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- 16:33, 30 August 2023 RhubarbJayde talk contribs created page Von Neumann ordinal (Redirected page to Ordinal#Von Neumann definition) Tags: New redirect Visual edit
- 16:31, 30 August 2023 RhubarbJayde talk contribs created page Limit ordinal (Redirected page to Ordinal#Ordinal arithmetic) Tags: New redirect Visual edit
- 16:28, 30 August 2023 RhubarbJayde talk contribs created page Principal (Redirected page to Additive principal ordinals) Tags: New redirect Visual edit
- 16:25, 30 August 2023 RhubarbJayde talk contribs created page Second-order arithmetic (Created page with "Second-order arithmetic, denoted \(Z_2\) is an extension of first-order (i.e. Peano) arithmetic by adding additional second-order variables as well as an induction scheme for \(\mathcal{P}(\mathbb{N})\), and a comprehension scheme. Proof-theoretically, \(Z_2\) is a very expressive system, as it can prove the consistency of Peano arithmetic and its extensions via the addition of iterated inductive definitions - an ordinal analysis of \(Z_2\) is considered the holy grail o...") Tag: Visual edit
- 15:34, 30 August 2023 RhubarbJayde talk contribs created page Additive principal ordinals (Created page with "An ordinal \(\gamma\) is called an additive principal or additively principal if, for all \(\alpha, \beta < \gamma\), we have \(\alpha+\beta < \gamma). The least additively principal ordinal is 1 since \(0 + 0 = 1\), and all additively principal ordinals other than 1 are limit ordinals. In particular, you can see from the Cantor normal form theorem (every ordinal has a CNF representation) that the additively principal ordinals are precisely the ord...") Tag: Visual edit
- 14:53, 30 August 2023 RhubarbJayde talk contribs created page OCF (Redirected page to Ordinal collapsing function) Tags: New redirect Visual edit
- 14:44, 30 August 2023 RhubarbJayde talk contribs created page Ordinal collapsing function (Created page with "An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as \( \Omega \) or <nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki> to smaller, recursive ordinals such as the SVO. The primary idea is that, at the point of epsilon numbers and beyond, especially at the level of Stron...") Tag: Visual edit
- 14:02, 30 August 2023 RhubarbJayde talk contribs created page Large cardinals (Redirected page to Large cardinal) Tags: New redirect Visual edit
- 14:02, 30 August 2023 RhubarbJayde talk contribs created page Buchholz's psi (Redirected page to Buchholz's psi-functions) Tags: New redirect Visual edit
- 14:01, 30 August 2023 RhubarbJayde talk contribs created page Buchholz psi (Redirected page to Buchholz's psi-functions) Tags: New redirect Visual edit
- 13:44, 30 August 2023 RhubarbJayde talk contribs created page Gap ordinals (Redirected page to Gap ordinal) Tags: New redirect Visual edit
- 13:34, 30 August 2023 RhubarbJayde talk contribs created page Omega^omega (Created page with "The ordinal \( \omega^\omega \) is relatively small compared to other countable ordinals, but has some interesting properties. In particular, \( \omega^\omega \) is: * The least \( \alpha \) so that \( \alpha \) is the \( \alpha \)th limit ordinal. * The least limit of additive principal ordinals. * The least ordinal which is, for all \( n < \omega \), an element of the class \( L^n(\mathrm{Ord}) \), where \( L \) is the limit point operator. * The proof-theoretic ordin...") Tag: Visual edit
- 13:17, 30 August 2023 RhubarbJayde talk contribs created page Mahlo cardinal (Created page with "A Mahlo cardinal is a certain type of large cardinal used in the study of reflection principles and consistency strength. These are much stronger than inaccessible cardinals, 1-inaccessible cardinals, hyper-inaccessible cardinals, and more. However, like with inaccessible cardinals, there are two primary types. == Weakly Mahlo == Analogously to how weakly inaccessible cardinals are more popular than strongly inaccessible cardinals in apeirological circles, but less...") Tag: Visual edit
