Godel, God, and knowledge
My basic premise of this thesis is this: If we can know God perfectly, we can prove everything in mathematics once we fully know him and Godel's theorem will not apply.
Religious ideas of God vary from religion to religion. Eastern Orthodox, many Lutherans, and even some Catholics believe that God knows us immanently. He is not just transcendent as most Catholics and the most traditional of Jews think of him. Many see God as the space in everything just as if it was the presence of Jesus walking on earth. For them he is throughout time and not just the cause of time. The orthodoxy call this God's "energy" (the study and practice of this is hesychasm).
For most of these Christians though God nature is unknowable in it's substance but paradoxically perfectly knowable through the energies. This is one of those mystical paradoxes religious writers speak of. But it is true not all Christians believe we can know God fully. Aquinas for example thought otherwise. The premise at the beginning is coming for the perspective of those who think that we can know God or anything at all with full conceptual understanding.
Now if we are filled with divine knowledge in a mystical experience, why would Godel's argument prove we couldn't prove all of mathematics while in that state of rapture. Or to put it differently for the more secular minded, does Godel prove that God or an alien cannot prove all of mathematics, or just the very specific species of humans that have existed for 200,000 years?
Religious ideas of God vary from religion to religion. Eastern Orthodox, many Lutherans, and even some Catholics believe that God knows us immanently. He is not just transcendent as most Catholics and the most traditional of Jews think of him. Many see God as the space in everything just as if it was the presence of Jesus walking on earth. For them he is throughout time and not just the cause of time. The orthodoxy call this God's "energy" (the study and practice of this is hesychasm).
For most of these Christians though God nature is unknowable in it's substance but paradoxically perfectly knowable through the energies. This is one of those mystical paradoxes religious writers speak of. But it is true not all Christians believe we can know God fully. Aquinas for example thought otherwise. The premise at the beginning is coming for the perspective of those who think that we can know God or anything at all with full conceptual understanding.
Now if we are filled with divine knowledge in a mystical experience, why would Godel's argument prove we couldn't prove all of mathematics while in that state of rapture. Or to put it differently for the more secular minded, does Godel prove that God or an alien cannot prove all of mathematics, or just the very specific species of humans that have existed for 200,000 years?
Comments (90)
God is a member of the Null Set.
Godot is coming, be patient!
Gödel diagnolizes us all.
I think he has a flawed argument. He get's into language games. Self-reference such as Russell's paradox is about how language can backfire. If we gave a barber the "barber paradox rule" he can ask for clarification in order to know what to do. Self-reference with pure numbers instead of language was what Godel was after yet we ultimately can clarify what we mean through language. Of course we can doubt axioms. But how would you prove a proposition was unprovable of itself such that God couldn't prove it to himself. And some of the Church Fathers spoke of the "deification" of man, which means that man will fully understand God and therefore know everything? How does Godel know they are wrong?
So Banno proved there is not God?
Nothing to be done.
[i]What are thoughts that we should capture
While in the throes of blissful rapture
That soothe the pain of incomplete
With certain knowledge
Now replete[/i]
I've read very little Austin. And mathematical logic is something I cannot do at all, it's beyond me I'm afraid. The only philosophy of language I can do are the people I mention in my profile and a little Wittgenstein, though nowhere near your level.
As for the OP, I can't even comment much. I don't understand what "prov[ing] everything in mathematics" would even entail.
So you can go ahead and go wild, if you like.
I don't think so. Just thought you might enjoy giving it a go. A something to give the impression we exist, if you like.
After all, that's what we do.
And I'd gladly participate if it didn't include Gödel.
I'm sure other threads will appear that will offer the opportunity to highlight a problem in language use.
You think god knows the proofs of those statements for which there is no proof.
You might like to think on that some more.
Giving out more rope again. I'll go on for now, but just for the encore of spasms.
He can prove everything I imagine. He is logical perfectly atomized into simplicity, as theists might say. So 1) God's existence cannot be disproved, and 2) and you can't disprove that we can fully understand what God is. Therefore Godel's theorems only apply to human thinking while in a natural state and not to it embedded with the divine. Theists say the full explanation of reality must be a God who is embedded in everything and even closer to me than I am to myself because he is all around me, and in a sense even more myself than I am myself. The conclusion seems to be the nearer one get's to God, the less Godel logic fully applies. My favorite philosopher is Hegel, who say everything can become rational. He follows the mystical philosophy of Jacob Boehme. The Absolute has full knowledge of everything covering everything but in a higher manner
Literally the laziest argument on the face of the earth.
It can't be argued against either, because God is simply a magical gap filler that can plug whatever hole, as needed. Bollocks.
You think god knows the proofs of those statements for which there is no proof.
