With difficulty. 2 + 2 = 4 is a simple theorem which you can deduce from the definition of addition. There are ways to make 2 + 2 = 5, however, you can expect them to be rather bizarre.

First, it’s easy to prove that 2 + 2 =4. You just need the rule x + Sy = S(x + y) where Sx means the successor of x, or x + 1. With some more steps, with simple natural axioms, you can also derive a contradiction from 2 + 2 = 5. Proof indented so it’s easy to skip.

Define 2 = SS0, 4 = SSSS0, and 5 = SSSSS0, then 2 + 2 = (SS0 + SS0) = S(SS0 + S0) = SSSS0 = 4.

So, if you add 2 + 2 = 5 (or SS0 + SS0 = SSSSS0) as an axiom to your axiomatization of arithmetic, then you deduce immediately that 2 + 2 = 4 = 5.

So then, so long as you have the rules that

• Sx = Sy -> x = y, and that
• 0 is not the successor of any number

you end up, after a few more steps, proving that S0 = 0, contradiction.

Here I'm using the rules of Robinson arithmetic - leaving out the rules about multiplication - rules 1 to 5 here. See Robinson arithmetic You don't need the rule 3 that every number has a predecessor to derive a contradiction from 2 + 2 = 5.

So, basically it’s just Robinson’s first three axioms, the fifth one, and the definition of 2, and 5, and the contradiction drops out. This is one of the simplest axiomatizations of arithmetic, and though you can add more powerful deduction rules, so long as you have addition at all, with any ordinary form of arithmetic, then this is going to drop out quickly as an easy result, that 2 + 2 = 4, and that this is inconsistent with 2 + 2 = 5.

So, you have to change something there, to make 2 + 2 = 5. Here are a few ideas:

1. change the definition of =. You could easily have 2 + 2 = 1 (mod 3) with modulo arithmetic, or “clock arithmetic”. E.g. on a clock, then 6 hours after 8 o’clock is 2 o’clock. So 8 + 6 = 2. That’s called modulo arithmetic, 8 + 6 = 2 mod 12, - means you treat numbers that are 12 apart as equal. You add a rule n + 12 = n, and drop the rule that 0 doesn’t have a successor.

So, you can do the same with modulo 3 arithmetic (say). Add the rule that n+3 = n. Drop the rule that no number has 0 as its successor. But that particular modulus of 3 won’t work because 5 = 2 mod 3.

It's not so easy to define an = with 2 + 2 = 5. You want to have 4 = 5 for that to work, but how can that be if the modulus is greater than 1?

I can only see one natural way to do this, and it's rather trivial. Modulo 1 arithmetic, where every number is equal to 0, and the successor of a number is itself, and bizarrely you decide to work with SS0 and SSSSS0 even though they are both equal to 0?

2, have non associative equality, that Sx = x for every x, and then 4 = 5, similarly 3 = 4, but you don't have 3 = 5 (otherwise it is just the same as modulo 1 arithmetic).

Or some other experiment along those lines, strange re-interpretations of =.

3. Redefine addition. That's the easiest way to do it, There are many binary operations and you can just make one up.

Define (+) so that x (+) y = x + y + 1. Easy peasy. Done!

4. Redefine 5 to be a symbol for 4. You now need a new symbol for 5, and might as well redefine 4 to be a symbol for 5.

5. Paraconsistent logic. Just add 2 + 2 = 5 as a new axiom, and accept that your system of axioms is inconsistent.

6. Strange deduction rules or conventions. E.g. use the convention that in any equation, you always add 1 to the right hand side, so that when you say 2 + 2 = 5, you mean the same as what ordinary folk mean when they write 2 + 2 = 4. Similarly you'd write 3 + 5 = 9 etc.where by 9, when it's on the right hand side of an equation, you mean 8. Then 3 + 5 + 7 = 8 + 7 (because 3 + 5 = 9) = 16.

7. Numbers that are time dependent, or for some other reason are fluid and changing, as you go through the equations. So, 2 (+) 2 = 5 because one of the objects you are counting split in two. For instance a number system to count clouds. Have a convention that when you say A = B then the = sign here represents the situation after a time step of ten minutes, say. Then 2 (+) 2 = 5 means that 2 clouds + 2 clouds, ten minutes later, became 5 clouds. It would be empirical without a deduction system, you are just noting down observations, not proving a result about numbers.

Blue Sky and White Clouds - if you had a system that’s based on counting clouds, and the convention that in an equation, time increases as you go to the right, then you could have numbers with bizarre seeming properties if you didn’t know the convention. This is a special case of redefining =.

Or maybe you have some rule about formation of new clouds, or whatever the objects are. E.g. that whenever you have at least 4 objects, a new one gets added, and more generally, you add as many new objects as there are multiples of 4 on the left hand side. E.g. 4 (+) 6 = 12 because 4 + 6 = 10, that’s two multiples of 4, so you get two new clouds (or whatever) so the right hand side needs 2 added to it. This is different from redefining addition because 12 doesn’t equal 4 + 6, rather, 10 = 6 + 6, because you have the convention that it’s a time series increasing from left to right. It’s more like a strange deduction rule combined with a reinterpretation of = than a way of redefining +. You’d also have 10 = 12 = 14 = 16 = 19 = …

Or perhaps the numbers just change randomly, so sometimes 2 clouds + 2 clouds = 4, sometimes they = 5, sometimes 1, sometimes 23 etc. If you were talking about clouds and understanding = with this time convention approach then that’s the most likely situation.

It’s fun to imagine what it might be like to be extra terrestrials who live in a constantly changing environment, where the numbers of objects are never the same, so you can’t count in the ordinary sense - not even themselves. Sometimes they may wake up and find they have split into a dozen individuals, sometimes a whole community becomes a single organism, and it’s constantly changing, even from one minute to another - you can’t complete a thought process without the numbers of everything in your environment changing, so much so that their maths has some rather weird version of topology perhaps as its fundamental idea, not arithmetic, and when they develop numbers, maybe they would come up with some strange notions of = and other operations on numbers. Maybe they live in the cloud decks of a gas giant - and can’t see anything around them except clouds and vortices, constantly changing, and are similar in nature themselves.

There may be other ways. I'd say that 3, redefine addition, is the easiest way to get lots of self consistent theories that have 2 + 2 = 5.

See also my answer to Is it possible that an alien civilization has completely different mathematics than ours?