- 12:49, 30 August 2023 RhubarbJayde talk contribs created page Inaccessible cardinal (Created page with "There are two main types of notion of inaccessible cardinal - weakly, or strongly inaccessible cardinals. They are the same assuming the generalized continuum hypothesis, but under axioms such as the resurrection axioms, it is possible that the amount of real numbers is weakly inaccessible but, by definition, being very far from being strongly inaccessible. == Weakly inaccessible == Weakly inaccessible cardinals were first invented. Essentially, cardinals such as \( \al...") Tag: Visual edit
- 12:38, 30 August 2023 RhubarbJayde talk contribs created page Takeuti-feferman-buchholz ordinal (Redirected page to Takeuti-Feferman-Buchholz ordinal) Tags: New redirect Visual edit
- 12:08, 30 August 2023 RhubarbJayde talk contribs created page Church-Kleene ordinal (Created page with "<nowiki>The Church-Kleene ordinal, commonly denoted \( \omega_1^{\mathrm{CK}} \) or \( \omega_1^{ck} \) is defined as the supremum of all "recursive ordinals". A recursive ordinal is the order-type of a well-order on the natural numbers which can be computed by a Turing machine. Note that all countable ordinals are the order-type of a well-order on the natural numbers, but there are only countably many Turing machines, and uncountably many countable ordinals, meaning the...") Tag: Visual edit
- 11:54, 30 August 2023 RhubarbJayde talk contribs moved page Stongly critical ordinal to Strongly critical ordinal (typo, I'm so stupid)
- 11:45, 30 August 2023 RhubarbJayde talk contribs created page Church kleene ordinal (Redirected page to Church-Kleene ordinal) Tags: New redirect Visual edit
- 11:43, 30 August 2023 RhubarbJayde talk contribs created page Bird ordinal (Created page with "The Bird ordinal (sometimes called Bird's ordinal) is an intermediate ordinal between the Takeuti-Feferman-Buchholz ordinal and Extended Buchholz ordinal which occurs occasionally in apeirological notations such as BMS. It was named by the apeirological community in honor of Chris Bird. This is because it is believed to correspond to the limit of his final system of array notations, and thus the growth rate of a natural extension of his...") Tag: Visual edit
- 11:31, 30 August 2023 RhubarbJayde talk contribs created page BMS (Redirected page to Bashicu matrix system) Tags: New redirect Visual edit
- 11:31, 30 August 2023 RhubarbJayde talk contribs created page Extended Buchholz ordinal (Created page with "The Extended Buchholz ordinal, sometimes known as OFP (short for omega-fixed-point), is the limit of an extension of Buchholz's original set of ordinal collapsing functions, defined by Denis Maksudov, which allows to collapse ordinals such as \( \Omega_{\omega + 1} \) (which corresponds to the Takeuti-Feferman-Buchholz ordinal), \( \Omega_{\omega^2} \) (which is believed to correspond to the BMS matrix (0,0,0)(1,1,1)(2,1,1)), or \( \Omega_{\...") Tag: Visual edit
- 11:28, 30 August 2023 RhubarbJayde talk contribs created page Bird's ordinal (Redirected page to Bird ordinal) Tags: New redirect Visual edit
- 11:27, 30 August 2023 RhubarbJayde talk contribs created page Takeuti-Feferman-Buchholz ordinal (Created page with "The Takeuti-Feferman-Buchholz ordinal, commonly abbreviated as TFBO, is the full limit of Buchholz's original set of ordinal collapsing functions. This name was assigned by David Madore in his "Zoo of Ordinals".<ref>A Zoo of Ordinals, David A. Madore, July 29 2017</ref> It is not particularly vastly larger than the Buchholz ordinal, although one could comparatively describe the difference in size as like that between \( \varepsilon_0 \) and the ...") Tag: Visual edit
