Consider number system which uses only digits ${0,1,3,5,7,9}$ (odd digits with the except of 0). What will be the result in such system of summation 1+1 ? As I understand it should collapse into 1 or 3. What exactly it will be and why ?
EDIT:
As far as I understood - there is no canonical answer. Then question is - What are best / reasonable alternative definitions of + in such system. Any suggestions ?
On
What is a "number system"? The Roman number system for example doesn't have any symbols for any of the numbers $2-4,6-9$ and yet has symbols of numbers Base 10 does not, such as $10, 50, 100, 500$ and $1000$.
So we can make up any system we want. $a^b$ mean $a$ to the $b$ power and $ab$ means $a + b$. Then the first several numbers can be written as
$0, 1,11,3,31,5,51,7,71,9,91,911,93,931,95,951,97,971,99,991,9911,993,9931,995,9951,997,9971,3^3,3^31, etc.$
Thing is ... this is a meaningless question. I assume by number system you meant a radix systime where $abc = a\times k^2 + b*k+ c$ for some base $k$. But that's impossible because that assumes you have $k-1$ symbols for each digit. Having only odd digits simply wouldn't work at all.
ANd if we don't do radix we can do whatever we want. Perhaps simplest would be to let $a' = a + 1$, Then $1 + 1 = 1^1$ which is the number two.
Here is a reasonable definition. Let's call the set $X = \{0,1,3,5,7,9\}$.
$\forall a,b\in (X\setminus \{0\})$, you have $a+b=0$ and $a+0=0+a=a$.
$0+0=0$
That makes sense. Odd numbers add together to the only even number. An odd number plus the only even number gives an odd number. This matches parity, but is not an easy addition to use. You do not even wind up with a basic mathematical structure like a group $(X,+)$. While you can create an addition that forms a group, it is unlikely to be "natural".