Is proving 1+1=2 the hardest problem in mathematics?

>>8219896

troll?

Ok, the proof starts from the Peano Postulates, which define the natural numbers N.

N is the smallest set satisfying these postulates:

P1. 1 is in N.

P2. If x is in N, then its "successor" x' is in N.

P3. There is no x such that x' = 1.

P4. If x isn't 1, then there is a y in N such that y' = x.

P5. If S is a subset of N, 1 is in S, and the implication (x in S => x' in S) holds, then S = N.

Then you have to define addition recursively...

Def: Let a and b be in N. If b = 1, then define a + b = a' (using P1 and P2).

If b isn't 1, then let c' = b, with c in N (using P4), and define a + b = (a + c)'.

Then you have to define 2:

Def: 2 = 1' 2 is in N by P1, P2, and the definition of 2.

Theorem: 1 + 1 = 2 Proof: Use the first part of the definition of + with a = b = 1.

Then 1 + 1 = 1' = 2 Q.E.D.

Note: There is an alternate formulation of the Peano Postulates which replaces 1 with 0 in P1, P3, P4, and P5.

Then you have to change the definition of addition to this:

Def: Let a and b be in N. If b = 0, then define a + b = a.

If b isn't 0, then let c' = b, with c in N, and define a + b = (a + c)'.

You also have to define 1 = 0', and 2 = 1'.

Then the proof of the Theorem above is a little different:

Proof: Use the second part of the definition of + first: 1 + 1 = (1 + 0)'

Now use the first part of the definition of + on the sum in parentheses: 1 + 1 = (1)' = 1' = 2 Q.E.D.

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>>8219914
>being this new

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Bertrand Russel wrote a 100+ page book about it iirc

>>8219896
'1+1=2' immediately implies the Riemann Hypothesis

There is nothing wrong with axioms

>>8219949
Yo dude, what's with all dem Greek letters and shit?

Yeah you 'mathematicians' find these things are our problems.

#DIEDIEDIE

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>>8219896
looks like math is a new modern "art".

. 1 dot
. 1 dot
.. How many dots are there?

>>8219917
We went over things like this on the first day of number theory.

Even though this feels far over complicated for something so simple, it really makes me happy.

>>8219949
The whole point of Russel and Whitehead's Principia Mathematica was to provide a 100% rigorous Symbolic Logic foundation for all of mathematics. Most of it was overly-formal proofs of elementary concepts, and it eventually proves that 1+1=2, after formally defining '1', '2', and '+' in terms of the formal framework. The book certainly isn't "about" proving this statement.

You can make any proof longer by adding formality. You can prove that inverses are unique in Group theory in about one line, or you can adhere to making one formal deduction at a time and prove it in more like 20 lines.

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>>8219896
done

>>8219896
>the hardest problem
No, but it took Russell and Whitehead
about 140 pages of development to get to it.

>>8219896
> true by definition
> lets prove it

mathematicians are retarded cucks

>>8221795
see >>8221749

>>8221335
10

>>8221784
it's evaluating 1+1 to 2, and then proving 2=2, rather than proving that 1+1 is actually 2

>>8219896
For someone as untalented as OP, every problem appears hard.

>>8221335
infinite?

>>8221795
But they weren't trying to get to it, it just eventually came about in their work as a corollary. There were single page proofs of 1+1=2 years before Russell and Whitehead.

>>8221812
It is only true by definition if you work in a mathematical framework which defines every single possible number and its additions axiomatically, which is a fucking awful idea.

>>8219896
Considering all math is made up fantasy lies based on arbitrary symbols that has nothing at all to do with the real world, no, that is not hard at all.

>>8222021
>all math is made up
Yes.

>lies
No.

>has nothing at all to do with the real world
Partly true.

It can't be proven. You can't prove an axiom.It is a logical element. It is what we use to prove other things.

>>8221891
Its right in binary number.

