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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).
- 11:28, 1 September 2023 RhubarbJayde talk contribs created page Taranovsky's ordinal notations (Created page with "Taranovsky's ordinal notations are a collection of ordinal notation systems invented by Dmytro Taranovsky. These include degrees of recursive inaccessibility (DoRI), degrees of reflection (DoR) and the main system (MS), as well as variants such as degrees of reflection with passthrough.<ref>https://web.mit.edu/dmytro/www/other/OrdinalNotation.htm</ref> They all use a binary or ternary function symbol \(C\), but the comparison algorithms and ot...") Tag: Visual edit
- 11:06, 1 September 2023 RhubarbJayde talk contribs created page Indescribable cardinal (Redirected page to Reflection principle#Alternate meaning) Tags: New redirect Visual edit
- 15:13, 31 August 2023 RhubarbJayde talk contribs created page Hereditarily finite set (Created page with "A set is hereditarily finite if the smallest transitive set containing it is finite. So, an ordinal is hereditarily finite if and only if it is finite. Any hereditarily finite set is finite, but not every finite set is hereditarily finite, e.g. \(\{\omega\}\). One advantage of using hereditarily finite rather than finite sets is that they form a set, rather than a proper class. The set in question is \(V_\omega = L_\omega\) in the cumulative/Constructible hierarchy...") Tag: Visual edit
- 15:10, 31 August 2023 RhubarbJayde talk contribs created page Axiom of infinity (Created page with "The axiom of infinity is a common mathematical axiom included in theories such as Kripke-Platek set theory or ZFC. It asserts that there exists an inductive set - i.e. a set \(x\) so that \(0 \in x\) and, if \(n \in x\), then \(n+1 \in x\). By using \(\Delta_0\)-separation, this implies that \(\omega\) exists. The axiom of infinity, obviously, drastically increases the strength of set theory, since else one is not at all able to define Ordinal|ordinal...") Tag: Visual edit
- 15:05, 31 August 2023 RhubarbJayde talk contribs created page Proper class (Created page with "In second-order set theories, such as Morse-Kelley set theory, a proper class is a collection of objects which is too large to be a set - either because that would cause a paradox, or because it contains another proper class. The axiom of limitation of size implies that any two proper classes can be put in bijection, implying they all have "size \(\mathrm{Ord}\)". Russel's paradox combined with the axiom of regularity ensure that \(V\), the class of all sets, is a proper...") Tag: Visual edit
- 14:52, 31 August 2023 RhubarbJayde talk contribs created page Successor ordinal (Created page with "An ordinal is called a successor if it is equal to \(\alpha + 1\) for some other \(\alpha\). An ordinal \(\alpha\) is a successor if and only if \(\max \alpha\) exists. The smallest successor ordinal is 1, which is also the only successor ordinal to also be additively principal. The least ordinal that is not a successor, other than 0, is \(\omega\). If \(\beta\) is successor, then \(\alpha+\beta\) is also successor f...") Tag: Visual edit
- 14:45, 31 August 2023 RhubarbJayde talk contribs created page Absolute infinity (Created page with "Absolute infinity was a concept originally defined by Georg Cantor, the founder of set theory. He denoted it \(\tav\) and defined it as a number greater than everything else, so large that any property it could have would already be satisfied by something smaller. This is clearly not well-defined, since "being absolute infinity" is a property that it and only it has - this is similar to Berry's paradox. However, this idea paved the way for Reflection principle|reflecti...") Tag: Visual edit
- 14:39, 31 August 2023 RhubarbJayde talk contribs created page Continuous function (Created page with "An ordinal function is continuous iff it is continuous in the order topology on the ordinals. If one adds the requirement of being increasing, one obtains the normal functions. However, non-normal continuous functions aren't as studied in the literature and have more complex behaviour. One corollary of the well-foundedness of ordinals is that there is no continuous decreasing ordinal function which is n...") Tag: Visual edit
- 14:21, 31 August 2023 RhubarbJayde talk contribs created page Well-ordered set (Created page with "A well-ordered set is a set \(X\) endowed with a relation \(\leq\) on \(X^2\), called a well-order, so that \(\leq\) has the following properties * Transitivity: If \(a \leq b\) and \(b \leq c\) then \(a \leq c\). * Antisymmetry: If \(a \leq b\) and \(b \leq a\), then \(a = b\). * Totality: For all \(a, b\), either \(a \leq b\) or \(b \leq a\). * Well-foundedness: For any \(S \subseteq X\), there is \(s \in S\) so that, for all \(t \in S\), \(s \leq t\). * Reflexivity:...") Tag: Visual edit
- 14:04, 31 August 2023 RhubarbJayde talk contribs created page Set (Created page with "A set is one of the basic objects in the mathematical discipline of set theory, upon which most of apeirology is built. Although there's no technical way to define a set, a set is usually considered a collection of objects, and visualized as a bag. For example, the bag can be empty, yielding the empty set. In some systems of set theory, one has urelements, objects which aren't sets - however, a vast majority of set theory is pure set theory where everything eventuall...") Tag: Visual edit
- 13:58, 31 August 2023 RhubarbJayde talk contribs created page 1 (Created page with "1 is the next natural number after 0. In the system of Von Neumann ordinals and Zermelo's formalization of the natural numbers, it is represented by the set \(0+1 = \{\{\}\}\), while in the logical formalization of natural numbers it is identified with the proper class of singletons. Also, as a Church numeral, it is identified with the lambda calculus expression \(\lambda f. \lambda x. f(x)\). 1 is the least Additive...") Tag: Visual edit
- 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