Expansion system: Difference between revisions

More precisely, the definition involves only a set \( S \) of well-formed terms, a function \( []: S\times\mathbb{N}\to S \) (where [](x,n) is written as x[n]), and a set \( X_0 \). Then with \( X \) being the closure of \( X_0 \) under \( x\mapsto x[n] \) for every \( n\in\mathbb{N} \), and with \( x<y \) equivalent to the existence of \( m,n_0,n_1,...,n_m\in\mathbb{N} \) such that \( x=y[n_0][n_1]...[n_m] \), the expansion system is the set \( X \) ordered by \( < \). Such an ordered set still needs to be well-ordered in order to be an ordinal notation system, and the definition simply having this form is not enough to imply well-orderedness.

ItExpansion stillsystems needsmay toalso be wellcalled "dom-orderedtype systems" or similarly, in orderreference to bethe anfact ordinalthat notationsome systemof them have a function \( dom \) assigning a "domain" to each element \( x\in S \), andso thethat definition\( simplyx[\alpha] having\) thisis formdefined precisely for \( y\in dom(x) \). However, restricting the domain of an element is often not enoughnecessary, toespecially implywhen well-orderednessthe domain can only be \( \varnothing, \{0\} \) or \( \mathbb{N} \). Many expansion systems are also [[sequence system | sequence systems]], although there can be exceptions.