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The question of whether 1+1=2 is a theorem or not has been debated in the realm of mathematics for centuries. In this article, we'll explore the idea of extending a proof to make it longer and more complicated than necessary.

A Simple Proof

A simple proof of 1+1=2 can be obtained by using basic arithmetic operations:

1+1 = 2

This proof is straightforward and does not require any advanced mathematical concepts. However, some mathematicians have argued that this proof is too short and lacks rigor.

Extending the Proof

To extend the proof of 1+1=2, we can repeat an axiom a large number of times:

1. Axiom: 0 + x = x
2. Axiom: 0 + (x+y) = x + y
3. ... (repeat this step many times)

After repeating this process n times, where n is a large number, we can conclude that:

1+1 = 2

This extended proof may seem more convincing to some mathematicians, but it is not necessarily more rigorous or accurate.

Alternatives

There are alternative ways to prove 1+1=2 using advanced mathematical concepts such as set theory and group theory. For example:

• We can define two sets A = {1} and B = {1}. Then, we can show that the union of these sets is equal to {1} + {1}, which is equivalent to 2.
• Alternatively, we can consider a group G under addition modulo 2. In this group, we have 1+1 ≡ 0 (mod 2), which implies 1+1 = 2.

These alternative proofs may be more appealing to some mathematicians due to their use of advanced concepts and rigorous mathematical frameworks.

Conclusion

In conclusion, the question of whether 1+1=2 is a theorem or not is a matter of interpretation. While a simple proof can be obtained using basic arithmetic operations, alternative proofs using advanced mathematical concepts may seem more convincing to some mathematicians. Ultimately, the validity of a proof depends on the mathematical framework and the axioms used to derive it.

References

• MJD (2013). "How would one be able to prove mathematically that 1+1=2?" Mathematics Stack Exchange.
• Dejan Govc (2013). "A proof is a finite sequence of formulas..." Mathematics Stack Exchange.