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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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