Well-written, Samrat. Good job.
There is something quite fascinating about math that is linked to this paradox.
It is discovered when putting the binary system next to the decimal system. We know that we can use either number system to declare anything numerical in the universe.
In the decimal system, one can easily be tricked to think that 1 is used to show either a unit, unity, or simply the whole of all parts.
The binary system can be used to expose that lie.
In the binary system, there are many 0s and 1s. Yet there is no 1 like the decimal 1 in the binary system. Unity is the easiest way to show this.
If we want to express Unity in the binary system, then we can create that with ease. For instance, we can say that 1110101001 represents Unity. As long as we all agree, then this combination of zeros and ones represents Unity.
A single 1 in the binary system does not mean anything all that interesting; it is a mathematical tool, the pen with which we write, the screw in the wall, the part that declares 'on' instead of 'off'. Binary 1 is not the same as Unity because Unity is far more complex than that much repeated binary 1 by itself. Even ‘unit’ is not described all that clearly with binary 1.
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With this information, we can start to investigate the decimal system. Low and behold, the decimal 1 can indeed be used for unit, but to use the decimal 1 for Unity or the Whole of the Parts, there is actually a step in between that must be answered:
Can we use decimal 1 to represent Unity?
It turns out that we have to agree first that decimal 1 can be used for Unity. As long as we agree, we are good to move forward.
Yet this should show right away that we are dealing with an assumption, and a mathematical assumption has to be declared to turn it into a useful tool. We can do it, but it ain’t necessarily so.
There is evidence that decimal 1 can indeed be declared a unit, simply because 2 means two of them, 3 means three of them. So, we do have an ‘it’ with decimal 1 for sure.
It turns out that both 0 and 1 in the decimal system have similar characteristics as the binary 0 and 1. Not identical, but showing similar issues.
To make 1 represent Unity, we have to throw away the 0 as functional. We have to get rid of the function that 0 can portray.
Actually, let me write that down in a different manner.
Of all the numbers, only 1 can represent something singular.
Many people think that 0 is singular, too, but that is incorrect.
With 010, we see two 0s that are both doing something different. The first 0 can be removed, and the total we see is then not changed. But remove the second 0, and the total changed dramatically. One 0 is just there, the other has a function.
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To walk this down to the barber, we see that the setup is where 1 and 1 are being confused as being just 1.
The 1 of the barber is straight out, and we should have a rule for that person that applies to that person.
The 1 of the community does involve the barber, but where the others in the community provide the barber a clear action on what to do (shave or not shave), we get into trouble when we mix the barber in with the community. The 1 for the community will include the barber, but the 1 of the barber made the barber special all by himself. He’s in his own category, his own system.
This is therefore a case of eating one's cake twice. One can eat a cake left to right, and one can eat it from right to left. But one cannot eat the cake two times.
The paradox does therefore not exist because what is seen as a single system is actually a combination of two systems, confused as if there is just one system.
The two words 'those' and 'himself' are words clearly pointing to two groups, not one group. Those is a finger pointing outwardly. Himself is a finger pointing inwardly. Turn the barber into a that person only and not a himself and the paradox disappears.
Like the binary system shows us: there are always two distinct positions. One cannot write the binary system with 1s only. Anytime we have a 1 by itself, it must be declared what it is, and it cannot have two different mathematical functions. If we have two different mathematical functions, then we have 1 and 1.
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Naturally, there is an even faster solution to the paradox:
The barber is a woman.
