Goedel Escher Bach
Bach "The Musical Offering" ###########################
Canon - single theme. Vary time, pitch, inversion.
- Three voices - one is free, the other two play.
Fugue - first starts, then second five notes up/down. First countersubjects.
Endlessly rising canon.
Escher
Goedel
Translation of the Epimenides paradox - Epimedenes (a cretan) - "All cretans are liars"
- One step strange loop
- Goedels Incompleteness Theorum
- All consistent axiomatic formulations of number theory include undecidable propositions.
Euclid codified geometry Non Euclidian Geometry -
Lewis Carroll and math problems:
- What exists in Alice in Wonderland mathematically
Theory of sets - George Cantor (1800's)
- Set of paradoxes here regarding this
- Russells Paradox - mosts sets are not members of themselves
- Grellings Paradox - adjectives. Self descriptive (awkwardnesful) and non (ediable)
The following sentence is false The following sentence is true
Principia Mathmatica was about exorcising loops logic, set and number theory.
- Lowest type set - only objects
- Higher Type set - objects or sets of the lowest types
Object language Metalanguage
This analysis on the "sentence statement" deems it meaningless
Meta-mathematics, metalogic.
Russel and Whitehead "Principia Mathmatica" - tried to derive all mathematics from logic.
Hilbert set down a challenge that the system proposed here could be defined as both consistent and complete.
Reasoning methods (e.g. finitistic modes)
Goedels paper demolished Hilberts and Principia Mathematica. It stated that there were irrepareable holes and that no axiomatic system could produce all number theory truths.
MU Puzzle #########
A formal system Axioms of the system
Look at plane geometry (Euclidian) and the axiomatic base it provides.