[Guest]

Why is 1+1=2 an axiom? In other words prove that 1+1 does not equal 2

[Guest]

In Binary 1+1=10

[Guest]

In Binary 1+1=10

Good answer and it might explain how its 1+1 doesn't equal to two but it doesn't prove that 1+1=2 is an axiom.

Edited February 19, 2010 by igotchu

[Guest]

1+1=2

a=b

a+b=2a

a+b-2b = 2a-2b

2(a-b) = a+b-2b

2(a-b)=a-b

2=1

This is wrong, of course, because the last step required diving by (a-b), which is 0 if a=b.

Silly, though.

[Guest]

1+1=2 cannot be (or as of yet has not been) proven wrong if 1+1=2 is an axiom

Wiki on "axiom":

"In mathematics...an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. Unlike theorems, axioms (unless redundant) cannot be derived by principles of deduction, nor are they demonstrable by mathematical proofs, simply because they are starting points; there is nothing else from which they logically follow (otherwise they would be classified as theorems)." <http://en.wikipedia.org/wiki/Axiom>

So, i state that if someone can legitimately prove that 1+1=2 is wrong, all mathematics derived from the axiom 1+1=2 is also wrong.

- V.Rock

[akaslickster]

Because,

11 yep eleven. By not adding and just placing them side by side would make the equation an axiom, no?

Why is 1+1=2 an axiom? In other words prove that 1+1 does not equal 2

[Guest]

1 + 1 = 2 is not an axiom, at least not an integer axiom:

A0 if A and B ∈ Z then (A + B) ∈ Z, ∀ A, B

A1 (A + B) + C = A + (B + C), ∀ A, B, C

A2 A + B = B + A, ∀ A, B

A3 ∃ 0 such that A + 0 = 0 + A = A, ∀ A

A4 A + -A = 0, ∀ A

M0 if A and B ∈ Z then (A * B) ∈ Z, ∀ A, B

M1 (A * B) * C = A * (B * C) ∀ A, B, C

M2 A * B = B * A, ∀ A, B

M3 ∃ 1 ≠ 0such that A * 1 = 1 * A = A, ∀ A

O0 (A + B) * C = A * C + B * C, ∀ A, B, C

P0 ∃ P ⊂ Z

P1 if A ∈ P, then (A + 1) ∈ P

P2 Exactly 1 of {A ∈ P, -A ∈ P, A = 0} is true, ∀ A

[Guest]

42 on this one.

[Guest]

1/3= .333.........*3=.999... Multiplying the expression 1/3 by 3 = 1 So .999=1 .999+.999=1.998, which is not 2

Edited March 2, 2010 by hoggardvikes

[Guest]

1/3= .333.........*3=.999... Multiplying the expression 1/3 by 3 = 1 So .999=1 .999+.999=1.998, which is not 2

Actually, 1 = 0.999... is true.

Also, 0.999... + 0.999... = 1.999... = 2

[Guest]

The question "Prove an axiom" a logical fallacy.

Axioms, by their definition, are the basis of all proof and are only validated by the results that they have with other axioms.

If there were some mathematical contradiction that consistently arose from using "1+1=2" then the axiom would be rendered useless. However, since the entire basis of math rests on these simple axioms and this math has been validated millions of times over throughout history in science, engineering, economics, and in the first grade, it is a fact, even an axiom, that 1+1=2.

[Guest]

The question "Prove an axiom" a logical fallacy.

Axioms, by their definition, are the basis of all proof and are only validated by the results that they have with other axioms.

If there were some mathematical contradiction that consistently arose from using "1+1=2" then the axiom would be rendered useless. However, since the entire basis of math rests on these simple axioms and this math has been validated millions of times over throughout history in science, engineering, economics, and in the first grade, it is a fact, even an axiom, that 1+1=2.

What if I was a robot from the distant future (the year 2000)? Then it would be 1+1=10.

[Guest]

this probably isn't the answer, but I remember in elementary school, we said 1+1= window. I color coded the different parts: 1 is red, + is black, = is blue, and window is green

[Guest]

What if I was a robot from the distant future (the year 2000)? Then it would be 1+1=10.

10_binary = 2_base10

so 1+1 = 2_base10

And robots from the year 2000 would not be from the distant future.

This topic is ridiculous.

[Guest]

Yes, the topic is ridiculous. You cannot prove or disprove an axiom using logic.

If you could prove the claim using logic, then the claim would be a theorem, not an axiom.

And the logic required to prove the claim would derive from other axioms that cannot be proved using logic.

See Godel's incompleteness theorems:

"Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250)."

In other words, it is pointless to try to disprove or prove axioms.

I side with Valbaca and vistalite on the answer to the original question.

However, if you want to just have fun thinking about ways in which the expression 1 + 1 = 2 is false, I would suggest considering Boolean Algebra.

In Boolean Algebra, the expression 1+1 is actually equal to 1.

It's kind of funny to go back to someone in high school and tell them, I went to college and learned that 1+1=1.

Of course these 1's are different than their normal algebraic counterparts.

More strikingly different is the + operator which in standard notation for Boolean algebra denotes an OR operation.

