A very basic take on Godel's Incompleteness Theorem
Suppose a mathematical theory/system T.
G=This sentence is not provable in T
Either G is provable or not provable
1. G is provable. So G is unprovable
2. G is not provable
So, there is G in the theory T
Have I got it right?
G=This sentence is not provable in T
Either G is provable or not provable
1. G is provable. So G is unprovable
2. G is not provable
So, there is G in the theory T
Have I got it right?
Comments (99)
G_1: sentence is true but not provable <-- this was offered in the original post
G_2: sentence is true and provable
G_3: sentence is false but not provable
G_4: sentence is false and provable
If G is false then "this math sentence is false or provable in T".
No.
I know the real proof is very complex but it seems to rely on a modified form of the Liar's paradox. Can you explain where I went wrong. Thanks
G is not provable but it's true. But I'm not really an expert on the fine points so I probably shouldn't have jumped in earlier.
Are you equating "true" with "provable"?
If you read the wikipedia article then this can't be done.
G is provable means it can be proven either true or false, how come "so G is unprovable"?
I don't know.
G = This sentence is not provable
If you can prove G then G is not provable. If you can't prove G then G is not provable
Since, you can either prove or not prove G, it follows that G is not provable.
I think the logic works like that.
You forgot to add that: T is consistent and G is a sentence in the vocabulary of T.
Good to see you Nagase and thanks.
1st one says a computerizable set of rules cannot prove all statements of itself. The 2nd is stronger, saying it cannot even prove itself as consistent.
Well that to me broke down any miracles maths had attained. Now if the plain arithmetic cannot be stated to be consistent then what can? nothing on earth. This is exactly a fatal blow to Hilbert as pioneer supporter of maths.
Finally if nothing is consistent then where should you place your trust on?
Don't give up on math. I think, as Galileo said(?), math is the language of the universe. Consistency may not be be so problematic. It could be that we can't prove every true mathematical statement BUT that may not be required.
Yes it broke Hilbert's program, but it didn't prove that nothing is consistent in maths. For example, classical propositional logic is provably consistent (though we obviously want to use stronger logics than a propositional theory).
That said, you can work without consistency. After all, in response to Godel's Incompleteness Theorems, one can choose to go the inconsistent route and adopt a Paraconsistent Logic. This allows one to develop a mathematical theory on inconsistent foundations, yet because Paraconsistent logics lack the Law of explosion, theory is non-trivial.
This can result in an interesting mathematical theory which proves true (and simultaneously false) some of Frege's Logicism, as logicism becomes provable here and one can prove that the Continuum Hypothesis is false (in this formalism).
I watched a video on it and the reasoning was...
Assume a consistent theory T
G is a statement in T
G = This sentence isn't provable from the axioms of T
Either G is true or false
Assume G is false. That means ''this sentence is provable from the axioms''. But only true statements are provable. That means G is true. But G is false (assumed). A contradiction.
But T is consistent. So it can't be that G is false.
What are we left with? G is true. In other words ther IS a sentence that can't be proved from the axioms.
Have I brushed against the truth or is this the real import of Godel's theorems?
So the following disjunctive conclusion is inevitable: Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified . . . (Gödel 1995: 310).
i.e I am against the scientific trend to treat human as a classical moving machine, especially what inside the brain or thoughts.
T = consistent mathematical theory
G = The mathematical proposition P is not provable from the axioms of T
G is either provable or not provable
1. G is provable. In other words ''the mathematical proposition P is not provable from the axioms'' is true. G is true
2. G is unprovable. G is true
For me the problem starts with 'This sentence is not provable'. This is meaningless. It does not state what is not provable. It would make no more sense to say 'This sentence is provable'.'This sentence' is not a statement and is not even a sentence. It is not provable or unprovable.
I wish someone would explain incompleteness in a way that it seems plausible to non-mathematicians. But explanations always it seems to depend on taking the liar paradox seriously, which try as I might I cannot do. , . . .
Thanks. But at least you understand what we're talking about. I need to find a professor of mathematics to sit down with me and help me understand it. But they aren't easy to come by.
