Difference between revisions of "Small countable ordinals"

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$$0,1,2,3,\ldots,\omega,\omega+1,\omega+2,\omega+3,\ldots,\omega\cdot 2,\omega\cdot 2+1,\ldots,\omega\cdot 3,\ldots,\omega^2,\omega^2+1,\ldots,\omega^2+\omega,\ldots,\omega^2+\omega+1,\ldots,\omega^2+\omega\cdot 2,\ldots,\omega^3,\ldots,\omega^\omega,\omega^\omega+1,\ldots,\omega^\omega+\omega,\ldots,\omega^\omega+\omega\cdot 2,\ldots,\omega^\omega\cdot 2,\ldots,\omega^{\omega^\omega},\ldots,\omega^{\omega^{\omega^\omega}},\ldots$$
 
$$0,1,2,3,\ldots,\omega,\omega+1,\omega+2,\omega+3,\ldots,\omega\cdot 2,\omega\cdot 2+1,\ldots,\omega\cdot 3,\ldots,\omega^2,\omega^2+1,\ldots,\omega^2+\omega,\ldots,\omega^2+\omega+1,\ldots,\omega^2+\omega\cdot 2,\ldots,\omega^3,\ldots,\omega^\omega,\omega^\omega+1,\ldots,\omega^\omega+\omega,\ldots,\omega^\omega+\omega\cdot 2,\ldots,\omega^\omega\cdot 2,\ldots,\omega^{\omega^\omega},\ldots,\omega^{\omega^{\omega^\omega}},\ldots$$
  
We shall give here an account of the ordinals below [[epsilon naught | $\epsilon_0$]].
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== Counting to $\omega^2$ ==
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We explain here in detail how to count to $\omega^2$. This is something that anyone can learn to do, even young children.
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== The ordinals below [[epsilon naught | $\epsilon_0$]] ==
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We shall give here an account of the attractive finitary represenation of the ordinals below [[epsilon naught | $\epsilon_0$]].

Revision as of 15:54, 29 December 2011


The ordinals begin with the following transfinite progression

$$0,1,2,3,\ldots,\omega,\omega+1,\omega+2,\omega+3,\ldots,\omega\cdot 2,\omega\cdot 2+1,\ldots,\omega\cdot 3,\ldots,\omega^2,\omega^2+1,\ldots,\omega^2+\omega,\ldots,\omega^2+\omega+1,\ldots,\omega^2+\omega\cdot 2,\ldots,\omega^3,\ldots,\omega^\omega,\omega^\omega+1,\ldots,\omega^\omega+\omega,\ldots,\omega^\omega+\omega\cdot 2,\ldots,\omega^\omega\cdot 2,\ldots,\omega^{\omega^\omega},\ldots,\omega^{\omega^{\omega^\omega}},\ldots$$


Counting to $\omega^2$

We explain here in detail how to count to $\omega^2$. This is something that anyone can learn to do, even young children.


The ordinals below $\epsilon_0$

We shall give here an account of the attractive finitary represenation of the ordinals below $\epsilon_0$.