Not all statements in a given language can be given a truth value, in that they don't refer to anything that allows to determine a truth value — leo
If it can't be given a truth value, its not a statement would be a simple rule to adopt. — Devans99
We can say things such as "This sound is purple" or "This smell is true", but they don't refer to anything in the range of what we experience, or at least I can't form a mental image of whatever these statements may refer to — leo
Like the paradoxes in the theory of relativity, they are a consequence of the postulates at the basis of the theory, we can choose to ignore them and just "shut up and calculate" and make predictions that fit somewhat with observations, or we can change the framework (change the theory, pick different postulates) so that the paradoxes disappear while making similar observable predictions, in the end it depends whether we're looking for mathematical 'elegance' with symmetries and so on or if we're looking for intuitive simplicity. I'm a bit like you on this, I prefer intuitive simplicity that can be grasped by many over mathematical elegance that leads to complexity, paradoxes and confusion — leo
So in the case of the Godel statement, ‘this statement is not provable’… means 'it is not provable that this is a statement'. If you can’t prove its a statement then you can even start to prove it. — Devans99
The so-called "Liar's paradox" seems quite silly — Leontiskos
I agree it's not much use to spend much time pondering about them
— leo
Me too. — Leontiskos
a statement declares a fact; it does not in addition instantiate that fact to a given truth value. — Devans99
I believe Godel's objections would go away though I need to look at that conjecture further — Devans99
1. this is a statement
2. and it is false
So 2 says 1 is false. IE it is not a statement. — Devans99
So in the case of the Godel statement, ‘this statement is not provable’… means 'it is not provable that this is a statement'. — Devans99
If you can’t prove its a statement then you can even start to prove it. — Devans99
Not all statements in a given language can be given a truth value — leo
These self-denying statements are acceptable according to formal logic and they lead to Godel's Incompleteness Theorems. — Devans99
they are a consequence of the postulates — leo
If we remove this class of malformed, contradictory statements then these limitations do not apply any more. — Devans99
'This statement is not provable' means:
1) this is a statement
2) this is not provable — Devans99
So what exactly did Godel add to our body of knowledge? — Gregory
You can perfectly know the construction logic of a system but that does still not allow you to know its complete truth. So, even if we manage to figure out the perfect theory of the physical universe, we will still not be able to predict most of its facts. — Tarskian
That is ridiculously overbroad and vague. — TonesInDeepFreeze
https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics
What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.
Unlike your post, that quote seems at least fairly clear and doesn't make an overbroad mischaracterization of incompleteness. — TonesInDeepFreeze
http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf
There are more true but unprovable statements than we can possibly imagine.
We have come a long way since Gödel. A true but unprovable statement is not some strange, rare phenomenon. In fact, the opposite is correct. A fact that is true and provable is a rare phenomenon. The collection of mathematical facts is very large and what is expressible and true is a small part of it. Furthermore, what is provable is only a small part of those.
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