Alternatively, wiki's page on the Halting Problem conveys in the simplest and quickest fashion nearly all of the relevant computational and philosophical implications of Godelian incompleteness without the torture of Godel numbering.
In fact, the popular misconception of Godel's theorem as amounting to saying that "a sentence is 'true', assuming that it is unprovable" is a more apt (but still ultimately misleading) conception of a weaker form of incompleteness that is directly implied by the undecidablity of the Halting problem, namely that there cannot be a sound and complete axiomatization of the natural numbers, since that would imply the existence of a computation on the natural numbers that halts if it doesn't halt.
Godel's actual proof, that doesn't appeal to a model or intended interpretation of the axioms of Peano Arithmetic, is a stronger result since it directly proves syntactical incompleteness without appealing to any notion of soundness. His proof can be intuited geometrically and trivially, as showing that the forest of mathematical proofs that constitutes Peano Arithmetic doesn't describe the background that isn't part of the forest of Peano Arithmetic. Is that really an interesting and deep result?
So the heuristic argument of there being a "true but unprovable statement" that the public use as a heuristic for understanding or remembering Godel's theorem is very misleading. Their catchphrase is more suitable as a moderately misleading heuristic for remembering the proof of the Halting problem (which also doesn't refer to models, but to external assumptions concerning halting conditions, that with much charity might be interpreted as comprising a weak model of something or other).
Many mathematicians and logicians cannot themselves be bothered to master Godel's incompleteness proof, for there isn't any payoff for doing so, and will probably content themselves with a technical understanding of the weaker version I mentioned above that is straightforward to prove and remember, and loses practically nothing.
It would help if I could find a statement that is true but provably undecidable, but I've never seen one. Do you have an example?
Your piece of paper example doesn't do it for me since the two statements make contradictory claims and clearly cannot both be true. We can invent these paradoxical circular arguments easily enough but where do they occur naturally?
.
Yes, that would be because Godel wanted to reinforce the point that his theorems have entirely constructive proofs, that don't appeal to the law of excluded middle or to meta-mathematical assumptions. Unfortunately, that point was lost on Douglas Hofstadter, who only caused misunderstanding of the theories, by linking them to the meta-mathematical concept of self-reference that isn't of any proof-theoretic relevance to the theorems.
Peter Smiths book is much longer because it also covers related topics that developed after Godel that are of relevance to his now ancient theorems, even if the book is badly outdated in certain respects and unhelpfully assumes a background of classical logic for presumably historical reasons.
Quoting tim wood
Well syntactical incompleteness does at least imply that any system of mathematical logic is an unfinishable open-system, such that any sufficiently expressive axiomatic system can be consistently extended in an infinite number of mutually inconsistent ways, leading to an infinite number of conflicting rival interpretations with respect to any theory of arithmetic. A problem with your meta-mathematical heuristic however, aside from the fact that it isn't relevant to Godel's constructive proof of the theorems, is that it assumes the existence of mathematical truth prior to system construction - which might or might not be epistemically appropriate in a given situation.
Essentially, if one accepts Godel's completeness conjecture , namely the conjecture that first order logic is sound and complete with respect to the theorems that are tautologies, i.e. the theories that are satisfied by every model of the logic, then either a theorem is syntactically provable in such a logic or there must exist a model in which the theorem is false. So the heuristic you mention is a potentially useful heuristic for automated theorem proving - if the completeness conjecture is assumed, which is generally appropriate for classical logic, then a theorem can be disproved by merely showing that it's negation is satisfiable in a model of the logic, meaning that one can "observe" the theory to be false, without having to laboriously work "within the system" to produce a syntactical proof of the negated theorem.
Syntactically however, G is merely the fixed-point of an alternating fix-point equation, whose iteration causes an increase in the number of negation signs at the front of the fixed-point. It is structurally similar to a coalgebra representing a potentially infinite list of oscillating boolean values, objects that are normally called "impredicative" rather than "self-referencing" in the literal sense implied by Hofstadter.
