User blog:LittlePeng9/Encoding ordinals using binary strings

Encoding ordinal as binary relation
Imagine we have an infinite countable ordinal \(\alpha\), equipped with standard \(<\) relation. Because \(\alpha\) has cardinality of \(\mathbb{N}\), there is an ordering of \(\mathbb{N}\) which "resembles" ordering of \(\alpha\). For example, if we take ordering of \(\omega2+4\), that is \(0<1<2<3<...<\omega<\omega+1<...<\omega2<\omega2+1<\omega2+2<\omega2+3\), we can take following ordering of natural numbers: \(4<6<8<10<...<5<7<9<...<0<1<2<3\). Technically, this "resemblance" means that they have the same order type, and there is order-preserving mapping between them. Order-preserving mapping is bijection \(f:\alpha\rightarrow\mathbb{N}\), such that \(\beta<\gamma\rightleftarrows f(\beta)\triangleleft f(\gamma)\), where \(\triangleleft\) is this non-standard ordering of natural numbers. In our example, we have \(f(n)=2(n+2), f(\omega+n)=2(n+2)+1,f(\omega2+m)=m\) for \(n\in\mathbb{N}\) and \(m\in\{0,1,2,3\}\).

Encoding binary relation as subset of \(\mathbb{N}^2\)
Soon.

Encoding subset of \(\mathbb{N}^2\) as subset of \(\mathbb{N}\)
Soon.

Encoding subset of \(\mathbb{N}\) as binary string
SOON.