Is there a consistent axiomatic system where $1+1 = 2$, $2 \not = 1+1$?

Question

Is it possible to create a consistent axiomatic system where $1+1 = 2$, $2 \not = 1+1$?

Answer 1

You seem to intend that we're allowed to redefine "$+$" and "$=$" to mean whatever we want. Then certainly we can have $1+1=2$ and $2 \neq 1+1$: for example, we could interpret "$+$" to mean $-$ and "$=$" to mean $<$.

Answer 2

Equality is an equivalence relation, so it satisfies the following three necessary conditions:

1. Reflexivity, $a = a$;
2. Symmetry, $a = b \implies b = a$;
3. Transitivity $a = b, b = c \implies a = c$.

These properties hold regardless of the math going on on either side of the relation. So you can define $+$ to be standard real addition, or addition modulo $p$, or whatever. If you make the statement $ 1 + 1 = 2$ within whatsoever arithmetical system you so desire, then by using $=$, you are making a claim that necessarily carries these properties, so $2 = 1+1$, always.

You could, however, define a relation that is not an equivalence relation such that $1 + 1 \sim 2$, but $2 \not\sim 1+1$.

For example, $\textrm{C} + \textrm{O}_2 \to \textrm{CO}_2$, but $\textrm{CO}_2 \not\to \textrm{C} + \textrm{O}_2$, leading to global warming.

Answer 3

You could but firstly the $+$ and $=$ ect. would have to hold different functions and/or, secondly you could have an axiom that for instance works with the principles of the time arrow (direction of time) and the law of entropy in a dynamic system, which would state that once an operation has been carried out then it cannot be reversed or broken down into its original constituents. i.e. as per your example $1+1=2$, $2\not=1+1$

P.S. I know that I used physics examples but that is for two good reasons, firstly it is easier to picture, secondly (more importantly) the examples have were originally derived from mathematical equations.