Which Godel statements are seen to be true by humans?

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SUMMARY

The discussion centers on the philosophical implications of Gödel statements as articulated by Roger Penrose in his works, particularly "The Emperor's New Mind" and "Shadows of the Mind." Penrose argues that while Turing machines cannot enumerate all Gödel sentences, humans can intuitively recognize their truth. The standard Gödel sentence is typically considered true within standard arithmetic, but alternative interpretations exist. Critics, including Solomon Feferman, highlight logical flaws in Penrose's reasoning, coining the term "Penrose-Lucas fallacy" to describe these errors.

PREREQUISITES
  • Understanding of Gödel's incompleteness theorems
  • Familiarity with Turing machines and their limitations
  • Knowledge of standard and nonstandard models of arithmetic
  • Basic grasp of philosophical arguments in artificial intelligence
NEXT STEPS
  • Read Solomon Feferman's critique of Penrose's arguments
  • Explore the implications of Gödel's theorems on artificial intelligence
  • Investigate nonstandard models of arithmetic and their interpretations
  • Study the philosophical debates surrounding the Penrose-Lucas fallacy
USEFUL FOR

Philosophers, computer scientists, mathematicians, and anyone interested in the intersection of artificial intelligence and mathematical logic will benefit from this discussion.

Tosh5457
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My questions arose when reading an article about artificial intelligence, and the argument of Penrose that says that humans can see the truthfulness of statements that machines cannot. But it doesn't say what those "Godel sentences" are.

http://en.wikipedia.org/wiki/Philosophy_of_artificial_intelligence#Lucas.2C_Penrose_and_G.C3.B6del

"Roger Penrose expanded on this argument in his 1989 book The Emperor's New Mind and his 1994 book Shadows of the Mind. He presents a complex argument, and there are many details that need to be considered carefully. However the essence of it is that

  1. It is impossible for a Turing machine to enumerate all possible Godel sentences. Such a program will always have a Godel sentence derivable from its program which it can never discover
  2. Humans have no problem discovering these sentences and seeing the truth of them"
 
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The Goedel sentences can be considered either true or false. In the standard model of arithmetic, the usual Goedel sentence is usually considered true. However, it is possible to consider it false and get nonstandard models of arithmetic.

http://boolesrings.org/victoriagitman/2012/01/18/an-iphone-app-for-a-nonstandard-model-of-number-theory/
 
Beware of Penrose's arguments: Penrose is an excellent physicist, but not much of a logician. You can find Solomon Feferman's excellent article that points out the logical errors in Penrose's arguments either by googling "Feferman Penrose", or go to math.stanford.edu/~feferman/papers/penrose.pdf . In fact, because Penrose's arguments are nothing more than those of Lucas (which were wrong) with a few smokescreens (pretending to "correct" the errors of Lucas, but not quite doing so), there is the standard term: the Penrose-Lucas fallacy (which you can also google). There are much better sources for reading both about the logical points as well as for the meta-mathematical implications.