Could Set Theory Actually Prove 1+1=3?

[ilmareofthemai]

Hello all!
I recently read A Universe in Zero Words (it actually has words), a book about the history and influence of important equations. It discussed (if I understood correctly) that our current arithmetic operations are based on set theory, and that since set theory isn't entirely consistent, that a proof of the sum of one and one being equal to three might be produced.
Thoughts?
R

 

[MLP]

2+2

I would question what exactly is meant by saying that "set theory isn't entirely consistent".

 

[micromass]

Hello all!
I recently read A Universe in Zero Words (it actually has words), a book about the history and influence of important equations. It discussed (if I understood correctly) that our current arithmetic operations are based on set theory, and that since set theory isn't entirely consistent, that a proof of the sum of one and one being equal to three might be produced.
Thoughts?
R

That book is not entirely correct then. Set theory might be completely consistent, but the problem is that we don't know. We can never actually prove that set theory is consistent or not. So while most mathematicians guess that set theory is consistent, we can never know for certain. This is one of Godel's incompleteness theorems.

So if the book says that set theory isn't entirely consistent, then that is false. The right thing to say is that we don't know whether it is consistent or not. And if it is consistent: then we will never be able to prove that it is consistent. But yes, it can be that set theory is inconsistent. So it might happen that we produce a proof of 1+1=3.

 

[0xDEADBEEF]

Well and then maybe they meant naive set theory, which is inconsistent due to the "set of all sets... " stuff. But that has kind of been resolved.

 

[blue_raver22]

I can provide some insight into this topic. Set theory is a foundational theory in mathematics that deals with the concept of sets and their properties. It is widely used in many branches of mathematics, including arithmetic. However, like any other mathematical theory, set theory has its limitations and potential inconsistencies.

One of the most famous examples of these inconsistencies is Russell's paradox, which shows that a set cannot contain itself as an element. This paradox and others like it have led to the development of different versions of set theory, such as Zermelo-Fraenkel set theory, which aim to address these inconsistencies.

So, is set theory inconsistent? The answer is not a straightforward yes or no. It depends on the specific version of set theory being used and the assumptions and axioms it is based on. Some versions may be inconsistent, while others may not be.

In regards to the proof of the sum of one and one being equal to three, it is highly unlikely that this would ever be proven using any version of set theory. This is because the basic arithmetic operations, such as addition, are well-defined and consistent within set theory. However, if there were to be a proof of this, it would likely be due to a mistake in the logical reasoning rather than an inconsistency in set theory itself.

In summary, while set theory has its limitations and potential inconsistencies, it is still a crucial and widely used theory in mathematics. It is constantly evolving and being refined to address any potential issues. So, while we should remain critical and aware of its limitations, we can still trust in the reliability and validity of set theory in our mathematical calculations and equations.