Continuing to perform transfinite induction beyond produces more ordinal numbers , each represented as the largest surreal number having birthday . (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is . There is another positive infinite number in generation : . The surreal number is not an ordinal; the ordinal is not the successor of any ordinal. This is a surreal number with birthday , which is labeled on the basis that it coincides with the sum of and . Similarly, there are two new infinitesimal numbers in generation : and .
continuinG tO perForm tRanSfInite inDUction bEYonD pRoducEs more ORdinaL numBERS , eaCh REpresENteD as thE Largest SUrreal NUmber Having BirtHdAY . (This iS eSseNtIaLLY a DeFiNItiOn of the orDinal NUmBerS rEsuLting fRom TRansfinIte InductiOn.) tHe first suCH OrdiNaL is . There is anOtheR PosiTIve InfinIte number in GEneratiON : . the surREal NumBeR IS not an ordinal; the ordInal is NOT thE sucCEssoR of aNy ORdInaL. this is a surreaL Number wITh bIRthday , WhicH iS laBEled oN THE baSIS THAT iT coinCiDes With ThE sum of and . simILaRly, therE ArE tWo nEw INfInITesimAL nuMbErs In geneRAtIOn : aNd .