Thank you very much for your effort. I'll read that many times, and see if I can understand. But, honestly, the very first step makes no sense to me. I could say automobiles are 1, bicycles are 2, airplanes are 3, etc. Why would I think that makes any sense?
My interest is in the philosophical implications, but there is a variety of views on this and I have no way to distinguish the wheat from the chaff.
2^3 x 3^4 x 5^5 = 2,025,000.
Now I have a relationship between leaves and green.
Why would I think I've done something useful or significant? As the saying (adjusted for inflation) goes, that and $3 will get me a cup of coffee.
And why would I be surprised if something weird, like Gödel's incompleteness theorem, shows up? I would expect any number of inconsistencies and oddities to show up when I'm doing something unnatural like this. Why is it not equally unnatural for Gödel to assign number values to things like "and", "negation", and "all"?
I guess there's no hope for me in this.
You might find this helpful: https://en.wikipedia.org/wiki/G%C3%B6del,_Escher,_Bach
If nothing else it's a great read.
It is, indeed, a great read. At least the few hundred pages I managed to read many years ago.
I never quite grasped what Hofstaedter was trying to say, however, and this may be directly connected with my inability to see what Godel was saying. . .
. .
Yes. Most of us are content with knowing a consistent system is incomplete. And most of us never encounter this situation.
'“Is white” is white.'
'"Preceded by itself in quote marks yields a full sentence” preceded by itself in quote marks yields a full sentence.'
Quoting tim woodThis is what's tripping me up. Take the number 144. It doesn't refer to itself. Although it is the square of 12, it doesn't [I]mean[/I] the square of 12, or refer to squares in general. 144 is also the sum of the the primes 47 and 97, but it doesn't [I]mean[/I] the sum of those two primes, or refer to the idea of even numbers > 2 being the sum of two primes. And it doesn't refer anything else.
Fibonacci numbers are be derived from a certain process. But that doesn't mean they are about that process. How are Godel numbers otherwise? Yes, a given equation can only produce one specific number. But that doesn't mean that number is [I]about[/I] that equation, any more than 25 is [I]about[/I] 4!+1.
What am I not understanding? How did Godel make numbers self-referential?
Quoting tim woodI assume you mean his original paper? Any particular translation?
Thank you very much. I'll try these kindle version.
Possibly no ebook of Davis. We'll see what happens.
Context:
(1) Godel's theorem was strengthened by Rosser. Subsequently, when we say 'Godel's incompleteness theorem' we usually actually mean the Godel-Rosser theorem. And Godel's notation is old-fashioned and hardly used anymore. And we now have much more streamlined expositions of the proof. For those reasons, to best understand the theorem, it is much better not to first study the original proof but rather more modern expositions of it. (Though, of course, after understanding the proof, we can deepen our appreciation and understanding, both mathematically and historically, by also going back to study the original proof).
(2) A proper understanding of Godel-Rosser requires understanding basic symbolic logic and mathematical logic. For that I recommend this sequence of books:
Logic: Techniques of Formal Reasoning - Kalish, Montague and Mar. For basic symbolic logic.
Elements Of Set Theory - Enderton. For understanding the set theoretic rubrics and methods of mathematical logic.
A Mathematical Introduction To Logic - Enderton. For a systematic treatment that works up to and includes a proof of Godel-Rosser.
/
Without at this juncture going into the proof of Godel-Rosser, I will outline (in rough oversimplifications that would need to be made rigorous for a truly proper exposition) some starting points, and pleading that I am rusty on the subject myself. This will at least clear up the matter of numbers and referring and give some crucial overall context.
(3) We have formal languages (I'll just say 'languages'). A language has a set of symbols, a classification of the kinds of symbols, and rules for concatenating symbols into sequences of symbols that are the terms (names, both definite and indefinite) and the formulas (open statements, i.e. expressions that would be sentences if we filled in definite names for the variables). Certain of the formulas are sentences. The symbols, terms, formulas and sentences have no meaning merely by themselves.
Some of the symbols are logical symbols: the sentential connectives (representing 'not', 'and', 'or' and 'implies') and the quantifiers ('all' and 'some'). Also, usually '='. The logical symbols are in every language. Non-logical symbols (such as '0', '+', etc.) are not in every language.
