Bitcoin 21 million limit and Zeno's paradoxes

[remotemass]

I think that if the bitcoin number of decimal places can indeed be extended, theoretically ad infinitum, than the number of bitcoins is not limited to 21 million, actually.
Can anyone prove otherwise, that is, that such sum up series of mining rewards is convergent to a such a limit?

If not, how many years would it take, theoretically, to reach 21 000 001 bitcoins, given the number of decimals places could always be extended?

[EmperorBob]

 

1. Assume we already have infinite precision right now.
2. A halving period is 210,000 blocks.
3. The first halving period produced 50 bitcoins per block, (50*210,000 total).
4. Each period is half the previous one.
5. http://www.wolframalpha.com/input/?i=sum+210000*50%2F2^n%2C+n%3D0+to+infinity

Or are you worried about the impact of rounding errors (which shouldn't be a problem given that we're not using floating point calculations)?

[ECore]

Are you talking about * "poof!" * more bitcoins are added?

How would that happen unless everybody changes their bitcoin software?  You can do whatever you want to do with your software, but I doubt you will be able to change everybody else's .  Good luck with that.

[remotemass]

Indeed, the series does converge. Period. 

[coastermonger]

The math works out differently

Adding more coins beyond 21 million would devalue all of the existing coins. 

Adding more decimal places would simply allow a higher degree of precision when exchanging and setting prices. 

[bg002h]

The OP's point is subtle and very intuitive (if mathematically unsophisticated...but hey, not many people are, right?). He is asking will the series converge (sum of all created bitcoins) even if "we keep shifting the decimal point."

That's a very valid question and I suspect many won't understand it. Anyhow, the issue of shifting the decimal place is mathematically identical to adding more decimal places...so the question is "does the series converge even if there is infinite precision?"  The answer is yes and it will converge such that you will have the same fraction of the final overall total of bitcoins created that you have today, regardless if how the decimal gets moved about.

[niko]

In other words, even with infinite precision, you might be adding fractions of a coin, and halving that fraction periodically, but you will never have a total more than some certain maximum, which is easy to calculate.

[DeathAndTaxes]

In other words, even with infinite precision, you might be adding fractions of a coin, and halving that fraction periodically, but you will never have a total more than some certain maximum, which is easy to calculate.

Yeah its 21M BTC.

Here is a simpler series that may illustrate the dynamic

1/2 + 1/4 + 1/8 + 1/16 .......

The sum of the series with any finite number of steps will approach but never reach 1.  The more steps the closer you get to 1.

Bitcoin follows the same series.

1/2 of all Bitcoins will be mined in first 210K blocks
1/4 of all Bitcoins will be mined in the second 210K blocks
1/8 of all Bitcoins will be mined in the third 210K blocks.
etc

Due to the limit of the number of digits in the protocol there is a finite number of steps in the series (30 to be exactly).  This could be extended.  With a larger finite precision we could have 60, 90, 5000 steps but the number in this simplified series will never exceed 1.   Same thing with Bitcoin the, the number of bitcoins mined will always be <21M.  The more finite precision we have the closer we will get to but never reach 21M.  With infinite number of blocks and infinite precision the series converges on 21M.

[BombaUcigasa]

I think that if the bitcoin number of decimal places can indeed be extended, theoretically ad infinitum, than the number of bitcoins is not limited to 21 million, actually.
Can anyone prove otherwise, that is, that such sum up series of mining rewards is convergent to a such a limit?

If not, how many years would it take, theoretically, to reach 21 000 001 bitcoins, given the number of decimals places could always be extended?

It is true, there are not going to be 21 million bitcoins.

At some point, the division is not able to provide a rounded value above 0 so the halving stops in 0. At that point there will be 2,099,999,997,690,000 satoshis, and then no more will be added. The paradox is solved by binary computing and floor rounding.

In conclusion it is impossible to reach 21.0M, and in no case 21 000 001 bitcoins.

[tclo]

There are some smart people on these forums.

[DeathAndTaxes]

At some point, the division is not able to provide a rounded value above 0 so the halving stops in 0. At that point there will be 2,099,999,997,690,000 satoshis, and then no more will be added. The paradox is solved by binary computing and floor rounding.

I would just add the above value is with current precision.  OP was asking about increasing precision.  If precision is increased then we would get closer to 2.1 quadrillion satoshis (but never reach it).

[BombaUcigasa]

Also I would like to point out that sometime in the past someone credited themselves or their pool a block with a smaller reward than the maximum possible. Not only we have lost and discarded coins, we have less than theoretically possible.

[dancupid]

Cut a cake in half and then cut it into quarters and then eights etc - how many cakes do you have? It's still just one cake.
Zeno's parodox is not a paradox really - one cake is one cake irrespective of how much time you spend cutting it up.
However you are right - it will take an infinite amount of time to produce an infinite number of blocks.

[niko]

Cut a cake in half and then cut it into quarters and then eights etc - how many cakes do you have? It's still just one cake.
Zeno's parodox is not a paradox really - one cake is one cake irrespective of how much time you spend cutting it up.
However you are right - it will take an infinite amount of time to produce an infinite number of blocks.

...and that is all that really matters
BTC is here to stay.