Ah, now I have to remember what I was thinking of back in September! Well, let's see what I can come up with now.

One thing that I could do is to redefine every term in the expression. You tried to forestall this by insisting

1 is still the successor of 0 and 3 is still the successor of the successor of 1, where 0 is the additive identity

[Note: I originally interpreted this as "3 is still the successor of 2" for some dumb reason.] But you never insisted that 2 is the successor of 1, so I'll redefine 2 to be 1 and redefine 3 to be 2, and your conditions are met, while my theorem holds. (I could also, or instead, redefine equality.)

But this is silly; nobody uses the terms in this way.

For another method, I'll be a little more precise. Since you mentioned the successor of 0, let's work in Peano Arithmetic (first-order, classical logic, starting at zero), supplemented with the axiom that 0 = 1. Then 1 + 1 = 3 can be proved as follows:

• 1 + 1 = 1 + S(0) by definition of 1;
• 1 + S(0) = 1 + S(1) by substitution of equality;
• 1 + S(1) = 3 by any ordinary proof in PA;
• 1 + 1 = 3 by transitivity of equality (twice).

Of course, this is also silly, since PA with my new axiom is inconsistent. Anything in the language can be proved (by going through the axiom that 0 = S(n) is always false, combining this with my new axiom, and using ex contradictione quodlibet).

Here is a slightly less silly way: Modular arithmetic is very useful, not silly at all, and in arithmetic modulo 1, 1 + 1 = 3 is true.

But however useful modular arithmetic in general may be, arithmetic modulo 1 is silly (for roughly the same reasons that an inconsistent set of axioms is silly); everything is equal to everything else, so any equation at all is true. In other words, arithmetic modulo 1 is trivial.

You can get arithmetic modulo b by replacing the Peano axiom that 0 = S(n) is always false with the axiom that 0 = b and b − 1 additional axioms stating (altogether) that a = b is false whenever (in ordinary arithemetic) 0 < a < b. But you could instead add an arbitrary axiom of the form b = c (and another finite set of inequalities between smaller numbers). So let us use the arithmetic given by the axiom that 2 = 3. Then 1 + 1 = 3 is easy to prove (since the proof that 1 + 1 = 2 doesn't rely on the axiom that we've removed, and we still have transitivity of equality). And yet this system is not trivial; it is basically (0, 1, 2, 2, 2, …).

Actually, this example is minimal; let's go for a little overkill and instead use the axiom that 1 = 2. Of course, we can still prove that 1 + 1 = 3 (this time leaving the formal proof entirely to the reader). This system is a bit more trivial than the last one, but not quite trivial; it is basically (0, 1, 1, 1, …).

Now, although these systems of arithmetic are nontrivial, I really ought to give some mathematical reasons why anybody would be interested in them at all. I can give several profound reasons for the last one, which I will skip on the grounds that you can find them elsewhere; it boils down to this: this system is the system of truth values in classical logic (Boolean algebra). I don't even have to tell you how to interpret 0, 1, and + (much less =) in this system; I'm simply using them with their standard meanings in this context.

So now that you see that 1 + 1 = 3 in Boolean algebra, I need to turn around (as meta-contrarian) and explain why it is still silly. One reason is that nobody doing Boolean algebra (or even the slightly less trivial system of arithmetic based on 2 = 3) should ever want to write "3"; they should just write "1" (or "2" in the other system) instead. Another reason is that you shouldn't just throw "1 + 1 = 3" or even "1 + 1 = 1" out without explanation; the default meanings of those terms are in Peano arithmetic (or an extension thereof), not Boolean arithmetic. So in a general context, you really ought to say "1 + 1 = 3 in Boolean arithmetic", or something like that, instead. Just saying "1 + 1 = 3" and expecting people to know what the heck you're talking about is, well, silly.

I have no idea if that's what I was thinking in September, but that's what I thought of now. I hope that you like it.

Edit: Read-o fixed.