## A very basic take on Godel's Incompleteness Theorem

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Why is it not equally unnatural for Gödel to assign number values to things like "and", "negation", and "all"?

It's not a question of "natural". It is utterly rigorous though. We define a certain function from the set of symbols into the set of natural numbers. There is nothing suspect about that in the least. Not even suspect in terms of common sense, such as assigning numbers to the players on a sports team, let alone the purely mathematical context of incompleteness. ADDED EDIT: For that matter, we don't begrudge that the computers were using to type these messages code symbols - letters, numerals and characters - as strings of 0s and 1s. And no sense in begrudging a mathematician from encoding symbolic mathematical notation with natural numbers.
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I assume you mean his original paper? Any particular translation?

If I'm not mistaken, the only authoritative translation is the one in 'From Frege To Godel'.
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How did Godel make numbers self-referential?

Forget about the idea that's been presented in this thread, in connection with incompleteness, of numbers referring to themselves.
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knowing a consistent system is incomplete

Just to be clear, some consistent systems are incomplete while others are complete.
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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.

This is clearly true, and it must be frustrating trying to talk to people who can't follow the calculations. Still, I find it odd that it's so difficult to express the basic idea in a non-specialist language.

My interest is philosophical, but the implications of incompleteness for metaphysics seems to be a matter a debate among scholars and there is little agreement.
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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.

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 epistemology and metaphysics, even wider divergence about the theorem.

But again, no matter the philosophical responses, the theorem is rock solid mathematics.

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.
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it must be frustrating trying to talk to people who can't follow the calculations.

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.
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I've followed your math posts for a long time, and I appreciate the care you take to get things right. I'm not a logician, but I have studied math, and I can see you know what you are about. It'd be a digression here, but if you feel like discussing the real numbers (including perhaps constructions or the intuitive foundations thereof ), that might be fun.
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Thank you for that.

I am not an expert, but sometimes I have good (rigorous) understanding of some of the basics, though, over time I've become rusty on some of the more advanced details.

The reals are constructed in set theory usually in one of two ways: As equivalence classes of Cauchy sequences or as Dedekind cuts. Also, there is a way to have reals as actual individual denumerable sequences, but I don't know the details of that, and it is not a common approach.

What did you have in mind about the reals?
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Good to see you back, Tones. :up:
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Thanks, jgill. Good to see you too. Probably only a brief stopover for me though. My lifecoach pandemic response guru (just kidding) tells me that my path to cyberlife pandemic era self-actualization (just kidding) is better charted away from forums.
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What the theorems tells us is that the knowable consistency of our choice of axioms depends at least on us limiting ourself to a less-than-complete set of axioms, it should be intuitive that out of the set of all sound mathematical conclusions there are axioms in addition to those on which these conclusions were built which if coupled with them would yield contradictions and that at a certain complexity it would be impossible to know if it were consistent.

If this is not the essence of the Godel Inc Theorems then I don't know what I'm talking about.
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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.

Okay. I get this.

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.

Okay. But what is an example of G for some system T? .

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.

I get this. I just don;t see it's significance beyond mathematics. Stephen Hawking used to have as essay online titles 'The End of Physics', arguing that incompleteness means physics cannot be completed, but he later took it down. It ought to mean that metaphysics cannot be completed, but I;I've not seen this argued. .

In the philosophy of mathematics and philosophy of computability, there are different diverging perspectives about the theorem.

Amen to this.

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.

I'd half agree. My view is that if mathematicians are unable to work out and clarify the implications of incompleteness for philosophy then philosophers can ignore it, since they're hardly likely to do any better. But I don't want to ignore it. I feel it's important but cannot pin down the reasons. I don't find mathematicians helpful on this issue since they don't seem able to make clear why philosophers should even be interested. Perhaps they needn't be. Do you have an opinion?
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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.

I can see this point. 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.

. .
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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.

But is it important beyond mathematics ? Do people care much whether a Turing machine halts ? ( I find it interesting, but I find constructions of the real numbers interesting. ) It's a bit like quantum mechanics. There's a kind of mystic fog that hangs around it.
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But is it important beyond mathematics ?

I don't know. I wish a mathematician would tell me but they don't seem to know either/ .
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The reals are constructed in set theory usually in one of two ways: As equivalence classes of Cauchy sequences or as Dedekind cuts.

:up:

Yes, I've studied both. I've even developed twists on both.

What did you have in mind about the reals?

I have no doubt about the formal correctness of the various popular constructions. I guess I'm interested in the relatively subjective and philosophical issue of meaning. What do we mean with our formalism ? I feel comfortable with the rational numbers. The reals weird and (historically) controversial. The set of computable and therefore countable reals has measure 0. But these are the ones we can know relatively directly. So R is mostly a black and seamless sea in the darkness.