In this context, we restrict discussion to languages such that there is an algorithm to determine whether any given symbol is a symbol of the language, whether any given sequence of symbols is a term, whether any given sequence of symbols is a formula, and whether any given formula is a sentence.
(4) For a given language, we may provide an interpretation to give denotations for the symbols, meanings for the formulas, and a specification of what it means for a sentence to be true or to be false per the interpretation. Such interpretations are called 'models for the language'.
(5) A set of axioms is a set of sentences from a given language. Some of the axioms are logical axioms, in the sense that they are true per every interpretation. Some of the axioms are non-logical axioms, in the sense that it is not the case that they are true per every interpretation.
A proof system (I'll just say 'system') for a language is a specification of a set of axioms and a set of rules for proofs, which are certain finite sequences of formulas.
In this context, we restrict discussion to systems such that there is an algorithm to determine whether any given sequence of formulas is a proof.
Any sentence that is the last entry in a proof is called a 'theorem' of the set of axioms.
A theory is a set of all theorems of a set of axioms.
(6) A model is a model of a given theory if an only if every member of the theory is true in the model.
(7) For every formula P, there is the formula ~P, which is the negation of P. With an interpretation, a sentence P is either true or false and not both, and ~P is true if and only if P is false.
A theory is consistent if and only if there is no sentence P such that both P and ~P are in the theory.
A theory is complete if and only if, for every sentence P, either P is in the theory or ~P is in the theory. A theory is incomplete if and only if it is not complete.
A system is complete if and only if, for every sentence P that is true in every interpretation, P is a theorem of the axioms of the system. A system is complete if and only if it is not complete.
So note that there are two different senses of 'complete', and two different senses of 'incomplete'.
We talk about theories that are "sufficient to express a certain amount of arithmetic". However, usually, that is not given as a rigorous notion. Instead, in any exact context, we would specify which particular theory we mean.
(8) Godel-Rosser is a theorem of mathematics about mathematical systems. It is usually presented with some rubrics and methods of set theory, but actually it can be carried out within merely the rubrics and methods of arithmetic. Moreover, it can be carried out within merely finitistic and constructive arithmetic. So, it is impeccable arithmetical reasoning that in principle could be reduced to primitive reckoning about finite sequences of stroke marks on a page. I don't know of any mathematical methods that a person would trust but that would leave the reasoning of Godel-Rosser as not trusted.
(9) Godel-Rosser is the theorem that any consistent theory sufficient to express a certain amount of arithmetic is incomplete. Note that thereby any consistent extension of such a theory is incomplete.
But rather than rely on the unrigorous notion "sufficient to express a certain amount of arithmetic", instead we mention particular theories such as PA (first order Peano arithmetic), Q (Robinson arithmetic), etc. Here I'll discuss PA.
(10) The non-logical symbols for PA are:
0 (intuitively, zero)
S (inutivively, the successor of a number)
+ (intuitigely, addition)
* (intuitively, multiplication)
The non-logical axioms of PA (stated here semi-formally) are:
For all n and m,
0 is not Sn (intuitively, 0 is not a successor)
If Sn = Sm, then n = m (intuitively, successor is one-to-one)
n+0 = n
n+Sm = S(n+m)
n*0 = 0
n*Sm = (n*m)+n
For every formula P, if P(0) and for all n, if P(n) then P(n+1), then for all n, P(n). (intuitively if 0 has property P, and if for every n, n having property P implies that the successor of n has property P, then every number has property P)
(11) One model of PA is called 'the standard model'. With the standard model:
the universe of discourse is the set of natural numbers
'0' stands for zero
'S' stands for the successor operation
'+' stands for addition
'*' stands for multiplication
When we say 'true' in context of arithmetic, that can be understood formally as 'true in the standard model'.
