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Godels First Incompleteness Theorem

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mmanning02474

The document discusses Godel's incompleteness theorems and Hilbert's program. It provides background on key figures like Hilbert, Godel, Russell and Cantor. It then explains Hilbert's program to formalize all of mathematics and prove its consistency. Godel showed that any theory capable of elementary arithmetic cannot be both consistent and complete. Specifically, for any formal theory T including basic arithmetic truths, T includes a statement of its own consistency if and only if T is inconsistent.

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Uncertainty Trumps Determinism: Godel’s Incompleteness Theorem’s and Hilbert’s Program BAE Technical Seminar
David Hilbert Kurt Godel Bertrand Russell Alfred Whitehead Georg Cantor The Founders of the Modern Mathematical Foundation
[object Object],[object Object],[object Object],Grundlagenkrise der Mathematik
Avoiding Paradox or Embracing it? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Hilbert’s Second Problem (1900) ,[object Object]
Hilbert’s Program (circa 1921) ,[object Object]
Hilbert’s Program ,[object Object],[object Object]
Hilbert’s Program ,[object Object],[object Object]
Godel’s Answer to Hilbert’s Program (1931) ,[object Object],[object Object]
A Simple Language for a Godelian Problem – Is Printable ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Is Printable ,[object Object],[object Object],[object Object],[object Object]
Is Printable ,[object Object]
Is Printable ,[object Object],[object Object]
Is Printable ,[object Object],[object Object],[object Object]
Is Printable ,[object Object],[object Object],[object Object],[object Object]
Unprovable, Unrefutable, Undecidable ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
A Simple Godel Numbering ,[object Object],[object Object],[object Object],[object Object]
An Abstract Language, L , for Godel logic ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object]
[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object]
Godel Numbering and Diagonalization ,[object Object],[object Object],[object Object],[object Object],[object Object]
Why Diagonal ?
Why Diagonal ? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Georg Cantor published his diagonal argument in 1891. It’s a method for demonstrated that there are undenumerable sets of numbers.
Why Diagonal ? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],It’s possible to construct a sequence that is different in the from all other sequences in the matrix, e.g.: s ω+1 = (1, 0, 1, 1, 1, 0, 1, …)
Why Diagonal ? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],It’s possible to construct a sequence that is different in the from all other sequences in the matrix, e.g.: s ω+1 = (1, 0, 1, 1, 1, 0, 1, …)*
The Diagonalization Function ,[object Object],[object Object],[object Object],[object Object]
Theorem 1 ,[object Object],[object Object],[object Object]
Proof of Theorem 1 ,[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object]
The Set T , Expressibility ,[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object]
Decidability ,[object Object],[object Object],[object Object],[object Object]
Incompleteness ,[object Object],[object Object],[object Object]
Refutability ,[object Object],[object Object]
Alternative to Theorem 1 ,[object Object],[object Object]
Expressibility as a Metaphor for Arithmetic ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Correctness versus Consistency ,[object Object],[object Object],[object Object]
Godel’s 2 nd Incompleteness Theorem ,[object Object],[object Object]
How’d it Work out for Hilbert’s Program? ,[object Object],[object Object],[object Object],[object Object]
Godels First Incompleteness Theorem

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Godels First Incompleteness Theorem

  • 1. Uncertainty Trumps Determinism: Godel’s Incompleteness Theorem’s and Hilbert’s Program BAE Technical Seminar
  • 2. David Hilbert Kurt Godel Bertrand Russell Alfred Whitehead Georg Cantor The Founders of the Modern Mathematical Foundation
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