Cardinality and Bijection of finite sets

[Edgard Neuman]

well on that sad note, I will leave now.. I'll come back in six month with another amazing but uncommon idea to share, as usual 

Edited October 19, 2018 by Edgard Neuman

[Strange]

1 minute ago, Edgard Neuman said:

well on that sad note, I will leave now..

Why?

[uncool]

5 hours ago, Edgard Neuman said:

It's simpler to just consider that numbers works the same in this axiomatic excepts they are limited by ω

The axioms are designed to work together. Just saying "limited by ω" doesn't really work.

For the rest of your post, the idea of formal sums covers what you are trying to do. You are trying to deal with the concept of formal sums over an ordered set.