(12) PA is one of theories that is incomplete by Godel-Rosser. So there is a sentence in the language of PA such that neither it nor its negation is a theorem. But either the sentence or its negation is true in the standard model. So there is a true sentence of arithmetic that is not true in PA. And note, that any consistent extension of PA is incomplete. But to do basic mathematics, we would want to have at least PA, so Godel-Rosser provides that in a common sense, we can't even a complete basic arithmetic, let alone a complete comprehensive mathematics (such as includes calclus, etc.) that subsumes arithmetic.
(13) The proof of Godel-Rosser uses the arithmetization of syntax, which maps each symbol of PA to a different number. Then it maps each finite sequence of symbols to yet another number. Then it maps each finite sequence of finite sequences of symbols to yet another number. Such numbers are called 'Godel numbers'.
So, each sentence in the language is mapped to a different Godel number.
So, with the incompleteness proofs, numbers are used to refer to (to "code") symbols, strings of symbols, and strings of strings of symbols.
Without going further into the proof at this juncture, we can preview by saying that the Godel sentence named 'G' is a sentence in the language of PA that has a Godel number n such that G "says" that the sentence with Godel number n is not provable in PA.
So G "says": G is not provable in PA.
That is the "self-reference" in play.
The number n is the Godel number of G. And G mentions its own Godel number. So, the number of the sentence G is n and G also mentions n. This is not a number referring to itself. It is a sentence referring to its own Godel number.
So we can dispense with inapposite puzzlements about "numbers referring to themselves".
Quoting TheMadFool
Either G is provable in T or G is not provable in T.
Quoting TheMadFool
No.
And the incompleteness proof doesn't say "this sentence". It's more complicated than that, and thoroughly rigorous.
G is not a number. G is a formula. Then G is also coded by a number, which is its Godel number. The formula G and the number that codes G are two different things, though they can, in a certain sense, serve interchangeably in the particular context of the arithmetization of syntax.
You need to read a textbook. Then you could ask about what you don't understand. A few chats with a mathematician might give you an overview, but to actually understand the actual content, you need to study it.
No. As an illustration, it should be "The statement on the list at location n is not provable" and the location on the list for that statement is n.
G is not a location nor a number. You are conflating the sentence G with the Godel number for the sentence G.
Yes, the one that we show in the proof of incompleteness.
By the way, it is not undecidable in every theory. It is undecidable in particular theories.
Also, earlier in this thread, there were mentioned more substantive statements. They are usually mentioned in overviews, such as in encyclopedia articles, of incompleteness. Moreover, there is the major result about Diophantine equations.
The completeness theorem is not a mere conjecture. It is a theorem of mathematics. For any given consistent theory in a countable language, the proof of completeness provides a construction of a model of the theory. For an uncountable language (which is an unusual situation and not relevant to the languages for theories of which the incompleteness theorem pertains), still we have a model for each consistent theory though the proof of that is not constructive.
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.
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.
If I'm not mistaken, the only authoritative translation is the one in 'From Frege To Godel'.
Forget about the idea that's been presented in this thread, in connection with incompleteness, of numbers referring to themselves.
Just to be clear, some consistent systems are incomplete while others are complete.
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.
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.
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'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.
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?
Good to see you back, Tones. :up:
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.
If this is not the essence of the Godel Inc Theorems then I don't know what I'm talking about.
Okay. I get this.
Okay. But what is an example of G for some system T? .
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. .
Amen to this.
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?
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.
. .
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.
I don't know. I wish a mathematician would tell me but they don't seem to know either/ .
:up:
Yes, I've studied both. I've even developed twists on both.
Quoting TonesInDeepFreeze
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 ?
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
Yrs, but should I believe them? They simply don't seem to know.
The one constructed in the proof. When you read the proof, you see the G that is constructed.
Quoting FrancisRay
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.
Quoting FrancisRay
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.
Quoting FrancisRay
Of course. Anyone, including advanced mathematicians, may ignore it and still work productively.
Quoting FrancisRay
Read 'Godel's Theorem' by Torkel Franzen. He discusses your question in layman's terms.
I don't see it as mystic or foggy.
I don't have anything immediate to say about people's subjective impressions of mathematics.
Quoting plaque flag
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.
But I can't follow the proof. Does this mean I can never know an example of G?
