All public logs
Jump to navigation
Jump to search
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).
- 16:25, 26 December 2023 RhubarbJayde talk contribs created page Nothing OCF (Created page with "Nothing OCF is a weak OCF, defined by CatIsFluffy. It is similar to many other OCFs in definition, but omits addition. Therefore, the growth rate is much, much slower. It is believed to correspond to a weak version of Extended Buchholz's function, also defined by omitting addition, and that it catches up to the ordinary version of EBOCF by EBO. However, no proof of either of these claims has been given and...") Tag: Visual edit
- 13:49, 30 September 2023 RhubarbJayde talk contribs created page Set theory (Created page with "Set theory is a branch of mathematics involving the study of sets. These are collections of objects. Set theory is often used as a foundation for mathematics, as many mathematical objects (such as natural numbers, groups, topological spaces, ...) can all be encoded as sets. In pure set theory, the primary objects of study are infinite sets, which include infinite ordinals and cardinals. Set theory is the basis for a lot of apeirol...") Tag: Visual edit
- 14:36, 24 September 2023 RhubarbJayde talk contribs created page Grand hotel (Redirected page to Hilbert's Grand Hotel) Tags: New redirect Visual edit
- 14:36, 24 September 2023 RhubarbJayde talk contribs created page Infinite hotel paradox (Redirected page to Hilbert's Grand Hotel) Tags: New redirect Visual edit
- 14:35, 24 September 2023 RhubarbJayde talk contribs created page Infinite hotel (Redirected page to Hilbert's Grand Hotel) Tags: New redirect Visual edit
- 16:50, 14 September 2023 RhubarbJayde talk contribs created page Ordinal-definable (Redirected page to Ordinal definable) Tags: New redirect Visual edit
- 15:16, 9 September 2023 RhubarbJayde talk contribs created page HOD (Redirected page to Ordinal definable) Tags: New redirect Visual edit
- 15:06, 9 September 2023 RhubarbJayde talk contribs created page Ordinal definable (Created page with "Ordinal definability is a concept which is key in certain aspects of inner model theory. We say a set \(x\) is ordinal-definable iff there is an ordinal \(\beta\) and a collection \(\alpha_1, \alpha_2, \cdots, \alpha_n\) of ordinals so that \(x \in V_\beta\) and there is a first-order formula \(\varphi\) such that, for all \(y \in V_\beta\), \(y = x\) iff \(V_\beta \models \varphi(y, \alpha_1, \alpha_2, \cdots, \alpha_n)\). In other words, it is definable at some ini...") Tag: Visual edit
- 14:47, 9 September 2023 RhubarbJayde talk contribs created page HOD dichotomy (Created page with "The HOD dichotomy theorem is a theorem which shows that HOD, the class of hereditarily ordinal-definable sets, must either be close to or far from the true universe, \(V\). It is formulated in analogy with Jensen's original dichotomy theorem, which asserts that one of the two following holds: * Every uncountable cardinal is inaccessible in \(L\). * For every singular \(\gamma\), \(\gamma\) is singular in \(L\) and \((\gamma^+)^L = \gamma^+\)....") Tag: Visual edit
- 13:59, 9 September 2023 RhubarbJayde talk contribs created page Extender (Created page with "An extender is a collection of ultrafilters which, when combined, are able to coherently code a single elementary embedding (this can be either a nontrivial elementary embedding between a universe and an inner model, or a cofinal elementary embedding between two models of ZFC minus the powerset axiom). Namely, an extender consists of ultrafilters \(E_a\), where \(a\) is a finite set of ordinals, which cohere in a certain way, so that one is able to take th...") Tag: Visual edit
- 13:52, 9 September 2023 RhubarbJayde talk contribs created page Kunen's inconsistency (Created page with "Kunen's inconsistency is a theorem proved by Kenneth Kunen which proves that certain large cardinals, which were previously believed to be natural generalizations of weaker large cardinals, are inconsistent. The proof uses the axiom of choice in a crucial way - it is believed that the large cardinals the theorem rules out may still be able to exist in ZF. It originally proved: "there is no nontrivial elementary embedding \(j: V \to V\), the...") Tag: Visual edit
