Representation of the ordinal numbers up to . Each turn of the spiral represents one power of . Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor).
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.[1]
Transfiniteinduction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its...
importance of well-ordering is justified by the possibility of applying transfiniteinduction, which says, essentially, that any property that passes on from...
dyadic fractions; a wider universe is reachable given some form of transfiniteinduction. There is a generation S0 = { 0 }, in which 0 consists of the single...
Transfinite may refer to: Transfinite number, a number larger than all finite numbers, yet not absolutely infinite Transfiniteinduction, an extension...
as the rank parameter in the construction, and the integrity of transfiniteinduction, by which both the ordinal numbers and the von Neumann universe...
{\displaystyle [A]^{k}} is also common, especially when k {\displaystyle k} is a transfinite cardinal number. A set A is a subset of B if and only if their intersection...
Morgan Teleological argument – Argument for the existence of God Transfiniteinduction – Mathematical concept Turtle Island (Native American folklore) –...
a3, a5, ...⟩). Process all possible strategies of S1 and S2 with transfiniteinduction on α. For all sequences that are not in A or B after that, decide...
Infinite descending chain Limit ordinal Successor ordinal Transfiniteinduction ∈-induction Well-founded set Well-order Power set Projection Quasi-set...
{\displaystyle \mathrm {Ord} } can be defined with no problem. Transfiniteinduction works on stratified statements, which allows one to prove that the...
well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing...
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map...
induction in the same fashion that primitive recursion is related to ordinary induction, or transfinite recursion is related to transfiniteinduction...
{\displaystyle T_{\delta \sigma }=(T_{\delta })_{\sigma }.} Now define by transfiniteinduction a sequence Gm, where m is an ordinal number, in the following manner:...
with Leopold Kronecker, who was philosophically opposed to Cantor's transfinite numbers. Primary literature in English: 1890. "Letter to Keferstein"...
not show transfiniteinduction for such large ordinals. For example, the usual first-order Peano axioms do not prove transfiniteinduction for (or beyond)...
{\displaystyle x} , so is the union y ∪ { y } {\displaystyle y\cup \{y\}} . Transfiniteinduction can be used to show each ordinal α {\displaystyle \alpha } is in...
The process involves defining, for each countable ordinal, via transfiniteinduction, a set by "throwing in" all possible countable unions and complements...
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