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Prove: 1 + 1 = 2

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Reply 20

cobra

JamesF

Click

Reply 21

JamesF

It just follows from the definitions, what a let down eh...

Reply 22

I was pretty sure some professor somewhere proved that 1+1 = 2 using like hundreds of sheets of paper. And only a select few people actually understood it.

It was a few years ago.

Reply 23

Zapsta

Widowmaker

I did this in yr 10

let a = b

a² = ab Multiply both sides by a

a² + a² - 2ab = ab + a² - 2ab Add (a² - 2ab) to both sides

2(a² - ab) = a² - ab Factor the left, and collect like terms on the right

2 = 1 Divide both sides by (a² - ab)

You can't divide by zero :rolleyes:

Reply 24

fishpaste

It practically is an axiom though.

Reply 25

Evolin

Zapsta

You can't divide by zero :rolleyes:

xactly wot i woz gonna say!
therfore 1 does not equal 2

and yeh...i think 1+1=2 by definition!
i dunno how could prove it or why!

Reply 26

JamesF

2776

I was pretty sure some professor somewhere proved that 1+1 = 2 using like hundreds of sheets of paper. And only a select few people actually understood it.

It was a few years ago.

Actually I'm pretty sure i read that too, which is why the above proof has to use Peano's axioms, not the field axioms.

Reply 27

Willa

is the principia an axiom!?

when i mean axiom i mean basic properties which are assumed in order to analyse and describe other bodies. 2 being 1 greater than 1 is itself pretty much an axiom, and so you cant prove it, unless you decide to invent even simpler axioms and prove 1+1=2 on those basis.

then again, i might be wrong

Reply 28

Squishy

Willa

when i mean axiom i mean basic properties which are assumed in order to analyse and describe other bodies.

That's what Peano's axioms are for.

There is a natural number 0.
Every natural number a has a successor, denoted by S(a).
There is no natural number whose successor is 0.
Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b).
If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This axiom ensures that the proof technique of mathematical induction is valid.)

Reply 29

dan2004

Widowmaker

Your first line clearly ststes that a=b. therefore, as previously mentioned, when you try to divide by a²-ab this is the same as a²-a² which is zero. This is undefined. :smile:

Reply 30

wanderer

I have a problem with the proof linked to above. When he defines addition, doesn't he use the concept of addition at the end: (a + c)'? How can a definition hold if it uses the concept it is intended to define?

Reply 31

fishpaste

wanderer

I think he defines it recursively, and so it holds up.

Reply 32

wanderer

How about:

Define 1 logically as 'a variable representing any single object'

Define the + sign as 'the logical opreator AND'

Define any number greater than one as 'a variable representing a series of single objects'. 2 can then be defined as 1 AND 1, 3 as 1 AND 1 AND 1, etc.

Hence:

1 + 1 = 2

is equivalent to:

1 AND 1 = 1 AND 1

which is a tautology. There you go.