User blog:Rgetar/Number distinction

Here I want to share a thought that came to me long time ago, but I still have not seen anything like it anywhere else.

Square root of four:

41/2 is ± 2.

That is, we take a number (4), apply to it an operation (square root), and get two numbers (2 and -2).

So, from one number (4) we get two numbers (2 and -2).

But, if we get two different numbers, maybe, initially we also had two different numbers, but just did not distinct them?

Maybe, 4 is actually two different numbers (let's denote them 4+ and 4-):

22 = 4+

(-2)2 = 4-

So,

4+1/2 = 2

4-1/2 = -2

Then, 2 itself should be two different numbers 2+ and 2-, so, 4+ should be two different numbers (2+2 and 2-2), as well as 4- (now we distinct four different numbers in 4, we may denote them 4++, 4+-, 4-+, 4--), and so on.

Then, if we distinct numbers, which we did not distinct, maybe, we can do reverse operation - begin not to distinct numbers, which we distincted?

If from one number (4) we get two numbers (2 and -2), maybe, these two number are actually one, and we may consider 41/2 as single number? Or we can take any two numbers, say, 1 and 3, and begin to consider that it is a single number. We can do arithmetic operations with them:

1 + {1; 3} = {2; 4}

{-1; 1} + {1; 3} = {0; 2; 4}

(Here all objects inside {} we consider as single object, just designated in different ways).

So, why we think that two objects are two separate objects, not a single object? Because they behave differently in the same situation. For example, 1 and 3. If we add to them 1, we get different results, 2 and 4 respectively. But how do we know that 2 and 4 are two separate objects? And so on.

And vice versa, if there is a situation in which a single object behaves differently, we can regard it as a set of objects, the difference between which is manifested in this situation.

We can "collapse" and "expand" sets this way.

Maybe, we should consider any math objects not as single objects, but as sets of different objects, which difference can be manifested in some situations.

(For example, a function has infinite number of antiderivatives. Maybe, this is not a single function, but infinite number of functions, and each has only one antiderivative?)

Different ones:

1+, 1-

1++, 1+-, 1-+, 1--

1+++, 1++-, 1+-+, 1+--, 1-++, 1-+-, 1--+, 1---

...

So, we get multi-dimensional sets of numbers.

I wonder, can such process be ended, and can we construct set of numbers such as all arithmetic operations (including powers, roots and logarithms) always give one and only one result, and if such set exists, how to designate its numbers, and how large this set is.