- 11:17, 30 August 2023 RhubarbJayde talk contribs created page Extended buchholz ordinal (Redirected page to Extended Buchholz ordinal) Tags: New redirect Visual edit
- 11:16, 30 August 2023 RhubarbJayde talk contribs created page Bachmann howard ordinal (Redirected page to Bachmann-Howard ordinal) Tags: New redirect Visual edit
- 11:15, 30 August 2023 RhubarbJayde talk contribs created page Large veblen ordinal (Redirected page to Large Veblen ordinal) Tags: New redirect Visual edit
- 11:15, 30 August 2023 RhubarbJayde talk contribs created page Small veblen ordinal (Redirected page to Small Veblen ordinal) Tags: New redirect Visual edit
- 11:11, 30 August 2023 RhubarbJayde talk contribs created page Feferman-Schütte ordinal (Redirected page to Veblen hierarchy) Tags: New redirect Visual edit
- 11:11, 30 August 2023 RhubarbJayde talk contribs created page Feferman-Schutte ordinal (Redirected page to Veblen hierarchy) Tags: New redirect Visual edit
- 11:11, 30 August 2023 RhubarbJayde talk contribs created page Feferman-schutte ordinal (Redirected page to Veblen hierarchy) Tags: New redirect Visual edit
- 11:03, 30 August 2023 RhubarbJayde talk contribs created page Buchholz ordinal (Created page with "The Buchholz ordinal is the limit of Wilfried Buchholz's original set of ordinal collapsing functions, with finite indices, and is equal to the limit of the sequence \( \omega \), \( \varepsilon_0 \), \( \mathrm{BHO} \), \( \psi_0(\Omega_3) \), ... - i.e. it is equal to \( \psi_0(\Omega_\omega) \). It is also equal to the proof-theoretic ordinal of second-order arithmetic with comprehension restricted to \(\Pi^1_1\)-formulae, or of Peano arith...") Tag: Visual edit
- 10:51, 30 August 2023 RhubarbJayde talk contribs created page Dimensional Veblen (Redirected page to Large Veblen ordinal) Tags: New redirect Visual edit
- 10:51, 30 August 2023 RhubarbJayde talk contribs created page Bachmann-Howard ordinal (Created page with "The Bachmann-Howard ordinal is a key milestone in the set of recursive ordinals. It is equal to the limit of the dimensional Veblen function, as well as the Buchholz hydra with only zero and one labels. It was originally discovered as the limit of a basic ordinal collapsing function, namely Bachmann's psi, which was used in ordinal-analysis. In particular, the Bachmann-Howard ordinal is exactly the proof-theoretic ordinal of basic Kripke-Platek s...") Tag: Visual edit
- 10:44, 30 August 2023 RhubarbJayde talk contribs created page Large Veblen ordinal (Created page with "The large Veblen ordinal is a large extension of the small Veblen ordinal. By using an entry-indexing notation (formally defined via finitely-supported ordinal functions), it is possible to further extend the multi-variable version of the Veblen hierarchy used to define the small Veblen ordinal to an array-like system with infinitely long arrays. In particular, the small Veblen ordinal can be denoted by \( \varphi(1,...,0,0) \), with \( \omega \) many zeroes. The...") Tag: Visual edit
- 10:40, 30 August 2023 RhubarbJayde talk contribs created page SVO (Redirected page to Small Veblen ordinal) Tags: New redirect Visual edit
- 10:39, 30 August 2023 RhubarbJayde talk contribs created page Small Veblen ordinal (Created page with "The Small Veblen ordinal is the limit of a finitary, variadic extension of the Veblen hierarchy. In particular, after the basic stage \( \varphi(\alpha, \beta) \), one lets \( \varphi(1,0,\alpha) \) enumerate fixed points of \( \beta \mapsto \varphi(\beta,0) \) - i.e. strongly critical ordinals - followed by \( \varphi(1,1,\alpha) \) enumerating its fixed points, and so on. The Small Veblen ordinal, very commonly abbreviated to SVO, is the least...") Tag: Visual edit
