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Ed Quanta
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http://www.math.hawaii.edu/~dale/godel/godel.html
Does this really prove it? Or is this just a rough sketch of a proof?
Does this really prove it? Or is this just a rough sketch of a proof?
http://www.math.hawaii.edu/~dale/godel/godel.html Liar Paradox. "Truth" for English sentences is not definable in English.
Proof. Suppose it is. Then so is its complement "False".
Let s be the sentence "This sentence is false" .
Since the phrase "This sentence" refers to s, we have
s iff "This sentence is false" iff "s is false" iff not s.
A contradiction.
Philocrat said:Well, the phrase "This sentence.." is metaphysically, epistemolically, and semantically doomed. For a start, it is Referentially Empty as it is semantically devoid of referents! It is referring neither to itself nor to anything esle in the world. For it to contain and mean something it must contain or refer to something concrete such as another self-standing sentence within it or outside it.
http://www.math.hawaii.edu/~dale/godel/godel.html There is a weak theological parallel in the Problem of Evil:
God doesn't exist since an ultimate ruler must be responsible for all things but a perfectly just being wouldn't be responsible for evil acts.
Tom Mattson said:Well Goedel didn't see it that way, but fine. If you prefer the paradox can be rephrased thusly:
The sentence below is false.
The sentence above is true.
gravenewworld said:I wouldn't recommend that you read On Formally Undeciable Systems By Godel. Although the proof is indeed brilliant, Godel himself is not all that good of a writer and people have cleaned up his proof to make it much shorter and much more refined. Intermediate logic books usually have a proof for the Godel theorems. To tell you the truth, you won't be able to just pick up a book on introductory logic and expect to understand his proof. It takes a lot of work. Become very familiar with first order logic, recursive sets, relations, and funcitons, godel numbering, arithmetization of syntax, metalogic, arithmetical definability before you try to tackle his proof. If you aren't looking to technically understand his proof, but just the result in a layman's sense there are plenty of books out there that will explain it in good plain old English such as Godel, Escher, and Bach: The Eternal Golden Braid. I am currently doing an independent study on logic and I am almost at the point where I can begin the chapter on Indefiniability, undecidability, and incompleteness (which proves Godel's Theorems) and I will admit this stuff is TOUGH. This is by far the most challenging material I have ever done. Recursion theory and arithmetization is very difficult to understand at times at the more advanced levels.
Well, the phrase "This sentence.." is metaphysically, epistemolically, and semantically doomed.
I don't get the part about S ignoring its input. If S ignores its input, how can it accept any input? You mean it passes its input to the verifier and accepts if the verifier accepts?Hurkyl said:Then to construct the sentence that 'says' "I am not provable", one designs an algorithm S that:
(1) Ignores its input.
(2) Constructs a sentence P (in the language of number theory) that is true if and only if S does not accept the string "0". (Involves the recursion theorem)
(3) For every possible string of symbols:
(3a) Run the proof verifier to see if it's a proof of P.
(3b) If a proof of P is found, accept the input.
Philocrat said:Well, the phrase "This sentence.." is metaphysically, epistemolically, and semantically doomed. For a start, it is Referentially Empty as it is semantically devoid of referents! It is referring neither to itself nor to anything esle in the world. For it to contain and mean something it must contain or refer to something concrete such as another self-standing sentence within it or outside it.
Godel's First Incompleteness Theorem is a mathematical theorem that states that in any consistent formal system that is powerful enough to represent basic arithmetic, there will always be statements that are true but cannot be proven within the system itself.
An adequate proof of Godel's First Incompleteness Theorem must follow the logical steps and assumptions outlined by Godel, and must also be accepted and validated by the mathematical community.
Kurt Godel, a mathematician, first discovered the First Incompleteness Theorem in 1931 while attempting to find a way to prove the consistency of mathematical systems. He published his findings in his famous paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems".
Yes, Godel's First Incompleteness Theorem applies to all consistent formal systems that are powerful enough to represent basic arithmetic. This includes systems such as Peano arithmetic, Zermelo-Fraenkel set theory, and many others.
Godel's First Incompleteness Theorem has significant implications in mathematics and philosophy. It shows that there are inherent limitations in formal systems, and that there will always be statements that are true but cannot be proven within those systems. This has also led to further research and developments in logic and the foundations of mathematics.