From that it is apparent that you don't know what Godel's theorem is. Your commentary is relevant to what you think Godel's theorem is but not relevant to Godel's theorem.
What's to be done? Will @Gregory stop and do some more work on understanding Gödel, or double down with more poor argumentation?
Quoting Banno
The latter. I guarantee it.
If you want to post on this thread at least address what I say and if you don't like the idea of God at least be up to saying so
So, according to you, a necessary condition for making sense of your idea is thinking spiritually. But meanwhile a necessary condition for making sense of Godel's theorem is knowing what it is.
Quoting Gregory
What I don't like is people spouting about Godel's theorem without knowing what it is. If you're not up to finding out what Godel's theorem is then at least be up to saying so.
Where did you read that?
I don't like the idea of god.
I'm pretty sure I've not hidden that view.
It seems that those who have thought spiritually see sense in nonsense.
I don't think that a good thing.
Together again at last! We'll have to celebrate this. But how?
Godel's theorems prove that, in the form in which we think now, mathematics is either inconsistent or has infinite propositions that can't be proven. It's undecidable which of these are true for Godel.
I don't see how someone can find this to be a satisfactory idea to rest in. Ideas of spiritually are not fairy dreams. They are some of the deepest thought you can have
Your way of thinking is contingent perhaps though, although it seems logical to you. Spiritual pursuits search for higher necessary knowledge and is still philosophy, actually is more philosophy than analytical philosophy.
Not everyone who comes to this forum is into analytical philosophy. You've called Hegel rubbish but some like him, as I do. He certainly thought every truth could be "sublated" until everything is known. If there will still be truths in mathematics that can't be proven, they will be seen as to why this is the case and the whole of truth can find a consistent point of rest. Even if we can't know every higher truth, there can be truth as a whole found in life. I don't see Godel's ideas as consistent with finding THE truth
And because the unprovable mathematical ideas would have to axioms known by intuition. If they are a connection of ideas and there is no way to get from one to the other, this blocks knowledge as a whole from find the truth of everything
...with a supreme effort @Banno succeeds in pulling off his boot. He peers inside it, feels about inside it, turns it upside down, shakes it, looks on the ground to see if anything has fallen out, finds nothing, feels inside it again, staring sightlessly before him.
Well?
Imagine some unprovable proposition. Can it be *understood* intuitively like axioms are and be taken as axioms?
Gödel offered a proof that math is either inconsistent or incomplete and that the dilemma is undecidable. So unless math is bogus, there will be unproven propositions. I don't know which ones these are but zero in on some mentally for me. Now I ask "are these propositions actually axioms for something else or something else entirely?"
We can't prove axioms
Quoting Gregory
Indeed, that's a generalisation of the first theorem. In a given formal system complex enough to do arithmetic there are statements which can neither be proved nor disproved. Assuming the system is consistent (surely not an unreasonable thing to do?) then it must be incomplete - it must contain unproven statement.
SO you asked Quoting Gregory
I just don't know what to make of this. An axiom in a formal system is a statement that is true within the system, and hence is not understood intuitively but in the terms given by the system. Given some set of axioms sufficient to the task of arithmetic, there will be some statements that are neither among the axioms nor among the theorems. But these cannot be taken as axioms without changing the formal system.
Perhaps you could set me right here?
Another axiom?
Is it being said that within this infinity of unprovable propositions in math each proposition can be analyzed and their meanings understood? What prevents someone from finding the golden thread going thru such a proposition?
Ok then
That's not Godel's theorem. You don't know what Godel's theorem is.
Previously asked:
Quoting TonesInDeepFreeze
According to Roger Penrose Godel was a “very strong”mathematical platonist, so even if proof leads to an infinite regress, you can read Godol’s theorem as perfectly compatible with an absolute god-given grounding for. math.
Well if all I've studied on this is wrong then we have a similar situation as with Bell's theorem where there is no consensus whatsoever of what theyre about. I've watched all the videos I could find on it, read about it in books, and discussed it with people who have computer science degrees. What you are saying is that there is massive misinformation on this but then why haven't you written a couple paragraphs here saying what Gödel really did. I don't believe in self reference in math or logic but maybe you can make a presentation of it will be interesting and fruitful
You did all that and managed still not to know what Godel's theorem is.
Quoting Gregory
It's not required for pointing out that you don't know what Godel's theorem is.
But I will indulge you:
The Godel-Rosser theorem may be given a modern statement as:
If a theory T is a consistent, recursively axiomatizable extension of Robinson arithmetic, then there is a sentence G in the language for T such that neither G nor ~G is a theorem of T.
You don't know the actual nature of the "self-reference" in Godel's proof. The proof may be formulated in finite combinatorial arithmetic. If you have a problem with the proof, then you have a problem with finite combinatorial arithmetic.