- 13:37, 9 September 2023 RhubarbJayde talk contribs created page Covering property (Created page with "The covering property is a property of inner models, in particular core models, which is a measure of how "closely they approximate" the real universe \(V\) of sets. Namely, we say an inner model \(N\) has the covering property iff it is able to "cover" uncountable sets of ordinals: for every uncountable set \(X\) of ordinals, there is \(Y \in N\) so that \(X \subset Y\) and \(|X| = |Y|\). Of course, \(V\) has the covering property, and so \(V = L\...") Tag: Visual edit
- 12:49, 9 September 2023 RhubarbJayde talk contribs created page Extendible (Created page with "Extendible cardinals are a powerful large cardinal notion, which can be considered a significant strengthening of weakly compact cardinals, or a combination of supercompact and superstrong cardinals. A cardinal \(\kappa\) is called \(\eta\)-extendible iff, for some \(\theta\), there is some elementary embedding \(j: V_{\kappa+\eta} \to V_\theta\) with critical point \(\kappa\). We say \(\kappa\) is extendible iff it is \(\eta\)-extendible...") Tag: Visual edit
- 18:30, 8 September 2023 RhubarbJayde talk contribs created page Extender model (Created page with "Extender models are inner models, which have similar fine structure to Gödel's \(L\), but which are able to accommodate large cardinals, typically at the level of measurable cardinals and above. Extender models are typically either constructed - where they typically have the form \(L[\vec{E}]\) (here, \(\vec{E}\) is an extender or a coherent sequence of them) and their fine structure analysed - or defined...") Tag: Visual edit
- 17:38, 8 September 2023 RhubarbJayde talk contribs created page Supercompact (Created page with "Supercompact cardinals are a kind of large cardinal with powerful reflection properties. The construction of an inner model accommodating a supercompact cardinal is considered the holy grail of inner model theory, and is extremely difficult. Formally, a cardinal \(\kappa\) is called \(\lambda\)-supercompact iff there is some inner model \(M\) so that \(M^\lambda \subseteq M\) and there exists an elementary embedding \(j: V \to M\) with critical point \(\kappa\) a...") Tag: Visual edit
- 19:12, 7 September 2023 RhubarbJayde talk contribs created page Filter (Created page with "A filter is a particular notion used to define ultraproducts/ultrapowers and various large cardinals above measurable cardinals, although they also have some relation to indescribable and greatly Mahlo cardinals. Formally, a filter on a set \(X\) is a collection \(F\) of subsets of \(X\) satisfying the following conditions: * \(X \in F\). * \(\emptyset \notin F\). * If \(A \in F\) and \(B \in F\) then \(A \cap B \in F\). * If \(A...") Tag: Visual edit
- 18:42, 7 September 2023 RhubarbJayde talk contribs created page Ultrafilter (Redirected page to Filter) Tags: New redirect Visual edit
- 18:18, 6 September 2023 RhubarbJayde talk contribs created page Cofinality (Created page with "The cofinality of an ordinal \(\alpha\), denoted \(\mathrm{cof}(\alpha)\) or \(\mathrm{cf}(\alpha)\), is the least \(\mu\) so that there is some function \(f: \mu \to \alpha\) with unbounded range. For example: * The cofinality of \(0\) is \(0\). * The cofinality of any successor ordinal is \(1\), because the map \(f: 1 \to \alpha+1\) defined by \(f(0) = \alpha\) has unbounded range. * The cofinality of any limit of ordinal is at least \(\omega\): if it's countabl...") Tag: Visual edit
- 11:45, 4 September 2023 RhubarbJayde talk contribs created page Banach-Tarski paradox (Created page with "The Banach-Tarski is a famous, counterintuitive consequence of the axiom of choice. It says that it's possible to decompose a ball in three-dimensional space into separate parts, which can be rearranged to form two balls, each with the same volume as the original. However, an actual such decomposition in the real world is not possible, since the separate parts aren't actual shapes. The proof requires the axiom of choice, and, therefore, the truth of the Banach-Tars...") Tag: Visual edit
- 11:23, 4 September 2023 RhubarbJayde talk contribs created page Hilbert's Grand Hotel (Created page with "Hilbert's Grand Hotel is an analogy and paradox used to explain the notion of countability. One starts off by imagining a hotel, with an infinite amount of rooms, and each is occupied. One's intuition says that it's not possible to fit any more people - however, due to the way infinite bijections work and the fact that they go against common sense, it is possible to still fit many more people. Firstly, if there is a single new guest who wants a room, i...") Tag: Visual edit
