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• 20:20, 11 July 2023 Yto talk contribs created page Proving well-orderedness (Created page with "Proving that an ordered set is well-ordered can be very challenging. The methods that can be used to do this vary depending on the type of ordered set. There are of course cases when none of this applies, but mainly in the context of pure apeirology, it often does apply. <h1>Proving totality</h1> As one of the two conditions for an order to be a well-order is that it is a total order, proving totality is a significant part of the proof of well-orderedness. In arbitraril...")
• 06:26, 11 July 2023 Yto talk contribs created page Pair sequence system (Created page with "'''Pair sequence system''' ('''PSS''') is an ordinal notation system defined by BashicuHyudora. It is also a sequence system with sequences of pairs of natural numbers, and an expansion system with the base of standard form being \( \{((0,0),(1,1),(2,2),...,(n,n)) : n\in\mathbb{N}\} \) and with the expansion \( S[n] \) defined in the following way: - The parent of a pair \( x \) in \( S \) is the last pair before it with a smaller first element. The ance...")
• 06:16, 11 July 2023 Yto talk contribs created page Primitive sequence system (Created page with "'''Primitive sequence system''' ('''PrSS''') is an ordinal notation system defined by BashicuHyudora. It is also a sequence system with sequences of natural numbers, and an expansion system with the base of standard form being \( \{(0,1,2,...,n) : n\in\mathbb{N}\} \) and with the expansion \( S[n] \) defined in the following way: If \( B \) is the subsequence of \( S \) such that the first element of \( B \) is the last element of \( S \) strictly smaller...")
• 04:07, 11 July 2023 Yto talk contribs created page Expansion system (Created page with "An '''expansion system''' is an ordinal notation system defined in a special way. It is defined through expansion, with standard form constructed from a specified set called the base of the standard form (usually with order type \( \omega \)). 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 \( X_0 \) u...")
• 04:03, 11 July 2023 Yto talk contribs created page Sequence system (Created page with "A '''sequence system''' is an ordinal notation system in which sequences are well-ordered. Typically, it is an expansion system, with the expansion chosen so that x[n] is always lexicographically smaller than x, and additionally, so that x[0] is x without its last element and x[n] is always a subsequence of x[n+1]. If all of these hold, then as long as the base of its standard form is totally ordered, the order of the sequence system is identical to the lexicogr...") Tag: Visual edit: Switched
• 22:11, 10 July 2023 Yto talk contribs created page Ordinal notation system (Created page with "An '''ordinal notation system''' (also called an '''ordinal notation''' informally) is a system of names for ordinals, or equivalently, a well-ordered set of objects that "can be considered names". There may not be a formal restriction of what objects can be considered names, but most common examples include strings, sequences, trees/hydras, and terms built up from constants and functions. Most authors also require the well-order to be recursive, and this requirement is...")
• 22:36, 3 September 2022 Yto talk contribs created page User:Yto (Created page with "Hi, i've been an apeirologist for approximately 4 years. I thought about writing articles on this wiki, but realized that i'm bad at finding sources, so i'll just use this page for unsourced explanations and philosophy related to apeirology. Hopefully that's ok.")
• 16:05, 31 August 2022 User account Yto talk contribs was created automatically