Monday, September 30, 2013

Veblen hierachy of ordinal numbers

We already know that a wide variety of ordinal numbers be expressed in cantor norm formal such as 0,1,2,3,4, $\omega$, $\omega+1$, $\omega^2+2\omega+1$, $\omega^\omega$, and $\omega^{2\omega^3+1}+2\omega^5+3\omega^2+2$.

We can use the generalized veblen function of one argument to express these same ordinals so we get $\phi(1)$, $\phi(1)+1$, $\phi(2)+2\phi(1)+1$, $\phi(\phi(1))$, $\phi(2\phi(3)+1)+2\phi(5)+3\phi(2)+2$. The first ordinal that cannot be expressed as such a number in Cantor normal form is $\epsilon_0$ which equals $\phi(0,1)$.

The first ordinal which cannot be expressed through addition and the veblen function of two arguments is the Fefermann-Schutte ordinal which is equal $\phi(0,0,1)$. A considerable amount of countable ordinal numbers can be expressed by the generalized veblen function.

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