Since there are a finite number of bitcoins (~21 million) why not simply use integers 1 -> 21,000,000 to identify the bitcoins rather than go to all the trouble of calculating what seems like an arbitrarily difficult to calculate hash value?
The difficulty setting, which as I understand is fairly arbitrary, is what "restricts" the supply of possible hash values. Also, apparently it is possible to start new "genesis blocks" which create a complete new set of hashes to calculate.
This means there is no real limit to the number of bitcoins that can be created so why not dispense with the complexity and just number them, 1 to 21 mil.
Clarification: I know about the hashes being proof of work. My question is, that since "difficulty" is arbitrary then surely bitcoin scarcity is arbitrary too ?
Please read this: https://en.bitcoin.it/wiki/Protocol_specification
Also, at least a bit of this: http://www.bitcoin.org/bitcoin.pdf
Then you can come back and answer your own questions.
As the previous poster mentioned, a "bitcoin" is not actually an atomic unit. It's broken down to 8 decimal places (currently), so there isn't a "Bitcoin #1" that is still identifiable in the network. Pieces of it went everywhere and now make up various other Bitcoins.
Difficulty isn't arbitrary. It's a direct correlation to the amount of hashing power in the network, which then influences the rate at which blocks are found. If blocks are being found too quickly (i.e. lots of hashing power in the network) the difficulty increases. If blocks aren't being found quickly enough (i.e. not enough hashing power in the network) then the difficulty will lower. It's definitely not "arbitrary".
New Genesis Blocks can be created, but they begin a new Block Chain, and have nothing to do with the current Block Chain. That essentially makes a new variant of Bitcoins, but one that is completely different (Namecoins for example). So, you cannot create a new Genesis Block for the current Block Chain.
There is a limit to the number of "bitcoins", currently. It is: 2,100,000,000,000,000 (21,000,000.00000000)
This number comes from moving the decimal place 8 places to the right, in order to "split" the number of "atomic" bitcoins in the network. This could be done at a time when Bitcoin values are so high that no one actually spends or transfers a single coin, only fractions. By moving the decimal to the right you can create a unit that is more understood by users (e.g. "I'll take 123 BTC for that", instead of "I'll take 0.123 BTC for that"). When/If the decimal is moved to the right, you effectively do a 10:1 split of the number of "coins" in the network.