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

Physiks lover
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  • I do not know what you mean with consistent, but my first question is: what is $2$ in your system? If $2$ is $1+1$, then is your statement valid only for $2$ or all possible $1+1+...+1$ (is $+$ associative?) ?Etc...etc... :) – Avitus Jul 09 '13 at 13:16
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    In other words, there will be $a$ and $b$ such that $a=b$ but $b\neq a$? I am curious to know your definition of "$=$". – Start wearing purple Jul 09 '13 at 13:16
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    This of course depends on the logic. In first-order logic with equality, it's obviously impossible. In first-order logic wihout equality (with = being treated as a nonlogical relation symbol), it's equally obvious that it's possible. – Chris Eagle Jul 09 '13 at 13:21
  • @O.L. Yes, that's what I had in mind. I only have a basic grasp of maths, but I think it boils down to generalising "=" in an abstract way, analagous to the way the addition operator was generalised as a binary operator of a group. – Physiks lover Jul 09 '13 at 14:32
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    Sure, because $1,2,+,=,\neq$ are all just symbols. However, we almost always use the symbol "=" to be an equivalence relationship, so if $X=Y$ then $Y=X$. But that is convention - you can make up whatever you want. Would it be useful? Probably not, and certainly not with this notation. – Thomas Andrews Jul 09 '13 at 14:33
  • @ChrisEagle Thanks for the informative comment. – Physiks lover Jul 09 '13 at 14:35

3 Answers3

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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.

Emily
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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 $<$.

Chris Eagle
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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.

genty4321
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