There is no requirement for claims
Provisional applications are not examined, and therefore have no formality requirements. As long as they are validly filed, it doesn't matter precisely what format they are in.
But including claims in a provisional is good
One key purpose of a provisional is to for a later non-provisional or Convention application to be entitled to an earlier date. This only occurs if the invention claimed in each claim of the non-provisional was present in the provisional.
If the claims of the non-provisional match the claims of the provisional, then this is very clear. And this is where the key benefit lies. By having claims in your provisional, you are making abundantly clear that you are entitled to an earlier date.
Conversely, if there are no claims (or claim-like language) in the provisional, there is an increased risk that the claims of the non-provisional will not be entitled to the earlier date. But it all comes down to risk, and if you are careful with your drafting, you can minimise this risk. But why take the risk if you can avoid it simply with claims?
Is it true that you cannot mention the claims in the non-provisional if you did not write them in the provisional?
No. While it may be good practice to match them up, there is no obligation to have claims in the provisional.
In fact, it is not uncommon for you to need to completely redraft the claims for the non-provisional (perhaps due to further development). This is fine. However, having some kind of claims in the provisional may still be beneficial: you're not obligated to use them in your non-provisional, but you can if they are helpful.
Claims are not easy to write
That's true: they are certainly the hardest part of any patent application. But there is a strong argument for starting with the claims before the description. The purpose of the description is to support your claims: it has no real value on its own. So by starting with the claims, you can ensure that the language in the claims is reflected in the language of the description.
If you start with the description (where you have much more literary freedom), you may end up with language which reads well but doesn't quite suit a good claim. It's not impossible to draft claims last, it's just more difficult.