# Questions tagged [transfinite-induction]

Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.

90 questions
Filter by
Sorted by
Tagged with
27 views

53 views

342 views

### $\omega$th iteration of Cayley-Dickson construction

The first nine Cayley-Dickson hypercomplex algebras are real (1D), complex (2D), quaternion (4D), octonion (8D), sedenion (16D), pathion (32D), chingon (64D), routon (128D), and the voudon (256D). The ...
96 views

### Is it possible to prove Regularity with Transfinite Induction only?

Let us assume that we have only statement of transfinite induction. (And maybe some other well-know axioms) My question: "Is it possible to derive from it a regularity axiom as a theorem?". Some of ...
95 views

### Does this ordinal (built by using ZFC + ordinal many large cardinals to attain yet larger ordinals) have a name?

Consider the following pair of functions: For α an ordinal, let $\nu(\alpha)$ be the least ordinal $\kappa$ such that $(V_{\kappa}, \in)$ is a model of ZFC in which there are $\alpha$-many ...
35 views

### Can I use mathematical induction here?

Given any digraph $D=(V,E)$ such that for every finite subset $S\subseteq V$ there exists a vertex $v_{S}\in V$ capable of reaching any vertex in $S$ via a directed path. Can one deduce there is a ...
122 views

### Choice and the principle of transfinite induction

Does the Axiom of Choice suffice to show Transfinite Induction in ZF? Also, if possible, could you please give some general remarks/insight on (i) worthwhile noting equivalencies or dependencies ...