- 18:06, 3 September 2023 RhubarbJayde talk contribs created page Continuum hypothesis (Created page with "The continuum hypothesis (CH) is the assertion that there are \(\aleph_1\) many real numbers, or, equivalently, that \(2^{\aleph_0} = \aleph_1\). This is formulated in the context of the axiom of choice, and \(\aleph_1\) is the smallest uncountable cardinal. It is equivalent to the following assertion: "for every \(A \subseteq \mathbb{N}\), either \(A\) and \(\mathbb{N}\) have the same size, or \(A\) and \(\mathbb{R}\) have the same size". In the con...") Tag: Visual edit
- 17:46, 3 September 2023 RhubarbJayde talk contribs created page Aleph fixed point (Created page with "An aleph fixed point, also referred to as an omega fixed point (OFP), is a fixed point of the function \(f(\alpha) = \aleph_\alpha\). In other words, it is a cardinal \(\kappa\) so that \(\aleph_\kappa = \kappa\). The existence of such a \(\kappa\) is guaranteed by the axioms of infinity, powerset and replacement combined with Veblen's fixed point lemma, and therefore it is provable in ZFC. Aleph fixed points are large in that they are unreachable from below via the...") Tag: Visual edit
- 17:36, 3 September 2023 RhubarbJayde talk contribs created page Epsilon null (Redirected page to Epsilon numbers) Tags: New redirect Visual edit
- 17:35, 3 September 2023 RhubarbJayde talk contribs created page Aleph naught (Redirected page to Aleph 0) Tags: New redirect Visual edit
- 17:24, 3 September 2023 RhubarbJayde talk contribs created page Cantor's diagonal argument (Created page with "Cantor's diagonal argument is a method for showing the uncountability of the set of real numbers. It is a proof by contradiction - one assumes that, towards contradiction, there is a bijection from the natural numbers to the real numbers, and then one constructs a real number not in the range of this function, which contradicts surjectivity. It may be rephrased as the assertion that every function from the naturals to the reals is non-surjective,...") Tag: Visual edit
- 17:17, 3 September 2023 RhubarbJayde talk contribs created page Powerset (Created page with "The powerset of a set \(X\), denoted \(\mathcal{P}(X)\), is the collection of all subsets of \(X\). It is easy to see that, for \(X\) finite, the powerset of \(X\) has cardinality \(2^{|X|}\), and the same fact holds when \(X\) is infinite, although this is because cardinal arithmetic was defined to have that behaviour. Cantor's diagonal argument proves that the powerset of the natural numbers, \(\mathcal{P}(\mathbb{N})\), is uncountable. The questi...") Tag: Visual edit
- 16:35, 3 September 2023 RhubarbJayde talk contribs created page Bijection (Created page with "A bijection between two sets, \(X\) and \(Y\), is a "one-to-one pairing" of their elements. Formally, it is a function \(f: X \to Y\) (which can be encoded as a subset of \(X \times Y\)) so that: * Different elements of \(X\) are sent to different elements of \(Y\). * Every element of \(Y\) has some element of \(X\) which is sent to \(Y\). The first property is known as injectivity, or being 1-1, and can be formally be written as \(f(x) = f(y)\) only if \(x = y\). The...") Tag: Visual edit
- 16:35, 3 September 2023 RhubarbJayde talk contribs created page Supertask (Created page with "Supertasks are a hypothetical mechanism which can be used to simulate infinite time Turing machines and may be related to the divergence or convergence of infinite sums. Furthermore, supertasks can be used to draw matchstick diagrams for infinite ordinals.") Tag: Visual edit
- 16:26, 3 September 2023 RhubarbJayde talk contribs created page Aleph null (Redirected page to Aleph 0) Tags: New redirect Visual edit
- 13:53, 1 September 2023 RhubarbJayde talk contribs created page Template:Disambiguation (Created page with "<noinclude> <languages/> </noinclude><templatestyles src="Disambiguation/styles.css"/> <div class="disambiguation metadata plainlinks"> <div class="disambiguation-image">30px|<translate><!--T:1--> disambiguation</translate></div> <div class="disambiguation-text"><translate><!--T:2--> This is a [[<tvar name=cat>Special:MyLanguage/Category:Disambiguation pages</tvar>|disambiguation page]], which lists pages which may be the intended target.</...")