I'd say computing is mathematics, so I'm not sure this is a valid example. .
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.
Not in metaphysics, or so I believe.
Thanks. I'll check the reviews but am not hopeful. . ,
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.
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.
Quoting plaque flag
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:
: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.
I like that. I was a rock climber.
Quoting 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.
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.
[b]If G was provable in T then this requires a sequence of inference steps in T
that prove that they themselves do not exist.[/b]
Copyright 2023 PL Olcott
I said I don't have anything immediate to say about subjective impressions of mathematics. I am not thereby like a "chatbot" that is not interested in anything other than the syntactical aspects of mathematics. Specifically, the fact that I don't have anything immediate to say about "R is mostly a black and seamless sea in the darkness" does not entail that I am uninterested in intuitions involved in the construction of the real number system. And the poster quoted above had said that he had been reading my posts over time; but in some of my posts I had written that indeed I am interested in intuitions about mathematics.
Shouldn't G be in the form of arithmetic calculus propositions for the incomplete theorem to apply?
G is a sentence in the language of arithmetic.
There are many ways to couch incompleteness proofs. Here is one outline:
Let T be a recursively axiomatized, consistent theory that is "sufficient for a certain amount of arithmetic".
We adduce a sentence G that is is true (to be more precise, it is true in the standard model for the language of arithmetic) if and only if G is not provable in T.
Then we prove that G is not provable in T. So G is a true sentence that is not provable in T. Moreover we show also that ~G is not provable in T. So T is incomplete.
Could you demonstrate and prove the provability and unprovability of G in real arithmetic sentences in T?
If all it proves is that every T has the true and unprovable sentence "this sentence is true and unprovable" then it seems vacuous.
Or does it prove that every T has a "natural" example of a true and unprovable sentence, like the strengthened finite Ramsey theorem in Peano arithmetic?
If T is consistent, then T does not prove both that G is provable in T and G is not provable in G.
We prove that if T is consistent then T does not prove G and T does not prove the negation of G.
How that is all done is quite complicated, especially with the proofs of all the lemmas.
"This sentence is true and unprovable" is not the sentence we prove is not provable in T.
Rather, "This sentence is not provable" is the sentence we prove is not provable in T and we prove that it is a true sentence.
Don't forget that the predicate 'provable' can be emulated in T, but the predicate 'true' cannot be emulated in T.
I don't know what you mean by "vacuous" here. G is a sentence of arithmetic. It makes a certain true claim about natural numbers. Granted, the particular claim it makes about natural numbers is probably not of interest to anyone. But that's not the point. Rather, the point is that there is no recursively axiomatized and consistent system for basic arithmetic that is complete and thus, for any given such system, there are true sentences about the natural numbers that are not provable in the system. Moreover, we can then see that there are infinitely many such true and unprovable sentences. Moreover, we then see that it is possible that some of the sentences about arithmetic that are of interest to us might be undecidable (neither provable nor disprovable) in the system. Moreover, hastened by the previous point, we do go on to show specific sentences that are of interest that are undecidable. That leads to the work showing that there is no algorithmic method for solving Diophantine equations, which is not just of interest but is basic to mathematics, even basic to high school algebra, especially for any lazy teenager like me who ever wondered, "Isn't there a step by step procedure I could use to solve any possible equation that I might be asked to solve, so I wouldn't have to think over all these problems but instead could just apply the procedure?" Moreover, we are then led to showing the undecidability of profound and fundamental questions such as the axiom of choice in ZF and the continuum hypothesis in ZFC. Moreover the techniques used in the incompleteness prove lead to the profound find that there is no solution to the halting problem, etc. And to top all of that, the P v NP problem may be the most economically valued in mathematics, as solution to it would have vast ramifications for computing and business; and incompleteness informs us that it is possible that P v NP does not have a solution (though, granted, there are a lot of people who do think it does have a not yet discovered solution).
That is a good question. We know that, for example, PA and set theory are such Ts. But, putting aside the ambiguity of 'natural' and assuming a general informal sense of it, I don't know whether it holds for every qualifying T.