Mathematical logic | Proof theory

Hilbert's program

In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early part of the 20th century, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics. In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system. In his second theorem, he showed that such a system could not prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger with certainty. This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of itself, and therefore could not prove everything else. (Wikipedia).

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Anthony Licata: Hilbert Schemes Lecture 7

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From playlist SMRI Course: Hilbert Schemes

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Joshua Ciappara: Hilbert Schemes Lecture 10

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From playlist SMRI Course: Hilbert Schemes

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Hilbert Curve

This shows a 3d print of a mathematical sculpture I produced using shapeways.com. This model is available at http://shpws.me/2toQ.

From playlist 3D printing

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Anthony Henderson: Hilbert Schemes Lecture 1

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From playlist SMRI Course: Hilbert Schemes

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Emily Cliff: Hilbert Schemes Lecture 6

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From playlist SMRI Course: Hilbert Schemes

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Anthony Henderson: Hilbert Schemes Lecture 9

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From playlist SMRI Course: Hilbert Schemes

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From playlist Math

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Emily Cliff: Hilbert Schemes Lecture 3

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From playlist SMRI Course: Hilbert Schemes

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Anthony Henderson: Hilbert Schemes Lecture 4

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Hilbert Space Techniques in Complex Analysis and Geometry (Lecture 9) by Dror Varolin

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The computational theory of Riemann–Hilbert problems (Lecture 4) by Thomas Trogdon

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Matrix Entanglement by Vaibhav Gautam

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Algebraically closed field | Euclidean geometry | Gödel's incompleteness theorems | Reverse mathematics | David Hilbert | Transfinite induction | Foundations of mathematics | Grundlagen der Mathematik | Formal language | Entscheidungsproblem | Peano axioms | Ordinal number | First-order logic | Zermelo–Fraenkel set theory | Proof theory | Real analysis | Characteristic (algebra) | Mathematics | Gaisi Takeuti | Cantor–Dedekind axiom | Complete theory | Axiom | Mathematical logic | Gentzen's consistency proof | Arithmetic | Analytic geometry