2019/09/30 by Anton Freund · 1 citation
Computer Science · Mathematics · #Advanced Topology and Set Theory #Axiom #Axiom of choice #Combinatorics #Computability, Logic, AI Algorithms #Computer science #Discrete mathematics #Equivalence (formal languages) #Equivalence class (music) #Geometry #Logic, Reasoning, and Knowledge #Mathematics #Reflection (computer programming) #Set (abstract data type) #Set theory #Transitive relation #Universal set #Zermelo–Fraenkel set theory #math.LO #msc:03B30 #msc:03E30 #msc:03F05
paper · pdf · doi:10.1090/proc/15103
published in Proceedings of the American Mathematical Society 148(10), 4503-4515 (American Mathematical Society)
openalex created_date 2019/09/12 · openalex publication_date 2020/03/25 · arxiv created 2021/07/06 · arxiv updated 2021/07/07 · openalex updated_date 2026/08/05
We show that induction over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Δ (\mathbb R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -definable well-founded classes is equivalent to the reflection principle which asserts that any true formula of first order set theory with real parameters holds in some transitive set. The equivalence is proved in primitive recursive set theory (which is weaker than Kripke-Platek set theory) extended by the axiom of dependent choice.