Gödel's incompleteness theorem applies to formal languages with countable alphabets. So it does not rule out the possibility that one might be able to prove 'everything' in a formal system with an uncountable alphabet OR expand the alphabet to account for new variables*. — Shawn
What Gödel proved was a truth that can only be seen in a formal system such as Peano Arithmetic. — Shawn
The rules governing formal systems do not apply to informal systems, — Shawn
where pretty much any instance of applying a new rule to an informal system, can be integrated into the system itself.[/quotes]
I don't know what that is supposed to mean.
— Shawn
There is no incompleteness in nature — Shawn
Einstein had discussions with Gödel about how singularities and indeterminism could partake in physics and nature, which made him think nature was "incomplete," for lack of a better word). — Shawn
until greater understanding is attained about the relation between logic and mathematics, which was an aspiration of many mathematicians and logicians during the war period, which Gödel had negated with his Incompleteness Theorems — Shawn
if we can get past the conclusions of Gödel's Incompleteness Theorem's. — Shawn
Gödel's incompleteness theorem applies to formal languages with countable alphabets. — Shawn
So it does not rule out the possibility that one might be able to prove 'everything' in a formal system with an uncountable alphabet — Shawn
OR expand the alphabet to account for new variables*. — Shawn
No, Godel proved a meta-theorem regarding formal systems of a certain kind, including PA. The proof of that metatheorem can be done in various formal systems or done in ordinary informal mathematics, as is the case with Godel's original proof. — TonesInDeepFreeze
Moreover, the proof make use of only finitistic, intuitionistically acceptable principles. — TonesInDeepFreeze
Incompleteness is a property of certain formal systems. I don't know what it means to say that nature is or is not complete. — TonesInDeepFreeze
Einstein had discussions with Gödel about how singularities and indeterminism could partake in physics and nature, which made him think nature was "incomplete," for lack of a better word.
— Shawn
Who used the word, for lack of a better one? And what is your source? — TonesInDeepFreeze
What "conclusions" do you have in mind? The incompleteness theorem is a mathematical theorem with mathematical corollaries. Of course, some people make philosophical inferences based on the theorem, but such inferences are not of the mathematical theorem itself. — TonesInDeepFreeze
The defintion of 'formal language' includes that the language is countable. — TonesInDeepFreeze
There is no incompleteness in nature — Shawn
This would imply that for every true statement about the physical universe, there exists a proof that can be derived from the supposedly canonical and categorical but unknown theory of the physical universe. — Tarskian
I do not like stating this in formal systems like Peano Arithmetic; but, rather in decidability. By framing the question in terms of decidability, we do away with the problem of the inherent limitations of human intuition devising formal systems. This is a question model theorists might be able to prove, in my opinion. — Shawn
https://en.wikipedia.org/wiki/Decidability_(logic)
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not.
The problem is logic itself: — Tarskian
I always thought the solution to the problem of insufficient logic needed to compute certain undecidable problems is solved by appealing to greater complexity class sizes, which avoids the inherent limitations of a formal system which is incapable of decidability given its inherent limitations. — Shawn
An undecidable problem in logic is undecidable irrespective of how much time or memory you throw at the problem. The P versus NP issue only applies to problems that are at least logically decidable. — Tarskian
I think you are referencing Rosner. — Shawn
it seems that it relies on a contradiction performed in the system — Shawn
the liar paradox, which Rosner utilizes. — Shawn
II hope that I might have gotten the gist of it. — Shawn
Moreover, the proof make use of only finitistic, intuitionistically acceptable principles.
— TonesInDeepFreeze
I am not denying the logical validity of Godel's Incompleteness Theorems. — Shawn
What Gödel proved was a truth that can only be seen in a formal system such as Peano Arithmetic. — Shawn
the impact or the conclusions mathematicians reached at the time were too profound to the field of mathematics. — Shawn
Well, at the time, many physicists were of the opinion that mathematics governs physics. So, I hypothesized that Einstein was aware of Godel's Incompleteness Theorems, since they had many discussions between each-other. I can only imagine that Einstein was interested in Godel's thoughts about physics according to Godel given his Incompleteness Theorems. — Shawn
What "conclusions" do you have in mind? The incompleteness theorem is a mathematical theorem with mathematical corollaries. Of course, some people make philosophical inferences based on the theorem, but such inferences are not of the mathematical theorem itself.
— TonesInDeepFreeze
Sure; well, I won't comment on the effect Godel had on logic and mathematics. There are intellectuals that still ponder about it to this day, since this is a fundamental problem of mathematics at the time when Godel made his discovery. — Shawn
The defintion of 'formal language' includes that the language is countable.
— TonesInDeepFreeze
There are potentially infinitely countable alphabets that could allow one to continuously expand the alphabet by including new terms in the formal system itself according to the principles of the formal system itself. — Shawn
Maybe model theory can actually simulate reality with this possibility in mind. — Shawn
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