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Combined display of all available logs of Apeirology Wiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).
- 13:54, 31 August 2023 RhubarbJayde talk contribs created page Empty set (Created page with "The empty set is a set with no elements. Its existence can be proven in Kripke-Platek set theory, even without collection, by applying \(\Delta_0\)-separation with a contradictory formula to an infinite set. The existence of an empty set may seem paradoxical to a beginner to set theory, yet it does not pose any definitional issues and is useful. In particular, the empty set is used in the Von Neumann ordinal system, in which it encodes the number 0. Also,...") Tag: Visual edit
- 13:46, 31 August 2023 RhubarbJayde talk contribs created page Sharp (Created page with "A sharp for an inner model \(N\) is an object whose existence implies that the inner model is "far from \(V\)": for example, \(N\) can be nontrivially elementarily embedded into itself, and uncountable sets of ordinals may not be able to be covered by sets in \(N\). For example, the sharp for \(L\) is \(0^\sharp\). In general, for a real number \(r\), \(r^\sharp\) denotes the sharp for \(L[r] =...") Tag: Visual edit
- 13:21, 31 August 2023 RhubarbJayde talk contribs created page Axiom of determinacy (Created page with "The axiom of determinacy is a powerful proposal for a foundational axiom inspired by Zermelo's theorem. Given any subset \(A\) of Baire space \(\omega^\omega\), let \(\mathcal{G}_A\) be the topological game of length \(\omega\) where players I and II alternatively play natural numbers \(n_1, n_2, n_3, \cdots\). Then player I wins iff \(\langle n_1, n_2, n_3, \cdots \rangle \in A\), and else player II wins. \(A\) is called the payoff set of \(\mathcal{G}_A\). AD states th...") Tag: Visual edit
- 12:49, 31 August 2023 RhubarbJayde talk contribs created page Buchholz's ordinal collapsing function (Redirected page to Buchholz's psi-functions) Tags: New redirect Visual edit
- 12:48, 31 August 2023 RhubarbJayde talk contribs created page Ordinal collapsing functions (Redirected page to Ordinal collapsing function) Tags: New redirect Visual edit
- 12:46, 31 August 2023 RhubarbJayde talk contribs created page Von Neumann cardinal assignment (Redirected page to Cardinal) Tags: New redirect Visual edit
- 12:46, 31 August 2023 RhubarbJayde talk contribs created page Admissible ordinal (Redirected page to Admissible) Tags: New redirect Visual edit
- 12:44, 31 August 2023 RhubarbJayde talk contribs created page The veblen hierarchy (Redirected page to Veblen hierarchy) Tags: New redirect Visual edit
- 12:44, 31 August 2023 RhubarbJayde talk contribs created page Extended buchholz psi (Redirected page to Buchholz's psi-functions#Extension) Tags: New redirect Visual edit
- 12:42, 31 August 2023 RhubarbJayde talk contribs created page Countable (Redirected page to Countability) Tags: New redirect Visual edit
- 12:39, 31 August 2023 RhubarbJayde talk contribs created page Ordinal exponentiation (Redirected page to Ordinal#Ordinal arithmetic) Tags: New redirect Visual edit
- 12:38, 31 August 2023 RhubarbJayde talk contribs created page Ordinal product (Redirected page to Ordinal#Ordinal arithmetic) Tags: New redirect Visual edit
- 12:38, 31 August 2023 RhubarbJayde talk contribs created page Ordinal sum (Redirected page to Ordinal#ordinal arithmetic) Tags: New redirect Visual edit
- 12:32, 31 August 2023 RhubarbJayde talk contribs created page Order type (Redirected page to Ordinal) Tags: New redirect Visual edit
- 12:32, 31 August 2023 RhubarbJayde talk contribs created page Cardinality (Redirected page to Cardinal) Tags: New redirect Visual edit
- 12:30, 31 August 2023 RhubarbJayde talk contribs created page Zero sharp (Created page with "Zero sharp is a sharp for the constructible universe \(L\), which is of importance in inner model theory. Zero sharp, denoted \(0^\sharp\), if it exists, is a real number which encodes information about indiscernibles in the constructible universe. In particular, "\(0^\sharp\) exists" is the assertion that there is a unique class \(I\) of ordinals which is club in \(\mathrm{Ord}\), containing all uncountable cardinals so that, for every un...") Tag: Visual edit
- 11:27, 31 August 2023 RhubarbJayde talk contribs created page Zermelo-Fraenkel set theory with choice (Redirected page to ZFC) Tags: New redirect Visual edit
- 11:27, 31 August 2023 RhubarbJayde talk contribs created page Zermelo-Fraenkel set theory (Redirected page to ZFC) Tags: New redirect Visual edit
- 11:26, 31 August 2023 RhubarbJayde talk contribs created page Zermelo-Fraenkel choice (Redirected page to ZFC) Tags: New redirect Visual edit
- 11:26, 31 August 2023 RhubarbJayde talk contribs created page Zermelo-Fraenkel with choice (Redirected page to ZFC) Tags: New redirect Visual edit
