Just to check what would take for that limit to be changed. (not that I want to be changed, quite the opposite). I know the total number of bitcoin reward is halved every 4 years.

So does the verifying nodes check if the number of rewarded bitcoins match the distribution schedule?

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Every node in the Bitcoin network checks every block it receives and validates any transactions in it. As coin generation is a transaction that is included in the block, it is also checked. Same as a node would not accept a transaction that does not balance its inputs and outputs (for example, by spending more coins than it has), neither would it accept a block reward different, than the one that is scheduled to occur (50 coins currently, half of that for some number of blocks, and then other fractions of that amount). The nodes know what rewards to expect based on the block number, as it is set to half based on that number.

To change it you would have to (convince people to) change all the clients, exactly like any other protocol change.

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The software in the bitcoin client verifies that each transaction follows the algorithm that is part of the protocol.

Now even if the developers of the client were to somehow conclude that increasing the limit was a good idea and were to then push that client out for distribution, those holding bitcoins would likely not accept the new client. There would then be a fork in the blockchain as the limit increase would devalue the currency already issued and therefore it would be rational to those holding bitcoins already would protect the value of their currency by not accepting transactions from the new fork.

So to change the limit you not only need to get the developers to make the changes that fork the blockchain but also you need to get exchanges and merchants to accept transactions using the fork -- something that, in this case, is irrational.

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