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June 14th, 2017, 06:31 AM   #1
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Godel's undecidability

first point: This topic is about logic but I have found no category in this forum concerning eg logic, set theory atc, so I decided to place it in the category Applied Math.

I have one quick remark (question): Is it so that Eucleid's fifth postulate (about parallels) is the "first" special example of Godel incompleteness theorem? (Becuase that postulate cannot be proved and also its negation cannot be proved.) This was known in 19. century. And if so why it was later believed (by David Hilbert eg) that in every theory every formula or its negation can be proved?
Thank you for any comments.
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June 14th, 2017, 06:43 AM   #2
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One of the consequences of Gödel is that no axiom system can ever be complete. Until he came up with this proof, it was thought that either:
  1. we now had complete axiom systems for the three geometries; or
  2. that in time we would achieve that aim.
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June 14th, 2017, 10:38 AM   #3
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Not an expert on these matters but ...

*Euclidean geometry is incomplete, but it's not an example of Gödel incompleteness, which applies only to systems complex enough to define the arithmetic of the natural numbers.

* Tarski's axiomitization of geometry is complete.

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