When you write: True OR True = True, then it doesn't look as strange to someone familiar only with normal algebra.

[Guest]

And robots from the year 2000 would not be from the distant future.

Haha! Sorry, it was a reference to a song by Flight of the Conchords. Probably an inappropriate place to make a joke but oh well.

[Guest]

1/3= .333.........*3=.999... Multiplying the expression 1/3 by 3 = 1 So .999=1 .999+.999=1.998, which is not 2

I call the 0.(9) = 1 the "Problem of Metainfinity". Why? "Meta" means "about" in greek. "Meta"infinity refers to the fact that 0.99999... has an unlimited number of 9s (Not the numbers, the amounts of numbers). If we multiply the number 10 times, we will get the same number again, because the number of 9s is not specified. That's why there are so many proofs that 0.(9) is equal to 1. And 1/3 is APPROXIMATELY equal to 0.333. Unlike in mechanics and/or other sciences, in maths each 10^-100 is of matter.

Edited March 8, 2010 by Ianis G. Vasilev

[Guest]

I call the 0.(9) = 1 the "Problem of Metainfinity". Why? "Meta" means "about" in greek. "Meta"infinity refers to the fact that 0.99999... has an unlimited number of 9s (Not the numbers, the amounts of numbers). If we multiply the number 10 times, we will get the same number again, because the number of 9s is not specified. That's why there are so many proofs that 0.(9) is equal to 1. And 1/3 is APPROXIMATELY equal to 0.333. Unlike in mechanics and/or other sciences, in maths each 10^-100 is of matter.

Quite right, it is an interesting fact that 0.9... = 1, by pure equality, not approximation.

0.9... = 0.9 + 0.09 + 0.009 +... = 0.9*(1 + 1/10 + 1/100 + ...) = 0.9/(1-1/10) = 0.9/0.9 = 1

Since 1/(1-x) = 1+x+x^2+x^3+x^4+... for |x|< 1

People tend to have more issues thinking about the infinite number of 9's producing a value of 1, than say for example, the infinite number of zeroes that follow

1.00000000......

But 0 isn't any better of a symbol than 9 is it?

[Guest]

Correct!

1 + 1 = 2

2*1 = 2 | /2

1*1 = 1

The above statement gives us the ability to change 1*1 with 1 without prooving that a*1 = a, so

1 = 1 (This is true, so 1 + 1 = 2)

[Guest]

1/3= .333.........*3=.999... Multiplying the expression 1/3 by 3 = 1 So .999=1 .999+.999=1.998, which is not 2

good one, i like that answer

[Guest]

good one, i like that answer

That does not prove that 1+1 is not 2.

That proves that 0.999 + 0.999 is not 2.

But 0.999 is not 1, so it says nothing about 1 + 1.

If you have an infinite number of 9's like 0.999.... that number is 1.

If you have a finite number of 9's it is not 1.

If the guy who posted this actually meant an infinite number of 9's, in that case you will never get the 8 after all the 9's that he predicts.

Instead you can separate and get 1.000000... + 0.9999999.....

where the dots mean an infinite number of digits.

But we already said that 0.99999... was the same as 1.00000.....

so in actuality we have 1.0000... + 1.0000.... = 2.00000....

And if you wanted you could say it can also be represented as 1.999999.....

But both representations refer to the same quantity.

Likewise, by equality and not approximation

5.0000... = 4.9999....

1.250000.... = 1.24999999....

etc

There is no point in trying to prove 1 + 1 is not 2.

Just like there is no point in trying to prove an apple is not an apple.

[Guest]

So the answer is that you can't prove that 1+1=2 because 1+1=2 is an axiom.

Okay, so what if the question was to prove that 2+2=4. Do I say that 2=1+1 (the axiom) so 1+1+1+1=4? Is that a proof? Or is 2+2=4 another axiom because we are defining "4" as the sum of 4 1s? For example, why is it wrong to say that 1+1+1+1=3?

Basically, what I'm wondering is, what axioms do we use to define our numbers? We say that 2=1+1. Is 3 defined with an axiom that say 3=2+1? Or is 3=2+1 not an axiom because it's really 3=1+1+1?

If there's anybody who can educate me on this topic of axioms, I would be very interested to learn.

I'm thinking that a good way to define the numbers with axioms would be this:

All axioms:

2=1+1

3=1+1+1

4=1+1+1+1

5=1+1+1+1+1

...

If you define the numbers with anything other than 1 then it wouldn't be an axiom, would it? For example, 5=2+3 would not be an axiom because you can prove that is true by using the axioms 2=1+1, 3=1+1+1, and 5=1+1+1+1+1. So is this right?

And also, whether or not I am correct with what I said above, what are the axioms for defining decimal numbers?

Edited April 23, 2010 by Use the Force

[Guest]

So the answer is that you can't prove that 1+1=2 because 1+1=2 is an axiom.

Okay, so what if the question was to prove that 2+2=4. Do I say that 2=1+1 (the axiom) so 1+1+1+1=4? Is that a proof? Or is 2+2=4 another axiom because we are defining "4" as the sum of 4 1s? For example, why is it wrong to say that 1+1+1+1=3?