- 10:31, 30 August 2023 RhubarbJayde talk contribs created page Gamma numbers (Redirected page to Veblen hierarchy) Tags: New redirect Visual edit
- 10:30, 30 August 2023 RhubarbJayde talk contribs created page Zeta zero (Redirected page to Epsilon numbers) Tags: New redirect Visual edit
- 10:30, 30 August 2023 RhubarbJayde talk contribs created page Epsilon naught (Redirected page to Epsilon numbers) Tags: New redirect Visual edit
- 10:30, 30 August 2023 RhubarbJayde talk contribs created page Epsilon zero (Redirected page to Epsilon numbers) Tags: New redirect Visual edit
- 10:29, 30 August 2023 RhubarbJayde talk contribs created page Zeta numbers (Redirected page to Epsilon numbers) Tags: New redirect Visual edit
- 10:29, 30 August 2023 RhubarbJayde talk contribs created page Stongly critical ordinal (Redirected page to Veblen hierarchy) Tags: New redirect Visual edit
- 10:28, 30 August 2023 RhubarbJayde talk contribs created page Stongly critical (Redirected page to Veblen hierarchy) Tags: New redirect Visual edit
- 18:22, 29 August 2023 RhubarbJayde talk contribs created page Weakly compact cardinal (Created page with "A weakly compact cardinal is a certain kind of large cardinal. They were originally defined via a certain generalization of the compactness theorem for first-order logic to certain infinitary logics. However, this is a relatively convoluted definition, and there are a variety of equivalent definitions. These include, letting \(\kappa\) be the cardinal in question and assuming \(\kappa^{< \kappa} = \kappa\): * \(\kappa\) is 0-Ramsey. * \(\kappa\) is \(\Pi^1_1\)-indes...") Tag: Visual edit
- 17:42, 28 August 2023 RhubarbJayde talk contribs created page Patterns of resemblance (Created page with "The patterns of resemblance (PoR) are a system of ordinal-notations introduced by TJ Carlson. It is superficially similar to stability, yet is a notation for recursive rather than nonrecursive ordinals, and uses elementary substructures between ordinals themselves, instead of between ranks of the constructible universe. It uses a structure also found in BMS known as respecting forests, and was originally believed to have the same limit as BMS.") Tag: Visual edit
- 15:21, 28 August 2023 RhubarbJayde talk contribs created page Uncountable (Redirected page to Countability) Tags: New redirect Visual edit
- 15:21, 28 August 2023 RhubarbJayde talk contribs created page Countability (Created page with "Countability is a key notion in set theory and apeirology. A set is called countable if it has the same size as the set of the natural numbers. The way this is formally defined is that there is a map \( f: x \to \mathbb{N} \), where \( x \) is the set in question, so that different elements of \( x \) are sent to different natural numbers, and every natural number has some element of \( x \) sent to it. Georg Cantor, the founder of set theory, proved that the set of inte...") Tag: Visual edit
- 12:49, 28 August 2023 RhubarbJayde talk contribs created page Large cardinal (Created page with "Large cardinals are cardinals typically defined as satisfying certain combinatorial or reflection-type properties. Their existence is asserted by various large cardinal axioms, which are usually unprovable in \( \mathrm{ZFC} \), assuming its consistency. This is because almost all large cardinals, if they exist, are worldly: a worldly cardinal is a \( \kappa \) so that \( V_\kappa \) satisfies ZFC, and thus Gödel's second incompleteness theorem applies. Due to issues...") Tag: Visual edit
- 11:45, 28 August 2023 RhubarbJayde talk contribs created page User:RhubarbJayde (Added my profile) Tag: Visual edit
- 11:39, 28 August 2023 User account RhubarbJayde talk contribs was created