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additivel
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There are 9 letters in ADDITIVEL ( A1D2E1I1L1T1V4 )
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ADDITIVEL - In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any ...
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In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any * * * * β * , * γ * < * α * * * {\displaystyle \beta ,\gamma <\alpha } * , we have * * * * β * + * γ * < * α * . * * * {\displaystyle \beta +\gamma <\alpha .} * Additively indecomposable ordinals are also called gamma numbers. The additively indecomposable ordinals are precisely those ordinals of the form * * * * * ω * * β * * * * * {\displaystyle \omega ^{\beta }} * for some ordinal * * * * β * * * {\displaystyle \beta } * . * From the continuity of addition in its right argument, we get that if * * * * β * < * α * * * {\displaystyle \beta <\alpha } * and α is additively indecomposable, then * * * * β * + * α * = * α * . * * * {\displaystyle \beta +\alpha =\alpha .} * * Obviously 1 is additively indecomposable, since * * * * 0 * + * 0 * < * 1. * * * {\displaystyle 0+0<1.} * No finite ordinal other than * * * * 1 * * * {\displaystyle 1} * is additively indecomposable. Also, * * * * ω * * * {\displaystyle \omega } * is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every infinite cardinal is additively indecomposable. * The class of additively indecomposable numbers is closed and unbounded. Its enumerating function is normal, given by * * * * * ω * * α * * * * * {\displaystyle \omega ^{\alpha }} * . * The derivative of * * * * * ω * * α * * * * * {\displaystyle \omega ^{\alpha }} * (which enumerates its fixed points) is written * * * * * ϵ * * α * * * . * * * {\displaystyle \epsilon _{\alpha }.} * Ordinals of this form (that is, fixed points of * * * * * ω * * α * * * * * {\displaystyle \omega ^{\alpha }} * ) are called epsilon numbers. The number * * * * * ϵ * * 0 * * * = * * ω * * * ω * * * ω * * * ⋅ * * * ⋅ * * ⋅ * * * * * ... |