looking only at the sentence itself - the sentence would not have the same force. — Pantagruel
It just kind of begs the question of thinking itself. — theRiddler
I mean isn't it really more so the ego's attachment to it's own thoughts. Thinking has always just been useful, but it's not a miracle worker. We work at this problem like we're going to find some underlying wisdom, but what if there is none? What if it's just the illusion of sense created by consonants and vowels? What's the difference between a paradox and gobbledygook? — theRiddler
This defines the quality of G. "This statement" could be any statement - but if it isn't unprovable, it's not G because we defined it as being unprovable.This statement (G) is unprovable. — TheMadFool
This is always true1. Either G is provable or G is unprovable. — TheMadFool
This is never true. We defined G as unprovable. There may be a statement that looks exactly the same like G; but it's not G because G is per definition unprovable.2. If G is provable then G is true i.e. G is unprovable. — TheMadFool
This is always true because G is defined as being unprovable.3. If G is unprovable then G is true i.e. G is unprovable. — TheMadFool
Yes, we said so in the beginning!5. G is unprovable (4 Taut) — TheMadFool
this assumes something that is impossible. It's an invalid argument. An error in definition.2. If G is provable then G is true i.e. G is unprovable. — TheMadFool
This defines the quality of G. "This statement" could be any statement - but if it isn't unprovable, it's not G because we defined it as being unprovable. — Hermeticus
This is always true — Hermeticus
This is never true. We defined G as unprovable. There may be a statement that looks exactly the same like G; but it's not G because G is per definition unprovable. — Hermeticus
This is always true because G is defined as being unprovable. — Hermeticus
Yes, we said so in the beginning! — Hermeticus
this assumes something that is impossible. It's an invalid argument. An error in definition. — Hermeticus
Since, the Gödel sentence is the liar sentence in some sense can't we do the same thing we did to the liar sentence: take away its status as a proposition? — TheMadFool
Nope, because the English sentence is arguably not a proposition in the sense desired, but Godel's G absolutely is, in that it makes a definite, well-defined statement about a definite, well-defined subject (namely itself), in definite, well-defined terms. — tim wood
As I (along with several other people) explained, this is wrong. L cannot take a truth value since it has no semantic content that can possibly be verified or denied.1. L is true (assume for reductio ad absurdum) — TheMadFool
Yes. This stuff is difficult. I am not an expert in this, but you can become better educated. Both @TonesInDeepFreeze & I gave you several excellent books that can point you in the right direction.I'm out of my depths. — TheMadFool
I guess I'm accusing Gödel of committing the fallacy of false dichotomy.
A penny for your thoughts... — TheMadFool
The Gödel sentence is a spin-off of the liar sentence (This sentence is false). The assumption that we make with the liar sentence is that it's a proposition and therefore that it has a truth value. Reject that assumption and no contradiction results as there are no truth values that come into opposition.
Since, the Gödel sentence is the liar sentence in some sense can't we do the same thing we did to the liar sentence: take away its status as a proposition? — TheMadFool
The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or Church's thesis. We add Priest's dialetheic semantics to ordinary Peano arithmetic PA, to produce a recursively axiomatized formal system PA★ that contains its own truth predicate. Whether one is a dialetheist or not, PA★ is a legitimate, rigorously defined formal system, and one can explore its proof‐theoretic properties. The system is inconsistent (but presumably non‐trivial), and it proves its own Gödel sentence as well as its own soundness.
Although this much is perhaps welcome to the dialetheist, it has some untoward consequences. There are purely arithmetic (indeed, Π0) sentences that are both provable and refutable in PA★. So if the dialetheist maintains that PA★ is sound, then he must hold that there are true contradictions in the most elementary language of arithmetic. Moreover, the thorough dialetheist must hold that there is a number g which both is and is not the code of a derivation of the indicated Gödel sentence of PA★. For the thorough dialetheist, it follows ordinary PA and even Robinson arithmetic are themselves inconsistent theories. I argue that this is a bitter pill for the dialetheist to swallow.
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