Parménides García Cornejo

arXiv2013

Higher Classes of Ordinals induced by <_1

In this article we want to see that it is possible to iterate and generalize the notions presented in "Carlson's <_1-relation on the class of epsilon numbers" such that we can obtain the higher or thinner classes of ordinals induced by the <_1-relation as solutions of conditions. Indeed, these results are so general that in a future article we will be able to characterize the ordinal |Pi_1^1 - CA_0 | as the smallest ordinal of the thinnest class induced by <_1.

math.LO↗

arXiv2013

Carlson's <_1-relation on the class of epsilon numbers

Based on the class of epsilon numbers, another binary relational <^1 in the ordinals is introduced. We will see that we can easily describe all the isomorphisms that are witnesses of <^1. Afterwards we will show that the isomorphisms witnesses of <^1 are "essentially the same" than the isomorphisms of the <_1-relation for finite subsets of ordinals that are closed under "the cover construction". Finally, through our understanding of <^1 we will see how it is that <_1 induces thinner kappa-club classes of ordinals: Thinner than the thinnest class of ordinals seen in the article "Carlson's <_1-relation on the class of additive principal ordinals"; indeed, the thinnest class of ordinals induced by <_1 that we will obtain in this article will have as it's first element the Bachmann's ordinal |ID_1|. An independent proof of this last fact will be provided in a coming article.

math.LO↗