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Section: New Results

Factorization of ordinal numbers

Participant : José Grimm.

Ordinal numbers have been designed at approximately the same time that the foundations of mathematics were being revisited, in the beginning of the 20th century. These objects cross the boundaries of set theory and pose especially difficult challenges when considering the task of formalizing mathematics. This is the reason why we concentrate on formal proofs concerning these objects.

An ordinal number $x$ is said to be prime if $x>1$ and for every factorisation $x=ab$, one of $a$ or $b$ is equal to $x$ (the other factor is not necessarily equal to 1). Prime ordinals are of three kinds; a power of a power of $\omega$, the successor of a power of $\omega$, or a prime natural number. Every ordinal can uniquely be written as a product of primes, with the following restriction: if $a$ is followed by $b$ in the factor list then: if $b$ is of the first kind, so is $a$ and $a\ge b$, if $a$ and $b$ are natural numbers, then $a\le b$. The proof can be found in an updated version of [20]