Halving means the amount of bitcoins will be slashed in half gained by the miners with every new block of transactions. This is done to drop the supply of bitcoins which equals to higher prices. This plays a vital part in the high value of bitcoins among so many crypto currencies.
The supply is not reduced by a halving. The supply increases 10 minutes to a maximum of 21 million bitcoins, both before and after a halving. There have been two halvings so far and the supply is about 60% higher now than before the first halving.
This also means that after the halving, it is likely that the value of bitcoin will rise because older bitcoins inherit additional rarity due to this change. They are the best coin in this market also. Their prices are also very good.
Bitcoins do not become rarer. There are more bitcoins every 10 minutes. They become more common. Also, there is no difference between old bitcoins and new bitcoins.