1+1=2 always true?

**tass** · December 21st 2010, 06:32 AM

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.

**emakarov** · December 21st 2010, 07:04 AM

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.

**Soroban** · December 21st 2010, 07:19 AM

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

**wonderboy1953** · December 21st 2010, 08:33 AM

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.

**chisigma** · December 21st 2010, 08:41 AM

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

Merry Christmas from Italy

$\chi$ $\sigma$

**DrSteve** · December 21st 2010, 08:45 AM

1 polar bear + 1 seal = 1 polar bear

**wonderboy1953** · December 21st 2010, 08:50 AM

Originally Posted by DrSteve

1 polar bear + 1 seal = 1 polar bear

Good one.

**dwsmith** · December 21st 2010, 09:20 AM

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

**wonderboy1953** · December 21st 2010, 09:38 AM

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

**FernandoRevilla** · December 21st 2010, 11:09 AM

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

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

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

Theorem $1+1=2$

Proof

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

Alternative proof:

Avoiding unnecesary simbolism:

$S+S=SS$

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

**emakarov** · December 21st 2010, 11:34 AM

Well, if we go to that level...

Originally Posted by FernandoRevilla

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

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

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

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 $1+1=2$

Proof

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

According to the definition, S(0)+S(0) = S(S(0) + 0). It does not follow that this is S(S(0)).

**FernandoRevilla** · December 21st 2010, 12:05 PM

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 $f_1^1$ (succesor) and $f_1^2$ (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

**jgv115** · December 21st 2010, 02:08 PM

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

**CaptainBlack** · December 21st 2010, 02:20 PM

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

**tass** · December 21st 2010, 06:46 PM

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

Merry Christmas.
T.

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