1.28 billion years
There are about 6 or 8 others, all with half-lives greater than that of U-
235.

Here are some isotopes, potentially category 1, that are not found on earth. 
I mention mainly those that are the most stable isotope of their respective 
elements, because these would be the easiest to find if, indeed, they were 
present:

plutonium-244 has a half-life of 80 million years
uranium-236 has a half-life of 23.9 million years (and would be a product of 
any plutonium decay!)
curium-247 has a half-life of 16 million years
neptunium-237 has a half-life of 2.14 million years
technetium-99 has a half-life of 0.212 million years

There are about a dozen other radioactive isotopes known to have half-lives 
greater than 1 million years but less than 50 million years, which are not 
found naturally on Earth. All short-lived radioactive isotopes that are 
found naturally on earth clearly fall into categories (2) and (3). 
Radioactive isotopes can usually be identified by their radiation signature 
in amounts much less than you would need for chemical analysis.

The absence of plutonium from the natural rocks of the earth indicates that 
the earth has been around for at least 30 half-lives of plutonium-244, which 
is 2.4 billion years. If its age were any less, we might have expected to 
detect the 1 part in 1 billion or more that would remain of any original 
plutonium that may have been present. 

The presence of uranium-235 at a level of just under 1% of all uranium 
indicates that the earth has been around for less than 20 half-lives, 
which is 14.2 billion years, in the absence of any other evidence. But the 
amount of its final decay product, lead-207, in natural lead limits the age 
to about 6 billion years maximum. If there had been U-235 decaying for 
longer than that, there would have to be a lot more lead-207 around.

There is a lot of other evidence and more complicated analysis that makes 
geologists and geochemists very confident that the age of the earth is very 
close to 4.5 billion years.

The figures in this answer were obtained from tables on pages B-253 & B-410 
of the CRC Handbook of Chemistry & Physics, 56th Edition. Careful cross-
checking between the two tables is needed because there are a few misprints 
in each of them.