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Everything about Transfinite induction
Mathematical concept
Transfinite induction is an extension of mathematical induction to ordinal numbers. Its correctness is a theorem of ZF, and relies on the fact that the ordinal numbers are well-ordered, and thus a statement that is not universally true for all ordinals must have a minimal counterexample. In fact, this principle is also true for arbitrary well-ordered sets, but since any well-ordered set can be indexed by ordinals in an order-preserving way, it suffices to establish the principle for ordinals. Wikipedia →
🧭 Start here: a 6-step reading path
- Read the overview: Transfinite inductionA 5-minute orientation
- Skim a recent paper: Induction of Bleb Structures in Lipid Nanoparticle Formulations of mRNA Leads to Improved Transfection PotencyInfluential work since 2022
- Pick up a classic book: Proof-theoretical analyses of transfinite recursion and inductive definitionsThe most-reprinted book on the subject
- Read the landmark paper: Transforming growth factor type beta: rapid induction of fibrosis and angiogenesis in vivo and stimulation of collagen formation in vitro.Cited 2,086 times
- Go sideways: Ordinal numberA closely related topic
- Go sideways: Transfinite recursion theoremA closely related topic
📖 Definitions
Noun An extension of mathematical induction to well-ordered sets of transfinite cardinality, such as sets of ordinal numbers or cardinal numbers.
🗂️ The subject at a glance
🏆 Landmark papers (most cited)
Transforming growth factor type beta: rapid induction of fibrosis and angiogenesis in vivo and stimulation of collagen formation in vitro.
Schema induction and analogical transfer
Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis
Transfinite induction within Peano arithmetic
A NOTE ON PREDICATIVE ORDINAL ANALYSIS I: ITERATED COMPREHENSION AND TRANSFINITE INDUCTION
Induction and transfinite induction in intuitionistic systems
Arithmetic transfinite induction and recursive well-orderings
On Some Analytic Sets Defined by Transfinite Induction
🆕 Influential since 2022
Induction of Bleb Structures in Lipid Nanoparticle Formulations of mRNA Leads to Improved Transfection Potency
Mathematical induction, transfinite induction, and induction over the continuum
Some examples of use of transfinite induction in analysis
RELATIVE LEFTMOST PATH PRINCIPLES AND OMEGA-MODEL REFLECTIONS OF TRANSFINITE INDUCTIONS
📚 Books worth reading
- 📕Proof-theoretical analyses of transfinite recursion and inductive definitionsJan Terlouw · 1986
- 📕The mathematics of infinityTheodore G. Faticoni · 2012
- 📕There's something about Gödel!Francesco Berto · 2009
- 📕There's something about GödelFrancesco Berto · 2009
- 📕There's something about GödelFrancesco Berto · 2009
Ranked by number of editions on Open Library, a rough measure of how lasting a book is.
🔗 Related topics
- Ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite…
- Transfinite recursion theorem In mathematics, the transfinite recursion theorem says a function can be defined using a recursion over a well-ordered set; for example, ℕ but also over…
- Constructible universe In set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular class of sets that can be described entirely…
- Axiom of limitation of size In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation…
- Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for…
- Bourbaki–Witt theorem In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed-point theorem for partially ordered…
- Beth number In mathematics, particularly in set theory, the beth numbers form a certain (unset) sequence of infinite cardinal numbers (also known as transfinite numbers)…
- Well-ordering theorem In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict…