Proving 2+2=4 Using Field Axioms

In summary, the conversation discusses various ways to prove that 2+2=4, including using field axioms and defining the numbers 2 and 4. Ultimately, it is concluded that 2+2=4 can be proven by showing that 1+1+1+1 = 4(1) and that 4(1) = 4, but this relies on the definitions of addition and numbers.
  • #1
rafasaur
1
0
I've looked around but haven't found anyway to prove 2+2=4. I'm pretty sure you need to use field axioms, but I just haven't found it yet. Is there a way to do it? Like showing a+a=2a? Or a+b=c? Like 1+1=2. Something like that.

Thanks!
 
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  • #2
Directly.

What is the definition of 2? Of 4?
 
  • #3
rafasaur said:
I've looked around but haven't found anyway to prove 2+2=4.

See Foundations of Analysis by Landau.
 
  • #4
You just need a definition of 2, of 4 and of +.
Defn: 0 = empty set. 1 = {0}. 2 = {0,1} = {0,{0}}, 3 = {0,1,2} = {0,{0},{0,{0}}},
4 = {0,1,2,3} = {0,{0},{0,{0}}, {0,{0},{0,{0}}}}.
Addition is defined recursively. I.e. first adding one is defined. n + 1 = n union {n}.

i.e. 1+ 1 = {0} union {{0}} = {0,{0}} = 2.
2+1 = 2 union {2} = {0,1} union {2} = {0,1,2} = 3.
3+1 = {0,1,2}+1 = {0,1,2} union {3} = {0,1,2,3} = 4.
Assuming we have defined n+m then n + (m+1) = (n+m)+1.

now you have enough to do it. or keep reading.
So 2 + 2 = (2+1)+1 = 3+1 = 4.

Tata!

Aren’t you glad you asked? Basically it seems 4 = ((1+1)+1)+1. and 2 = (1+1).
so 2+2 = (1+1)+(1+1), so it boils down to associativity of +.
 
  • #5
(1+1+1+1)=1+1+1+1
(1+1)+(1+1)=4(1)
2+2=4

Does that work? Sorry I'm a beginner when it comes to proofs.
 
  • #6
You haven't said why "1+ 1+ 1+ 1" would be equal to "4(1)" or why 4(1) would be equal to 4. That was Mathwonk's point- this whole thing depends upon exactly how you define "+", "1", "2", and "4". You have just assumed basic arithmetic without giving any definitions. That is no different from just assuming that 2+ 2= 4.
 

1. How can you prove that 2+2 equals 4 using field axioms?

To prove that 2+2 equals 4 using field axioms, we first need to understand the definition of a field. A field is a mathematical structure that consists of a set of numbers and two operations, addition and multiplication, that follow specific rules. With the help of these rules, we can prove that 2+2 equals 4 by using the properties of fields and the definition of addition.

2. What are the field axioms?

The field axioms are a set of rules that define the properties of a field. These axioms include closure, commutativity, associativity, identity elements, inverse elements, and distributivity. These axioms are essential in proving that 2+2 equals 4 using field axioms.

3. Can you explain the closure axiom in the context of proving 2+2 equals 4 using field axioms?

The closure axiom states that when two numbers are added, the result will always be a number within the same set. In the context of proving 2+2 equals 4 using field axioms, this means that when we add 2 and 2, the result must be a number within the set of real numbers, as defined by the field axioms.

4. How does the distributivity axiom play a role in proving 2+2 equals 4 using field axioms?

The distributivity axiom states that when a number is multiplied by a sum, it is the same as multiplying the number by each term in the sum and then adding the results. In the context of proving 2+2 equals 4 using field axioms, this means that we can distribute the multiplication of 2 and 2 into 2 times 1 and 2 times 1, which equals 4.

5. Can you provide a step-by-step proof of 2+2=4 using field axioms?

1. Start with the definition of a field: a set of numbers with two operations, addition and multiplication, that follow specific rules.

2. Use the closure axiom to show that 2+2 equals a real number within the set.

3. Use the identity element axiom to show that 2+2 equals 4.

4. Use the distributivity axiom to show that 2+2 equals 2 times 1 plus 2 times 1, which equals 4.

5. Therefore, 2+2=4 using field axioms.

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