- 13:45, 1 September 2023 RhubarbJayde talk contribs created page User:RhubarbJayde/REL-NPR (Created page with "Relativized nonprojectibility, abbreviated REL-NPR, is a systematic extension of a particular characterisation of nonprojectible ordinals. In general, we say \(\alpha\) is a \(\Gamma\)-cardinal iff, for all \(\gamma < \alpha\), there is no surjection \(\pi: \gamma \to \alpha\) with \(\pi \in \Gamma\). This is motivated by the fact that: #\(\alpha\) is a cardinal iff it is a \(V_{\alpha+1}\)-cardinal. #\(\alpha\) is a gap iff it is an \(L_{\alpha+1}\)-cardinal. #\(\alpha...") Tag: Visual edit
- 13:29, 1 September 2023 RhubarbJayde talk contribs created page Inner model theory (Created page with "Inner model theory is the study of the "fine structure theory" and construction of inner models, proper class-sized models of ZFC which satisfy the existence of large cardinals, covering, the generalized continuum hypothesis, and more. The smallest inner model is \(L\), which arguably has the most and the most detailed fine structure, but it is unable to accomodate measurable cardinals, in the sense that no cardinal, even if it really...") Tag: Visual edit
- 13:25, 1 September 2023 RhubarbJayde talk contribs created page Measurable (Created page with "A measurable cardinal is a certain type of large cardinal which possesses strong properties. It was one of the first large cardinal axioms to be developed, after inaccessible, Mahlo and weakly compact cardinals. Any measurable cardinal is weakly compact, and therefore Mahlo and inaccessible. Furthermore, any measurable cardinal is \(\Pi^2_1\)-indescribable, but not \(\...") Tag: Visual edit
- 13:10, 1 September 2023 RhubarbJayde talk contribs created page Axiom of choice (Created page with "The axiom of choice is a somewhat controversial axiom in set theory, which is included in the axiomatic system of ZFC. It asserts that every family \(X\) of nonempty sets has a choice function, i.e. a map \(f: X \to \bigcup X\) so that, for all nonempty \(x \in X\), \(f(x) \in x\). Essentially, for a collection of sets, it is possible to find a function which chooses one element from each "bag" in this collection of bags. In many cases, the axiom of choice is...") Tag: Visual edit
- 12:52, 1 September 2023 RhubarbJayde talk contribs created page Disjoint union (Created page with "The disjoint union of a collection of sets is a set so that each element of the collection injects into the disjoint union, and so that the collection of images of each element of the collection creates a partition of the disjoint union. For example, the disjoint union of disjoint sets is just their union, while, e.g. the disjoint union of \(\{1,2\}\) and \(\{2,3\}\) could be taken to be \(\{(1,1),(1,2),(2,2),(2,3)\}\). The cardinality of the disjoint union of two sets i...") Tag: Visual edit
- 12:45, 1 September 2023 RhubarbJayde talk contribs created page Cartesian product (Created page with "In set theory, the Cartesian product of two sets, \(X\) and \(Y\), is denoted by \(X \times Y\), and is equal to the set of ordered pairs whose first coordinate is an element of \(X\) and whose second coordinate is an element of \(Y\). Cartesian product is used to give an alternate characterisation of being infinite - that \(X\) is equinumerous with \(X \times X\). A bijection witnessing this is called a pairing function. However, note that, if \(X\) is an infinite [...") Tag: Visual edit
- 12:39, 1 September 2023 RhubarbJayde talk contribs created page Lambda calculus (Created page with "Lambda calculus is a simple system of computation introduced by Alonzo Church. in which functions, and the operations of abstraction and application, act as primitive operations and objects. Natural numbers can be encoded in the lambda calculus using a system known as Church numerals. It's been proven that lambda calculus and Turing machines are able to compute the same processes, which led to the independently formulated Church-Turing thesis that all Turing-complete...") Tag: Visual edit
- 12:27, 1 September 2023 RhubarbJayde talk contribs created page Peano arithmetic (Created page with "Peano arithmetic is a first-order axiomatization of the theory of the natural numbers introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bul...") Tag: Visual edit
- 11:41, 1 September 2023 RhubarbJayde talk contribs created page Aleph 0 (Created page with "Aleph 0, written \(\aleph_0\), is the cardinal corresponding to the cardinality of the natural numbers. As an initial ordinal, it is considered the same as \(\omega\), while it may not be the same while in the absense of the axiom of choice.") Tag: Visual edit
- 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