User contributions for RhubarbJayde
A user with 299 edits. Account created on 28 August 2023.
1 September 2023
- 13:2513:25, 1 September 2023 diff hist +1,911 N 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
- 13:2213:22, 1 September 2023 diff hist +251 Weakly compact cardinal No edit summary current Tag: Visual edit
- 13:1313:13, 1 September 2023 diff hist +294 Axiom of choice No edit summary Tag: Visual edit
- 13:1013:10, 1 September 2023 diff hist +2,667 N 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
- 13:0713:07, 1 September 2023 diff hist +142 Cartesian product No edit summary current Tag: Visual edit
- 12:5812:58, 1 September 2023 diff hist +4 Kripke-Platek set theory No edit summary current Tag: Visual edit
- 12:5712:57, 1 September 2023 diff hist +8 ZFC No edit summary current Tag: Visual edit
- 12:5412:54, 1 September 2023 diff hist +338 Disjoint union No edit summary Tag: Visual edit
- 12:5212:52, 1 September 2023 diff hist +748 N 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
- 12:4512:45, 1 September 2023 diff hist +768 N 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
- 12:4312:43, 1 September 2023 diff hist +60 Well-ordered set No edit summary Tag: Visual edit
- 12:3912:39, 1 September 2023 diff hist +818 N 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
- 12:2712:27, 1 September 2023 diff hist +2,552 N 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
- 11:4111:41, 1 September 2023 diff hist 0 Aleph 0 No edit summary Tag: Visual edit
- 11:4111:41, 1 September 2023 diff hist +28 Aleph 0 No edit summary Tag: Visual edit
- 11:4111:41, 1 September 2023 diff hist +253 N 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
- 11:3911:39, 1 September 2023 diff hist +31 Taranovsky's ordinal notations No edit summary Tag: Visual edit
- 11:2811:28, 1 September 2023 diff hist +2,683 N 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
- 11:0611:06, 1 September 2023 diff hist +54 N Indescribable cardinal Redirected page to Reflection principle#Alternate meaning current Tags: New redirect Visual edit
- 11:0511:05, 1 September 2023 diff hist +1,056 Cardinal No edit summary Tag: Visual edit
31 August 2023
- 15:1315:13, 31 August 2023 diff hist +600 N 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..." current Tag: Visual edit
- 15:1015:10, 31 August 2023 diff hist +622 N 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
- 15:0515:05, 31 August 2023 diff hist +1,432 N 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
- 14:5214:52, 31 August 2023 diff hist +563 N 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
- 14:4514:45, 31 August 2023 diff hist +21 Absolute infinity No edit summary Tag: Visual edit
- 14:4514:45, 31 August 2023 diff hist +1,093 N 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
- 14:3914:39, 31 August 2023 diff hist +498 N 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..." current Tag: Visual edit
- 14:3314:33, 31 August 2023 diff hist +2 Sharp No edit summary current Tag: Visual edit
- 14:3214:32, 31 August 2023 diff hist −10 Sharp No edit summary Tag: Visual edit
- 14:3114:31, 31 August 2023 diff hist +940 Well-ordered set No edit summary Tag: Visual edit
- 14:2414:24, 31 August 2023 diff hist −14 Sharp Ignore me miserably failing Tag: Visual edit
- 14:2314:23, 31 August 2023 diff hist +38 Sharp No edit summary Tag: Visual edit
- 14:2314:23, 31 August 2023 diff hist 0 Sharp No edit summary Tag: Visual edit
- 14:2114:21, 31 August 2023 diff hist +21 Ordinal No edit summary Tag: Visual edit
- 14:2114:21, 31 August 2023 diff hist +1,831 N 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
- 14:0414:04, 31 August 2023 diff hist +955 N 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
- 14:0114:01, 31 August 2023 diff hist +96 0 No edit summary Tag: Visual edit
- 14:0014:00, 31 August 2023 diff hist +88 1 No edit summary current Tag: Visual edit
- 13:5813:58, 31 August 2023 diff hist +645 N 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
- 13:5513:55, 31 August 2023 diff hist +10 Empty set No edit summary current Tag: Visual edit
- 13:5413:54, 31 August 2023 diff hist +674 N 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
- 13:4713:47, 31 August 2023 diff hist 0 Sharp No edit summary Tag: Visual edit
- 13:4613:46, 31 August 2023 diff hist +2,119 N 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
- 13:3313:33, 31 August 2023 diff hist +130 Zero sharp No edit summary Tag: Visual edit
- 13:2713:27, 31 August 2023 diff hist +491 Axiom of determinacy No edit summary Tag: Visual edit
- 13:2113:21, 31 August 2023 diff hist +2,074 N 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
- 12:4912:49, 31 August 2023 diff hist +38 N Buchholz's ordinal collapsing function Redirected page to Buchholz's psi-functions current Tags: New redirect Visual edit
- 12:4812:48, 31 August 2023 diff hist +23 Pair sequence system No edit summary current Tag: Visual edit
- 12:4812:48, 31 August 2023 diff hist +41 N Ordinal collapsing functions Redirected page to Ordinal collapsing function current Tags: New redirect Visual edit
- 12:4612:46, 31 August 2023 diff hist +22 N Von Neumann cardinal assignment Redirected page to Cardinal current Tags: New redirect Visual edit