Why is it not equally unnatural for Gödel to assign number values to things like "and", "negation", and "all"? — Patterner
I assume you mean his original paper? Any particular translation? — Patterner
How did Godel make numbers self-referential? — Patterner
knowing a consistent system is incomplete — jgill
Sure there is. Just read a textbook on the subject. But if you're not interested in doing that, then indeed there's little hope that you'll understand the subject.
This is a technical subject. It requires study. Just as, say, microbiology is a technical subject and you can't expect to understand results in microbiology without knowing at least the basics. — TonesInDeepFreeze
it must be frustrating trying to talk to people who can't follow the calculations. — FrancisRay
The basic idea of what the theorem says can be stated roughly in common language:
If T is a consistent theory that expresses basic arithmetic, then there are sentences in the language for T such that neither they nor their negations are provable in T; moreover, either such a sentence is true or its negation is true, so there are true sentences not provable in T. — TonesInDeepFreeze
The basic idea of the proof is not as easy to say in common language, but we have:
For a consistent, arithmetically expressive theory T, we construct a sentence G in the language of T such that G is true if and only if G is not provable in T. Then we prove that G is not provable in T. But this cannot really be understood and be convincing if one doesn't study the actual mathematics of it; otherwise it can seem, at such a roughly simplified level, as nonsense or illegitimate trickery, though it is not, as would be understood when seeing the actual mathematics, not the oversimplified common summary.
Mathematically, there is no legitimate debate about the theorem. It is as rock solid a mathematical proof as any mathematical proof. It can be reduced to methods of finitistic constructive arithmetic.
In the philosophy of mathematics and philosophy of computability, there are different diverging perspectives about the theorem.
In any case, one cannot reasonably philosophize about the theorem without actually understanding it mathematically as a starting point. I wouldn't make claims about the philosophy of mind based on studies about the electrical chemistry of the human brain without first really understanding those studies. Should be the same with metaphysics referring to mathematics.
To be clear, the problem is not people who can't understand the mathematics, but those (not necessarily ones lately in this thread) who refuse (through years and years of their ignorant, confused, and arrogantly prolific disinformational posting) to even read the first page of a textbook on the subject. Such people are a bane and toxic to knowledge and understanding. — TonesInDeepFreeze
But I feel mathematicians are somewhat to blame for not being able to explain why the issue is important beyond mathematics, which is where most people live. — FrancisRay
But is it important beyond mathematics ? — plaque flag
The reals are constructed in set theory usually in one of two ways: As equivalence classes of Cauchy sequences or as Dedekind cuts. — TonesInDeepFreeze
What did you have in mind about the reals? — TonesInDeepFreeze
I don;t know. I wish a mathematician would tell me but they don't seem to know either/ . — FrancisRay
I'd wager that most of 'em would say not so much. — plaque flag
But what is an example of G for some system T? — FrancisRay
I just don;t see it's significance beyond mathematics. — FrancisRay
It ought to mean that metaphysics cannot be completed — FrancisRay
My view is that if mathematicians are unable to work out and clarify the implications of incompleteness for philosophy then philosophers can ignore it — FrancisRay
I wish a mathematician would tell me but they don't seem to know either/ . — FrancisRay
There's a kind of mystic fog that hangs around it. — plaque flag
R is mostly a black and seamless sea in the darkness. — plaque flag
we can enumerate Q and all finite sequences in Q. — plaque flag
The one constructed in the proof. When you read the proof, you see the G that is constructed. — TonesInDeepFreeze
Of course you're free to investigate whatever you like. On the other hand, for example, the undecidability of the halting problem, which is another way of couching incompleteness, has implications for actual computing.
Metaphysics is not usually formulated as a formal system. On the other hand, if you state a formal system, we can see whether it has the attributes to which the incompleteness theorem applies.
Of course. Anyone, including advanced mathematicians, may ignore it and still work productively.
Thanks. I'll check the reviews but am not hopeful. . ,Read 'Godel's Theorem' by Torkel Franzen. He discusses your question in layman's terms.
I don't see it as mystic or foggy. — TonesInDeepFreeze
Q is enumerable and so is the set of finite sequences of members of Q. The set of infinite sequences of members of Q and R are not enumerable. Okay, some things are enumerable and others aren't. — TonesInDeepFreeze
I'd wager that most of 'em would say not so much. — plaque flag
Yrs, but should I believe them? They simply don't seem to know — FrancisRay
But I'll save that for others who aren't satisfied with the relatively trivial (however difficult at times ) syntactical part. — plaque flag
:up:I confess. In my own research I have never cared, being more concerned with the difficult trivia that goes on outside the hallowed halls of Foundations. For instance, I rarely came into contact with transfinite theory — jgill
I was suddenly grabbed by the beauty of it. It is like granite or marble — plaque flag
I'm assuming you also are 'speaking a language' in your work. You have the feeling (I hypothesize) that patterns are being revealed that aren't just computer-checkable patterns in dead symbols. — plaque flag
I love those symbols. They are an integral part of exploring an obscure path of conceptualization and discovery. It's all about exploration. — jgill
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