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So, by employing methods from model theory, I have found a way to actually find the largest natural number ever, thus setting a limit for all of our attempts of finding larger and larger numbers. Before you say that, I'll answer - you CAN add 1, but that won't make a difference.

In order to do this, I'm going to need axiomatic system for number theory in order to find the largest number ever. But I find Peano arithmetic too mainstream, let's deal with something weaker (after all, weaker theory can't give us larger numbers (can it?)). I'll be using, throughout this blog post, Robinson arithmetic. For these unfamiliar with it, here are the axioms:

  1. \(Sx\neq 0\)
  2. \((Sx=Sy)\Rightarrow x=y\)
  3. \(x=0\lor \exists y(Sy=x)\)
  4. \(x+0=x\)
  5. \(x+Sy=S(x+y)\)
  6. \(x\cdot 0=0\)
  7. \(x\cdot Sy=x\cdot y+x\)

where \(S\) is a unary function (successor), and \(+,\cdot\) are binary functions (addition and multiplication).

We also define \(x\leq y\Leftrightarrow_{def}\exists z: x+z=y\).

Now, in order to get the largest number ever, we need something larger than \(0,1,2,3,...\), after all we all know that this set has no largest element. But we want the largest element. So what should we do? Let's add the largest number! I'll denote this number \(N\). But what would be a successor of this largest number? Let's make it \(N\) itself! But we have to do addition and multiplication too... Let's make all of these operations equal to \(N\) if they involve \(N\)! Except for \(N\cdot 0\). This will be \(0\). \(0\) is special. But not very special. Fark the logic, I want \(0\cdot N=N\).

But now you can shout "YOU CAN'T JUST ADD THE LARGEST NUMBER, THIS MAKES NO SENSE!" And here, my dear, you are wrong. This isn't wrong. This is actually correct. My system satisfies all the axioms, so how can this be wrong? Axiomatic systems are designed to model reality after all. For those of you who don't believe me, here is verification of the axioms:

  1. Straightforward, \(SN\) isn't \(0\), and no other successor is \(0\).
  2. Similar, \(N\) isn't counterexample, so there is none.
  3. Ditto.
  4. Ditto.
  5. If \(x=N\), then \(x+Sy=N+Sy=N=SN=S(N+y)=S(x+y)\). If \(y=N\) then \(x+Sy=x+SN=x+N=N=SN=S(x+N)=S(x+y)\). 
  6. \(0\) is special.
  7. If \(x=N\), then \(x\cdot Sy=N\cdot Sy=N=x\cdot y+N=x\cdot y+x\), because \(Sy\) is never special, by 1. If \(y=N\) then \(x\cdot Sy=x\cdot SN=x\cdot N=N=N+x=x\cdot N+x=x\cdot y+x\) because fark logic.

So as you have seen above, all of my definitions are actually perfectly logical, so they must be right (for interested, deduction rules in propositional logic are sound, i.e. if axioms are logically valid, then so is all we can derive from them, which is a justification for my claim). Thus \(N\) must really exist.

One last thing is to verify that \(N\) is indeed the largest. By our definition of \(\leq\), we have to check that, for every \(x\), there is a \(z\) with \(x+z=N\). But \(z=N\) will do the trick! So \(N\) is indeed the largest natural number we can have. And it's not infinity, it's an element of Robinson arithmetic's universe of discourse, i.e. natural numbers.

For the sake of giving a name to \(N\), I'm gonna give it a name of Robinson number.


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