One of the signs that the Earth is old is the fact that many radioactive elements with relatively short half-lives (in the millions of years or less) no longer exist on Earth. (These are called "extinct nuclides").
The question is, why not?
Of course an OLD Earth is a perfect explanation.
**IF** a radioactive element with a half-life of, say, 15 million years (like Cesium-247) was created at the same time as the Earth - 4.5 billion years ago – then about 300 half-lives of that element would have passed. Therefore only 1/2**300 of that element would be expected to exist. That would work out to be NONE.
That's what we see.
On the other hand, if the Earth is only 6000 years old hardly any of that element should have decayed radioactively. So it should be fairly abundant.
A Young-Earth creationist web site at http://www.creation-science-prophecy.com/extinct.htm makes the argument very well. First of all the author says this about those elements:
"We know that these missing nuclides were once around because the evidence of their past existence is still in the rocks. In the past, these missing nuclides broke down into daughter nuclides until there was nothing left, so the reaction stopped. Today we still have the daughter nuclides in the rock, so we know that the parent nuclides were once present in the rocks. This is how we know them to be extinct."
So we are able to find the daughter elements, but not the radioactive elements themselves.
A small number of radioactive elements with short half-lives DO exist but those only exist because they continue to be created with well-understood processes.
The most well-known of these elements is Carbon-14 used for radiocarbon dating. But science has demonstrated that C-14 is continuously produced in the atmosphere.
Here's how that YEC web site explains it.
"Of the nuclide/isotopes that have short half-lives, only those who are being produced constantly are present in nature. Carbon 14 is a good example of a nuclide found in nature while having a short half-life. Carbon 14 is produced in the upper atmosphere. Beryllium-10 Manganese-53 and Chlorine-36 are also produced in the same way, so they are present in nature despite their having a short half-life.
All other nuclides/isotopes having short half-lives are not present in nature... So they are extinct nuclides."
So, in summary we find that we have strong evidence for the previous existence of radioactive elements with short half-lives (even in the millions of years) and NO evidence that they exist now.
None of them.
Even elements with half-lives in the millions of years.
The explanation of an OLD EARTH is perfect, even one that is easy to understand.
ON the other hand, this YEC site that I reference gives two possible explanations.
1. God is deceptive. God's basically testing the faith of the YEC. God made some things with apparent age - such as Adam and Eve themselves - but there is no explanation for why radioactive elements in the the Earth should not be present.
2. God created the Earth with pre-existing matter. In other words God found some very old rocks lying around somewhere and created the Earth from those old rocks. Specifically the web site says:
"when the first few chapters of Genesis are studied, it is clear that the word 'created' or 'made' does not exclude the possibility that preexisting matter was used... So using a literal reading of Genesis, we can understand that there could have been something here before Creation Week. The preexisting matter would be responsible for the rocks dating billions of years."
1. An Old Earth - explains everything.
2. A Young Earth and a deceptive God.
3. A Young Earth, but a preexisting solar system with lots of older rocks that God used to create the Earth. (This explanation also implies a deceptive God.)
Obviously a rational person would select alternative 1. But creationists aren't rational.