User contributions for RhubarbJayde
A user with 299 edits. Account created on 28 August 2023.
9 September 2023
- 12:5712:57, 9 September 2023 diff hist −1 Extendible No edit summary Tag: Visual edit
- 12:4912:49, 9 September 2023 diff hist +1,067 N 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
8 September 2023
- 18:3018:30, 8 September 2023 diff hist +2,059 N 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
- 17:3817:38, 8 September 2023 diff hist +2,833 N 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..." current Tag: Visual edit
- 17:1417:14, 8 September 2023 diff hist +8 Filter No edit summary Tag: Visual edit
- 16:0416:04, 8 September 2023 diff hist +1,607 Filter No edit summary Tag: Visual edit
7 September 2023
- 19:1319:13, 7 September 2023 diff hist −18 Filter No edit summary Tag: Visual edit
- 19:1219:12, 7 September 2023 diff hist +2,690 N 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
- 18:4218:42, 7 September 2023 diff hist +20 N Ultrafilter Redirected page to Filter current Tags: New redirect Visual edit
6 September 2023
- 18:1918:19, 6 September 2023 diff hist +10 Inaccessible cardinal No edit summary current Tag: Visual edit
- 18:1818:18, 6 September 2023 diff hist +1,395 N 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
- 18:1118:11, 6 September 2023 diff hist +1,649 Ordinal No edit summary current Tag: Visual edit
4 September 2023
- 11:4511:45, 4 September 2023 diff hist +548 N 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
- 11:2311:23, 4 September 2023 diff hist +1,165 N 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
- 11:1411:14, 4 September 2023 diff hist +350 Countability No edit summary Tag: Visual edit
- 11:0711:07, 4 September 2023 diff hist +4 Axiom of choice No edit summary Tag: Visual edit
3 September 2023
- 18:0718:07, 3 September 2023 diff hist +20 Continuum hypothesis No edit summary Tag: Visual edit
- 18:0618:06, 3 September 2023 diff hist +2,328 N 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
- 17:5217:52, 3 September 2023 diff hist +4 Countability No edit summary Tag: Visual edit
- 17:5217:52, 3 September 2023 diff hist +4 Powerset No edit summary Tag: Visual edit
- 17:5117:51, 3 September 2023 diff hist −173 Inaccessible cardinal No edit summary Tag: Visual edit
- 17:4617:46, 3 September 2023 diff hist +1,334 N 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
- 17:3817:38, 3 September 2023 diff hist +105 Countability No edit summary Tag: Visual edit
- 17:3717:37, 3 September 2023 diff hist +991 Cardinal No edit summary Tag: Visual edit
- 17:3617:36, 3 September 2023 diff hist +29 N Epsilon null Redirected page to Epsilon numbers current Tags: New redirect Visual edit
- 17:3517:35, 3 September 2023 diff hist +21 N Aleph naught Redirected page to Aleph 0 current Tags: New redirect Visual edit
- 17:3517:35, 3 September 2023 diff hist +85 Ordinal No edit summary Tag: Visual edit
- 17:3117:31, 3 September 2023 diff hist +72 Omega No edit summary Tag: Visual edit
- 17:2517:25, 3 September 2023 diff hist +233 Cardinal No edit summary Tag: Visual edit
- 17:2417:24, 3 September 2023 diff hist +2,881 N 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
- 17:1917:19, 3 September 2023 diff hist +3 Additive principal ordinals No edit summary Tag: Visual edit
- 17:1917:19, 3 September 2023 diff hist +21 Omega →Properties Tag: Visual edit
- 17:1717:17, 3 September 2023 diff hist +1,038 N 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
- 16:5316:53, 3 September 2023 diff hist +531 Aleph 0 No edit summary current Tag: Visual edit
- 16:5316:53, 3 September 2023 diff hist +1,204 Ordinal No edit summary Tag: Visual edit
- 16:4816:48, 3 September 2023 diff hist +4 Omega No edit summary Tag: Visual edit
- 16:3516:35, 3 September 2023 diff hist +697 N 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
- 16:3516:35, 3 September 2023 diff hist +328 N 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." current Tag: Visual edit
- 16:2616:26, 3 September 2023 diff hist +21 N Aleph null Redirected page to Aleph 0 current Tags: New redirect Visual edit
- 15:5815:58, 3 September 2023 diff hist +12 Cardinal No edit summary Tag: Visual edit
- 15:5715:57, 3 September 2023 diff hist +114 Cardinal No edit summary Tag: Visual edit
1 September 2023
- 13:5613:56, 1 September 2023 diff hist −174 Template:Disambiguation No edit summary current
- 13:5413:54, 1 September 2023 diff hist −936 Template:Disambiguation No edit summary
- 13:5313:53, 1 September 2023 diff hist +1,293 N 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.</..."
- 13:5113:51, 1 September 2023 diff hist +202 Infinite No edit summary current Tag: Visual edit
- 13:4813:48, 1 September 2023 diff hist +9 Countability No edit summary Tag: Visual edit
- 13:4613:46, 1 September 2023 diff hist −1 User:RhubarbJayde/REL-NPR No edit summary current Tag: Visual edit
- 13:4513:45, 1 September 2023 diff hist +1,426 N 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
- 13:3013:30, 1 September 2023 diff hist +1 Inner model theory No edit summary current
- 13:2913:29, 1 September 2023 diff hist +749 N 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