1. ## 1+1=2 always true?

Hi,

I am not sure wether I am in the right forum for this question and it may even turn out banal.
I think to remember that the often cited, unmovable truth 1+1=2 is not so absolute as it seems to be. Could someone explain me, if there are mathematical systems or preconditions under which 1+1=2 is false or not entirely true anymore and how this works in simple terms?

Thank you very much for your patience with a math idiot.
Cheers.

2. All mathematical facts are unmovable and absolute. The only way an appearance of change may come is from redefining concepts. For example, one may redefine plus so that x + y is always 1, regardless of x and y. Properly speaking, this is an abuse of notation because we did not bother to come up with a new symbol and instead reuse the same +, which has a different meaning in a different context. This is why it may seem that 1 + 1 = 2 is not true anymore.

To make more mathematical sense, we can consider the set {0, 1} instead of all natural numbers and define x + y = 0 if x = y = 0 or x = y = 1; otherwise, x + y = 1, by definition. This newly defined operation has all familiar properties of +, including 0 + x = x, x + y = y + x, (x + y) + z = x + (y + z), etc.

3. . . . . . $\displaystyle \boxed{\begin{array}{c} \\ 1 + 1 \:=\:3 \\ \\[-3mm] ^{\text{for large values of 1}} \\ \\ \end{array}}$

4. ## In my universe

Originally Posted by tass
Hi,

I am not sure wether I am in the right forum for this question and it may even turn out banal.
I think to remember that the often cited, unmovable truth 1+1=2 is not so absolute as it seems to be. Could someone explain me, if there are mathematical systems or preconditions under which 1+1=2 is false or not entirely true anymore and how this works in simple terms?

Thank you very much for your patience with a math idiot.
Cheers.
If you combine one lump of clay with another bigger lump of clay, guess what? You still wind up with one lump of clay.

5. In $\displaystyle GF(2)$ is $\displaystyle 1+1=0$...

Merry Christmas from Italy

$\displaystyle \chi$ $\displaystyle \sigma$

6. 1 polar bear + 1 seal = 1 polar bear

7. Originally Posted by DrSteve
1 polar bear + 1 seal = 1 polar bear
Good one.

8. $\displaystyle \left\lfloor 1.9999999999\right\rfloor + \left\lfloor 1.999999999\right\rfloor = 2 \ \mbox{too}$

9. 0/0 is equal to one (and just about anything else you can think of).

10. Notations: $\displaystyle S(0):=1,\;S(S(0)):=2,\quad (S:\;\textrm{succesor}\;\textrm{function})$

Definition: $\displaystyle x+S(y)=S(x+y),\;\forall x,y\in \mathbb{N}$

Axiom: $\displaystyle 0+z=0,\;\forall z\in \mathbb{N}$

Theorem $\displaystyle 1+1=2$

Proof

$\displaystyle 1+1=S(0)+S(0)=S(0+S(0))=S(S(0))=2$

Alternative proof:

Avoiding unnecesary simbolism:

$\displaystyle S+S=SS$

That is, time and/or space, or natural ability to create bijections as a consequence of time and/or space. That is all.

Fernando Revilla

11. Well, if we go to that level...

Originally Posted by FernandoRevilla
Notations: $\displaystyle S(0):=1,\;S(S(0)):=2,\quad (S:\;\textrm{succesor}\;\textrm{function})$
It should be, 1 := S(0). The left-hand side is being defined. Usually what is defined is a new symbol. If the thing that is defined is a composite expression, then it may be necessary to prove that the thing is well-defined.

Definition: $\displaystyle x+S(y)=S(x+y),\;\forall x,y\in \mathbb{N}$

Axiom: $\displaystyle 0+z=\mathbf{z},\;\forall z\in \mathbb{N}$
The definition does not specify the + operation completely. It is not clear why one clause is a definition and the other is an axiom; they both should be either one or the other (as in Peano axioms).

Theorem $\displaystyle 1+1=2$

Proof

$\displaystyle 1+1=S(0)+S(0)=S(0+S(0))=S(S(0))=2$
According to the definition, S(0)+S(0) = S(S(0) + 0). It does not follow that this is S(S(0)).

12. Originally Posted by emakarov
Well, if we go to that level...
It should be, 1 := S(0).
Right (a lapse)

The definition does not specify the + operation completely. It is not clear why one clause is a definition and the other is an axiom; they both should be either one or the other (as in Peano axioms).
Irrelevant. The functions $\displaystyle f_1^1$ (succesor) and $\displaystyle f_1^2$ (sum) are correctly defined. If I decide to name them as axiom or definition are only pedagogical strategies (good or bad).

According to the definition, S(0)+S(0) = S(S(0) + 0). It does not follow that this is S(S(0)).
Too easy: correct it.

Fernando Revilla

13. I still recall a lot of my friends telling me 1+1 = window. So 1+1 isn't always 2

14. Originally Posted by tass
Hi,

I am not sure wether I am in the right forum for this question and it may even turn out banal.
I think to remember that the often cited, unmovable truth 1+1=2 is not so absolute as it seems to be. Could someone explain me, if there are mathematical systems or preconditions under which 1+1=2 is false or not entirely true anymore and how this works in simple terms?

Thank you very much for your patience with a math idiot.
Cheers.
It took Russell and Whitehead 379 pages before they could prove this in PM (1st edition, sightly fewer in later versions).

CB

15. These were very useful and unexpectedly many contributions. Thank you very much for your time and creative explanations.

Merry Christmas.
T.