User blog:Vel!/NOT the largest number ever

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SHOCK! HORROR! My own friend Wojowu DARES to make treacherously false claims about the nature of mathematical logic itself!

In a scandalous blog post of his, he claims the existence of this number called "\(N\)" — which he also calls the "Robinson number" — that turns out to be larger than every natural number. He then goes on to show how the existence of \(N\) is derived from the axioms of Robinson's Q (which I will abbreviate as just Q from now on).

As googologists, we know this to be a load of absolute balderdash! There is no "infinity" or any kind of "end" to the natural numbers! That's the whole point of our adventure, no?!

Recall the axioms of Q:


 * 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 operator and \(+,\cdot\) are binary operators. We also define \(x\leq y \Leftrightarrow \exists z: x+z=y\).

Wojowu, in a feat of dastardly metamathematical trickery, attempts to define his "\(N\)" and notes that any operation involving \(N\) results in \(N\), save for \(N \cdot 0 = 0\). A consequence is that \(SN = N\).

This is a load of baloney! We can easily show that for all \(x\), \(Sx \neq x\):


 * 1) \(Sx \neq 0\): This implies that \(S0 \neq 0\), \(S1 \neq 0\), etc. None of these contradict our statement, and in fact the first statement provides a special case of it.
 * 2) \((Sx=Sy) \Rightarrow x=y\): Again, no contradiction.
 * 3) \(x=0 \lor \exists y(Sy=x)\): Again, no contradiction, since there's no statement that \(y = x\).
 * 4) Ditto.
 * 5) Ditto.
 * 6) Ditto.
 * 7) Ditto.

As you can see, the notion that \(Sx \neq x\) is perfectly logical. This means that \(SN = N\) is wrong, and by the law of contrapositive \(N\) CANNOT EXIST!

I've shown to you from the axioms of Q that the existence of \(N\) is a gigantic truckload of hogwash! Bovine feces! I have EXPOSED Wojowu as a LIAR and a FAKE!