- 11:26, 31 August 2023 RhubarbJayde talk contribs created page Zermelo-Fraenkel (Redirected page to ZFC) Tags: New redirect Visual edit
- 11:25, 31 August 2023 RhubarbJayde talk contribs created page ZF (Redirected page to ZFC) Tags: New redirect Visual edit
- 11:25, 31 August 2023 RhubarbJayde talk contribs created page ZFC (Created page with "ZFC (Zermelo-Fraenkel with choice) is the most common axiomatic system for set theory, which provides a list of 9 basic assumptions of the set-theoretic universe, sufficient to prove everything in mainstream mathematics, as well as being able to carry out ordinal-analyses of weaker systems such as KP and Z2. The axioms are the following: * Axiom of extensionality: two sets are the same if and only if they have the...") Tag: Visual edit
- 11:22, 31 August 2023 RhubarbJayde talk contribs created page Z2 (Redirected page to Second-order arithmetic) Tags: New redirect Visual edit
- 11:13, 31 August 2023 RhubarbJayde talk contribs created page KP (Redirected page to Kripke-Platek set theory) Tags: New redirect Visual edit
- 11:05, 31 August 2023 RhubarbJayde talk contribs created page Constructible hierarchy (Created page with "The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the ordinals. Therefore, it is important in inner model theory, as well as in the study of \(\alpha\)-recursion theory, stability, Gandy ordinals and reflection principles. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \i...") Tag: Visual edit
- 21:09, 30 August 2023 RhubarbJayde talk contribs created page Stability (Created page with "Stability is a notion and very wide range of types of nonrecursive ordinals, inspired by the weaker notion of reflection. In general, stability is defined via ranks of \(L\) being similar to each other. The weakest type of stability is \((+1)\)-stable, i.e. \(L_\alpha\) being a \(\Sigma_1\)-elementary substructure of \(L_{\alpha+1}\). In general, \(\alpha\) is \(\beta\)-stable, or stable up to \(\beta\), if \(L_\alpha\) is a \(\Sigma_1\)-elementary substructure of \(L_\b...") Tag: Visual edit
- 20:55, 30 August 2023 RhubarbJayde talk contribs created page Gandy ordinal (Created page with "For an ordinal \(\alpha\), let \(\delta(\alpha)\) be the supremum of the order-types of \(\alpha\)-recursive well-orderings on a subset of \(\alpha\). An ordinal \(\alpha\) is called Gandy if \(\delta(\alpha)\) is equal to the next admissible ordinal after \(\alpha\), i.e. \(\delta(\alpha) = \alpha^+\). For example, \(\omega\) is trivially Gandy, and in general, any ordinal below the least recursively inaccessible ordinal is Gandy. An ordinal which is not Gandy i...") Tag: Visual edit
- 20:42, 30 August 2023 RhubarbJayde talk contribs created page Bad ordinal (Redirected page to Gandy ordinal) Tags: New redirect Visual edit
- 20:41, 30 August 2023 RhubarbJayde talk contribs created page Bad ordinals (Redirected page to Gandy ordinal) Tags: New redirect Visual edit
- 20:36, 30 August 2023 RhubarbJayde talk contribs created page EBO (Redirected page to Extended Buchholz ordinal) Tags: New redirect Visual edit
- 20:36, 30 August 2023 RhubarbJayde talk contribs created page TFBO (Redirected page to Takeuti-Feferman-Buchholz ordinal) Tags: New redirect Visual edit
- 20:35, 30 August 2023 RhubarbJayde talk contribs created page LVO (Redirected page to Large Veblen ordinal) Tags: New redirect Visual edit
- 20:34, 30 August 2023 RhubarbJayde talk contribs created page BHO (Redirected page to Bachmann-Howard ordinal) Tags: New redirect Visual edit
- 17:16, 30 August 2023 RhubarbJayde talk contribs created page Sound cardinal (Redirected page to Reflection principle) Tags: New redirect Visual edit
- 17:15, 30 August 2023 RhubarbJayde talk contribs created page Cumulative hierarchy (Redirected page to Reflection principle) Tags: New redirect Visual edit
- 17:15, 30 August 2023 RhubarbJayde talk contribs created page Reflection principle (Created page with "The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that, for all \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of...") Tag: Visual edit
- 16:52, 30 August 2023 RhubarbJayde talk contribs created page Kripke-Platek set theory (Created page with "Kripke-Platek set theory, commonly abbreviated KP, is a weak foundation of set theory used to define admissible ordinals, which are immensely important in ordinal analysis and \(\alpha\)-recursion theory. In terms of proof-theoretic strength, its proof-theoretic ordinal is the BHO, and it is thus intermediate between \(\mathrm{ATR}_0\) and \(\Pi^1_1 \mathrm{-CA}_0\). The axioms of KP are...") Tag: Visual edit
- 16:33, 30 August 2023 RhubarbJayde talk contribs created page Inaccessible ordinal (Redirected page to Inaccessible cardinal) Tags: New redirect Visual edit
- 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