In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precursor to second-order arithmetic that involves third-order parameters was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. The standard axiomatization of second-order arithmetic is denoted by Z2. Second-order arithmetic includes, but is significantly stronger than, its first-order counterpart Peano arithmetic. Unlike Peano arithmetic, second-order arithmetic allows quantification over sets of natural numbers as well as numbers themselves. Because real numbers can be represented as (infinite) sets of natural numbers in well-known ways, and because second-order arithmetic allows quantification over such sets, it is possible to formalize the real numbers in second-order arithmetic. For this reason, second-order arithmetic is sometimes called "analysis" (Sieg 2013, p. 291). Second-order arithmetic can also be seen as a weak version of set theory in which every element is either a natural number or a set of natural numbers. Although it is much weaker than Zermelo–Fraenkel set theory, second-order arithmetic can prove essentially all of the results of classical mathematics expressible in its language. A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom of which is a theorem of full second-order arithmetic (Z2). Such subsystems are essential to reverse mathematics, a research program investigating how much of classical mathematics can be derived in certain weak subsystems of varying strength. Much of core mathematics can be formalized in these weak subsystems, some of which are defined below. Reverse mathematics also clarifies the extent and manner in which classical mathematics is nonconstructive. (Wikipedia).
Reduction of Order - Linear Second Order Homogeneous Differential Equations Part 2
This video explains how to apply the method of reduction of order to solve a linear second order homogeneous differential equations. Site: http://mathispower4u
From playlist Second Order Differential Equations: Reduction of Order
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Visit http://ilectureonline.com for more math and science lectures! In this video I will introduce 2nd order linear homogeneous and non-homogeneous differential equations. Next video in the series can be seen at: http://youtu.be/aA4TNJKvFCQ
From playlist DIFFERENTIAL EQUATIONS 9 - 2nd ORDER INTRODUCTION
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Visit http://ilectureonline.com for more math and science lectures! In this video I will verview 2nd order differential equations, and the difference between 2nd order linear inhomogeneous and homogeneous differential equations with constant coefficients. Next video can be seen at: https
From playlist DIFFERENTIAL EQUATIONS 11 - 2nd ORDER, A COMPLETE OVERVIEW
2nd Order Differential Equation The Characteristic Equation
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From playlist Mathematical Physics I Uploads
The method of determining eigenvalues as part of calculating the sets of solutions to a linear system of ordinary first-order differential equations.
From playlist A Second Course in Differential Equations
Reduction of Order - Linear Second Order Homogeneous Differential Equations Part 1
This video explains how to apply the method of reduction of order to solve a linear second order homogeneous differential equations. Site: http://mathispower4u
From playlist Second Order Differential Equations: Reduction of Order
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Free ebook http://tinyurl.com/EngMathYT A lecture on how to solve second order (inhomogeneous) differential equations. Plenty of examples are discussed and solved. The ideas are seen in university mathematics and have many applications to physics and engineering.
From playlist A second course in university calculus.
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Solve second order differential equation by substitution, 2nd order differential equation with variable coefficients, Differential equation by substitution, second order linear differential equations, blackpenredpen
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Regularity and non-standard models of arithmetic #PaCE1
Follow-up video: https://youtu.be/7HKnOOvssvs Discussed text, including all links: https://gist.github.com/Nikolaj-K/101c2712dc832dec4991bf568869abc8 Curt's call: https://youtu.be/V93GQaDtv8w Timestamps: 00:00:00 Introduction 00:02:55 Wittgenstein and predicates (optional) 00:11:12 Skolems
From playlist Logic
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From playlist Mathematical Aspects of Computer Science
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From playlist Mathematics
Geometric Sequences (Precalculus - College Algebra 71)
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From playlist Precalculus - College Algebra/Trigonometry
From PSL2 representation rigidity to profinite rigidity - Alan Reid and Ben McReynolds
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From playlist Mathematics
Linear equations in smooth numbers - Lilian Matthiesen
Special Year Research Seminar Topic: Linear equations in smooth numbers Speaker: Lilian Matthiesen Affiliation: KTH Royal Institute of Technology Date: October 18, 2022 A number is called y-smooth if all of its prime factors are bounded above by y. The set of y-smooth numbers below x for
From playlist Mathematics
Arithmetic Sequences and Arithmetic Series - Basic Introduction
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From playlist New Precalculus Video Playlist
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From playlist Summer of Math Exposition 2 videos
Shortcut Reduction of Order - Linear Second Order Homogeneous Differential Equations Part 1
This video explains how to apply the shortcut formula for the method of reduction of order to solve a linear second order homogeneous differential equations. Site: http://mathispower4u.com
From playlist Second Order Differential Equations: Reduction of Order
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From playlist Global Noncommutative Geometry Seminar (Americas)
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A full lesson on an Introduction to Sequences to include Arithmetic and Geometric sequences and patterns. For more in-depth math help check out my catalog of courses. Every course includes over 275 videos of easy to follow and understand math instruction, with fully explained practice pr
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From playlist Further Proof by Mathematical Induction