Let's consider the construction from Cauchy sequences of rationals. We try to imagine a subset of all possible infinite 'streams' of rational numbers. We know we can't enumerate them, right ? But we can enumerate Q and all finite sequences in Q.

Note that this is more about feeling than anything technical. It's about motives for adopting this or that formal system. Does the system scratch the itch ? Capture an intuition of magnitude or continuous flow, for instance ?
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I don;t know. I wish a mathematician would tell me but they don't seem to know either/ .

I'd wager that most of 'em would say not so much. Why aren't people interested in the difference between R and Q ? That is so much more relevant, in some sense. We created a root for 2.

https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers
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I'd wager that most of 'em would say not so much.

Yrs, but should I believe them? They simply don't seem to know.
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But what is an example of G for some system T?

The one constructed in the proof. When you read the proof, you see the G that is constructed.

I just don;t see it's significance beyond mathematics.

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.

It ought to mean that metaphysics cannot be completed

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.

My view is that if mathematicians are unable to work out and clarify the implications of incompleteness for philosophy then philosophers can ignore it

Of course. Anyone, including advanced mathematicians, may ignore it and still work productively.

I wish a mathematician would tell me but they don't seem to know either/ .

Read 'Godel's Theorem' by Torkel Franzen. He discusses your question in layman's terms.
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There's a kind of mystic fog that hangs around it.

I don't see it as mystic or foggy.
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R is mostly a black and seamless sea in the darkness.

I don't have anything immediate to say about people's subjective impressions of mathematics.

we can enumerate Q and all finite sequences in Q.

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.
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The one constructed in the proof. When you read the proof, you see the G that is constructed.

But I can't follow the proof. Does this mean I can never know an example of G?

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.

I'd say computing is mathematics, so I'm not sure this is a valid example. .

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.

I can see no point in a metaphysical theory that is not a formal system. It wouldn't be a theory in the usual sense. Thus for me incompleteness ought to be important. I'd like to say that there is only one such theory that escapes incompleteness. but just can't understand the issue well enough to do so.

Of course. Anyone, including advanced mathematicians, may ignore it and still work productively.

Not in metaphysics, or so I believe.

Read 'Godel's Theorem' by Torkel Franzen. He discusses your question in layman's terms.
Thanks. I'll check the reviews but am not hopeful. . ,
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I don't see it as mystic or foggy.

Not you, of course, because you care about and have studied the details.

I'm talking about how math and physics can be (and often is ) taken from the outside in various metaphysical mystical ways.
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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.

I know all that, already said it, spent years writing proofs for professors. Not asking random internet guy about the basics of analysis. Tried to ask you about 'subjective' (maybe philosophical) responses to all the symbols that swim like fish in those textbooks you mentioned. You gave a disappointing response, like you are deaf and mute to anything that isn't mere chatbot correctness. I have loved math as a meaningful 'science' of form(s) with some intuitive validity. I care about various formalisms only because they strive to mean something, capture something beyond them. The continuum is a endlessly fascinating beast that great thinkers have wrestled with for centuries. I don't know if you know or care much about mathematical history, but I love the drama. But I'll save that for others who aren't satisfied with the relatively trivial (however difficult at times ) syntactical part.
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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

But I'll save that for others who aren't satisfied with the relatively trivial (however difficult at times ) syntactical part.

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 :cool:
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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
:up:

It's fascinating how different personalities prefer this or that aspect of mathematics. I got into it later than most (~30). I was suddenly grabbed by the beauty of it. It is like granite or marble, in which one could sculpt. So for me it 'has' to have meaning and shine for the intuition, because it's obvious that there's a teeming infinity of possible random formal systems that no one cares about. I

've studied a bit of theoretical computer science, and it affected me like studying Darwin affected me. It matters whether I can enumerate a set or not, matters to my intuition. 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.
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Many thanks for the book recommendation. It looks like a rare book and a much needed one.
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I was suddenly grabbed by the beauty of it. It is like granite or marble

I like that. I was a rock climber.

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.

I love those symbols. They are an integral part of exploring an obscure path of conceptualization and discovery. It's all about exploration.
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I love those symbols. They are an integral part of exploring an obscure path of conceptualization and discovery. It's all about exploration.

Oh I love those symbols too, because they speak to and for me. I think in those symbols. I truly love epsilontics. My understanding is that Weierstrass gave us that gift.

And I never go that long without obsessing over the real numbers, and I mean the weird basics of the system. The constructions and whether or not they really satisfy. It's arguably better to just take the axioms as describing ideal entities indirectly. Or at least I find nested equivalence classes less than convincing. Though a single stream of rational numbers is reasonable.
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