>>8222069
'1+1=2' is not an axiom in the current model of mathematics.

>>8222069
do not comment on mathematics if you have no fucking clue what you're talking about

it's pretty much the only field in which someone can literally prove you wrong without gathering mountains of scientific 'evidence'

did you even read that post about peano arithmetic?

>>8219896
It is sort of "just" by definition in the modern paradigm of Zermelo-Fraenkel axioms ( https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory ), but just like in Russel and Whitehead's formalism the annoying part is defining what that expression actually means.

1) What are the definitions of "1", "+1", "=" and "2"?

"=" is given to us:
Axiom of Extensionality: Two sets are "=" if and only if they have the same elements.

For the others we will define the set of Natural Numbers, using the following axiom:
Axiom of Infinity: There is a set I with the following properties:
- the empty set O is an element of I. (the Empty Set
Axiom posits that there actually *is* an empty set)
- if x∈I then its "successor" x∪{x} is also in I (the Pair Axiom is used to make {x} and the Axiom of Union tells you that ∪ makes sense).
Call these the "inductive properties," and make the abbreviation S(x)=x∪{x}.

Then the following definitions are pretty standard:
-Define the set of natural numbers N as the intersection of all sets with the inductive properties (first you need to prove that you can form arbitrary intersections of sets, but you can). Then N is non-empty and the smallest set with the inductive properties.
-Define the natural numbers 0=O, 1=S(0)={O}, 2=S(1)={O}∪{{O}}={O,{O}}. Note that these are indeed natural numbers because N has the inductive properties.
-Define the operation "+1" on the set of natural numbers by x+1=S(x). You can recursively define what it means to "add a natural number" by declaring "x+S(n)=S(x+n)", but we will only need to "add 1".

2)The proof.
1+1=S(1)=2, by definition

>>8222178
>modern paradigm
I should also point out that I'm not really an expert on Foundational maths, so there are probably paradigms more modern than this, I just learned this one in school

>>8219896
no. It follows directly from the standard definition of the natural numbers

>>8219896
1 finger + 1 finger =2 fingers

>>8222227
Showing 1 specific example that it works doesn't prove that it works on everything.

>>8222238
What the fuck are you talking about? In that concept of unit, it works fine just by definition. If you change the logical concept, of course another method should be used. But in the fundamental logic, its
irrefragably correct.

>>8221916
Same thing.

>>8222111
you use math to prove things ...
but how can you prove math ..?

>>8223443

really makes u think

>>8223443
>>8225517
And then you realize "math proves things relative to its assumed premises and rules of deduction" and then it's not confusing anymore.

>>8225595
proving axioms has driven the brigthest mathematicians into wasting their entire lives on it

>>8225599
protip: (one exists within zero)

;)

>>8219896
No. Prove that 2+2=4.

>>8225612
oh my god, that's gotta be at least twice as hard

>>8219896
objectively yes

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>>8225785

>>8219956
LOL

>>8219918
>implying anything is somehow actually proven
>maths
Fuck you

>>8222021
It's a system that has evolved and adjusted over time to reflect behavior in reality. There is a reason physics, some chemistry, engineering etc. use mathematics. Its fundamental purpose is to describe the world in objectivity. If it were pure fantasy lies, no one would use it, you could not use that computer to type you absolute cretin.

>>8219917
this still doesnt prove anything

your just inventing more and more stuff on the same unproven foundation

in much the same way you cant use a word in its own definition, you cant prove that 1+1=2 using this gobbledygook notation which affirms the consequent

nice try but number theory is to math as gender studies is to sociology

I thought this was the science board. You know, for actual science. Not the poorly thought out debate board.

>what is the purpose of pure mathematics? hmmmm...let's ask the pseudo-inellectuals of 4chan

http://lmgtfy.com/?q=what+is+the+purpose+of+pure+mathematics
http://mathoverflow.net/questions/43690/whats-a-mathematician-to-do/44213#44213