So first before we prove 2+2=4, lets first get to know what are natural numbers. In a non-mathematical language it would be just names given to the count we give. We call then zero, one, two, and so on……… . But at the same time we could’ve just named them some TOM, DICK, HARRY, and so on…… ;).
Now lets define them Set Theoretically -
Given that all the numbers from 1 to N exist, the number N+1 is defined as -
N+1=N?{N}
In other words we can say that the operator “+1” does nothing but unions the largest set smaller than the set we are operating upon with the set containing this largest set smaller than the set we were operating upon. (This is just the definition of the equation above :P )
We define 0 as an empty set,
0={}=?
Now we recursively define the next numbers -
1=0?{0}=??{0}={0}
2=1?{1}={0}?{1}={0,1}
3=2?{2}={0,1}?{1}={0,1,2}
4=3?{3}={0,1,2}?{3}={0,1,2,3}
and so on…….
We can actually this way prove that - N={0,1,2,3,...,N?1}
Now N+2=N+(1+1)=(N+1)+1=(N+1)?{(N+1)}={0,1,2,3...,N?1}?{N}?{(N+1)}={0,1,2,3...,N+1}
A good question would be why did the operator “+2” become equal to “+(1+1)”, this is simply because we defined 2 in that way. Also why did the brackets associate? Simply because we are dealing with sets(that too without any non-trivial operators), we can do that. ;)