Basically, what I'm wondering is, what axioms do we use to define our numbers? We say that 2=1+1. Is 3 defined with an axiom that say 3=2+1? Or is 3=2+1 not an axiom because it's really 3=1+1+1?

If there's anybody who can educate me on this topic of axioms, I would be very interested to learn.

I'm thinking that a good way to define the numbers with axioms would be this:

All axioms:

2=1+1

3=1+1+1

4=1+1+1+1

5=1+1+1+1+1

...

If you define the numbers with anything other than 1 then it wouldn't be an axiom, would it? For example, 5=2+3 would not be an axiom because you can prove that is true by using the axioms 2=1+1, 3=1+1+1, and 5=1+1+1+1+1. So is this right?

And also, whether or not I am correct with what I said above, what are the axioms for defining decimal numbers?

I don't think I can completely answer your question because I am not completely sure about some of the details.

The important thing to realize is that the authority to make claims such as 1+1=2 comes from axioms and definitions.

These are the parts in a mathematical theory that cannot be proven within the theory.

A definition is the specification of the essential characteristics of an idea or entity, such that all ideas which contain those characteristics are classified as the type of object being defined, and all ideas that do not meet all essential characteristics are not classified as such.

An axiom is a claim that is accepted and assumed always true in a theory.

By using the theory, you are in effect agreeing that for the sake of all arguments within the theory, the axioms and definitions are true. If you do not make this agreement, you are not actually using the theory.

You may be using some other more general theory with different axioms and definitions, but not this one.

Once these unprovable components are in place, you can then apply deductive logic to say things like,

assuming all the axioms and definitions in the theory are true, than by logic, it must be the case that "blah blah blah".

The claims that derive from the axioms and definitions are called theorems. Theorems are only as correct as the axioms and definitions themselves, because they are logically equivalent to them.

In math books you may see other words like lemmas and corollaries, but these are essentially just theorems.

A lemma is a statement considered as a less important theorem that is used as a stepping stone to a more important theorem.

A corollary is a theorem that follows almost immediately from a previous theorem.

I don't know if 1+1=2 is actually an axiom, it might actually be a theorem that derives from more fundamental algebraic axioms and definitions.

Either way, 1+1=2 logically must be true if you accept the axioms and definitions of algebra.

If you do not accept the axioms and definitions of algebra, than the statement could mean anything for all I know, because I do not know what theory or language the statement would belong to if not algebra.

Edited April 23, 2010 by mmiguel1

[Guest]

I think the best we can do is prove that 1 + 1 = A, is unique. That is to say, only the element A comes from 1 + 1.

Proof:

Let, 1 + 1 = A and

suppose that 1 + 1 = B, B not equal to A. So now there is another element that comes from 1 + 1 and is not A.

Then (1 + 1)+(1 + 1) = A + A = (1 + 1)+(1 + 1) = A + B

So A + A = A + B

(A + A) - A = (A + B) - A [i am not sure about this step]

A + (A - A) = (A + B) - A

A + 0 = (B + A) - A

A = B + (A - A)

A = B + 0

A = B

and we arrive at a contradiction.

From now on, we will write 2 instead of A, so that 1 + 1 = 2. So whenever we see 1 + 1, we can say it equals to 2.

Edited April 24, 2010 by marsupialsoup

[Guest]

here is a page about the basic axioms.

http://science.jrank.org/pages/505/Arithmetic-Axioms-operations-arithmetic.html

[Guest]

I don't think I can completely answer your question because I am not completely sure about some of the details.

The important thing to realize is that the authority to make claims such as 1+1=2 comes from axioms and definitions.

These are the parts in a mathematical theory that cannot be proven within the theory.

A definition is the specification of the essential characteristics of an idea or entity, such that all ideas which contain those characteristics are classified as the type of object being defined, and all ideas that do not meet all essential characteristics are not classified as such.

An axiom is a claim that is accepted and assumed always true in a theory.

By using the theory, you are in effect agreeing that for the sake of all arguments within the theory, the axioms and definitions are true. If you do not make this agreement, you are not actually using the theory.

You may be using some other more general theory with different axioms and definitions, but not this one.

Once these unprovable components are in place, you can then apply deductive logic to say things like,

assuming all the axioms and definitions in the theory are true, than by logic, it must be the case that "blah blah blah".

The claims that derive from the axioms and definitions are called theorems. Theorems are only as correct as the axioms and definitions themselves, because they are logically equivalent to them.

In math books you may see other words like lemmas and corollaries, but these are essentially just theorems.

A lemma is a statement considered as a less important theorem that is used as a stepping stone to a more important theorem.

A corollary is a theorem that follows almost immediately from a previous theorem.

I don't know if 1+1=2 is actually an axiom, it might actually be a theorem that derives from more fundamental algebraic axioms and definitions.

Either way, 1+1=2 logically must be true if you accept the axioms and definitions of algebra.

If you do not accept the axioms and definitions of algebra, than the statement could mean anything for all I know, because I do not know what theory or language the statement would belong to if not algebra.

Thank you very much. That was quite informative.