Prove that 1+1=2

Prove that 1+1=2

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en.wikipedia.org/wiki/Peano_axioms#Arithmetic
en.wikipedia.org/wiki/Peano_axioms
twitter.com/AnonBabble

Easy, expand the equation. First, define S as the function
[math] S(x) = x \cup \{ x\} [/math]

and then define the operation of addition a+b as performing the function S on a, b times.

Then, by definition this equation equals

[math] S(\{ \{ \} \}) = \{\{\},\{\{\}\}\} [/math]

and by evaluating the left hand side we get that indeed

[math] \{\{\},\{\{\}\}\} = \{\{\},\{\{\}\}\} [/math]

In simpler terms,
[math] \{0 ,1 \} = \{0,1\} [/math]

And two sets are equal if an only they contain the same elements, therefore indeed

[math]1+1 = 2[/math]
[math] \square [/math]

>And two sets are equal if an only they contain the same elements

Prove it :^)

>and then define the operation of addition a+b as performing the function S on a, b times.
This is vague, define addition the Peano way
a+0 = a
a+S(b) = S(a+b)

I see that we're going to be doing this every night this week.

I am sure we just agreed to define that as the standard equality. It is probably an axiom.

It's either an axiom or the equivalent
[math] A \subset B \land B \subset A \Leftrightarrow A = B [/math]

>apply function b times
>vague
no

...

x+x = 2x

You might want to consider engineering.
Your mind doesn't seem able to handle real maths.

Talking about math isn't against the rules.

>I am a freshman obsessed with being pedantically specific
>this is how real math is done
you don't know what you're talking about

> foundational shit
> not pedantic

not nearly as much as
>"applying a function b times" is ambiguous! not precise enough!

2-1=1

Let x = 1
x + x = 2x

2x = 2ยท1
2x = 2

x + x = 2
1 + 1 = 2

>using addition to define addition

Addition is a fundamental property that cannot be broken down

>what is the recursion theorem and the axiom of infinity

It's the inductive way to define addition
It holds for the base case, b = 0 is defined and the induction case: b being defined->S(b) is defined

>Addition is a fundamental property that cannot be broken down
fuck off
seriously why the fuck do you idiots come here and feel like talking authoritatively about shit you know nothing about? you aren't fooling anyone, holy fuck

>Addition is a fundamental property that cannot be broken down

No, addition is not a fundamental property. It is 2016, we have generalized anything. At this point there are no fundamental properties of anything because if something is ever fundamental some genius gets the idea to generalize it and now it is not fundamental anymore, it is just a one in a million case.

And addition is just that, a one in a million case. You could define plenty of groups with plenty of operations and call them addition.

false
In fact there is a physical example in front of you (your computer) which has the operation of addition broken down to logic gates/operations.

x+x only equals 2x if 1+1=2

That is not the Peano way. The Peano way is to define a recursive function like how user did. In case you haven't noticed but his function S is using the notion of counting and addition arises through the fact that 1+1 is actually just "one more than one more than the base case (zero)." What do you think happens when you add? You're taking a number which is just a representative of how many steps you are from zero and another number, and taking the total amount of steps away from zero. That's what the recursive function does. You just need induction and the basic properties of the natural numbers (unboundedness etc) to get the rest of the properties.

That is the inductive way, except its actually formally defined, so you can show addition properties.
en.wikipedia.org/wiki/Peano_axioms#Arithmetic

Here you go op

en.wikipedia.org/wiki/Peano_axioms

...

well, 1, is 1, and is there is another 1, it makes a 2

>what is recursion

>current year

...

1 + 1 = 11

it's an axiom, faggot

what's this meme about?

Great Job.