That is a question that could be asked only by someone unfamiliar with the basics of this subject.
If P is a closed formula, then there is a system S such that P is an axiom for S.
Bertrand Russell was famous for his mathematical ideas. But his paradox is false. Group items together, make a circle around them, and you have a set. A set containing itself is just bizarre, coming from a desire for exotic knowledge, and yes mathematicians aren't perfect. What I said about Gödel was based on what the majority of people have said about from what I personally have seen. Someone needs a really good background in math to read his actual papers so most of us are getting our ideas from second hand sources. Anyway, you can't prove that a set can contain itself from math itself, so rejecting Russell's paradox is a good way to start in approaching Godel. A set containing itself IS self reference
Every contradiction is false in every model. So what?
Meanwhile, it is a theorem of first order logic that there is not an x such that for all y, y bears relation R to x if and only if y does not bear relation R to y.
Quoting Gregory
So what? Neither Russell nor Godel depended on a claim that there is a set that is a member of itself.
Quoting Gregory
As Seinfeld put it, "Who are these people?" Whatever "majority of people" you talked with, your conversations did not supply you with a even a fraction of a decent understanding of Godel's theorem.
Quoting Gregory
His original papers are rather old-fashioned in their notation. More recent textbooks have pedagogically supplanted the original papers.
Quoting Gregory
Since I don't know your sources, I can't say whether the fault is in the sources or in your misunderstanding of them.
Quoting Gregory
In set theory, we prove that there is not a set that is a member of itself.
However, with set theory without the axiom of regularity, there is not a proof that there is not a set that is a member of itself.
And, dropping regularity, but adding a different axiom, there is a proof that there is a set that is a member of itself.
Quoting Gregory
Those two sentences alone are proof that you are completely mixed up and ignorant of what Godel's theorem is.
/
To understand this subject properly, one should learn basic symbolic logic, then a small amount of basic set theory, then an introductory course in mathematical logic - either in a class or by self-study.
Meanwhile, the best book about Godel's theorem for everyday readers:
Godel's Theorem: An Incomplete Guide To Its Use and Abuse - Torkel Franzen.
That book will disabuse you of your confusions.
Quoting TonesInDeepFreeze
Apparently even for you Gödel's theorem is hard to put into words. You can't provide what the theorem says, since you say I don't know it properly, in a few paragraphs. As I said, I've seen sources on this for years. I watched the Veratasium video two weeks ago and it said what I've heard everywhere else. My point in this thread is that if there are unprovable propositions, they don't exist in weird loopy ways but have a straightforward reason for why they are closer to axioms than from what is probable. When I studied Euclidean geometry in college our teacher kept telling us to see the golden thread in each proposition and how it runs from the first to the last. Russell's paradox is a different species of thinking. All it takes is a conversation to reveal what someone means if they present you with a paradox like that. It was a linguistic problem, not a logical one
What? It's not at all hard for me. I did it a few posts ago!
Quoting TonesInDeepFreeze
Please let me know that you see it now so that I may know that I'm not posting to an insane person. Then I'll see about correcting yet more of your ignorant confusion in your post above.
So you are saying that Godel's examples of things that are unprovable do not require a loop in them? As I see it, unprovable things can be 1) axoims which we understand intuitively as unprovable but which make sense ("common sense" comes in) as the basis of a system, or 2) propositions that are unprovable but which can be understood by intuition (thus knowledge is fully knowable), or 3) loopy statements like Russell's paradox that are really fallacious logically.
I am always willing to learn new things, but you wrote:
Quoting TonesInDeepFreeze
Couldn't you just have said "systems have axioms"? That is all that says! This is my problem with the whole symbolic logic stuff. They get into problems and call things paradoxes because they don't converse with adult conversation language. We should be truly speaking about truths, not fitting them into structures which confuses matters. We have crazy people try to PROVE there is a God from modal logic ("ontological argument"). It's just ridiculous that people would even consider trying to do this. I think very fluidly and I don't get a pleasant sensation from a paradox that just reverts back on itself. And you say:
Quoting TonesInDeepFreeze
In real human language, you are saying that a theory has a part of it is and is not a part of it. Again, key word is "recursive". I don't understand why anyone would want to think about logic eating itself like a snake eating its tail. That kind of stuff gives me a headache. It's not cool
If you were to know God perfectly, mathematics is irrelevant. Especially if that "everything" includes things that mere mortals have deemed false. Just relax and savor the bliss. :love:
Is this what you were referring too? It is not math but philosophy. There is nothing A unless B does not have a relation with itself? This what I'm talking about. Godel loops his arguments. Math and logic are different disciplines and combining them is a questionable enterprise. On the ladder of knowledge 1 plus 1 equaling two seems to come before the logicism used to prove this by Russell and Whitehead. So maybe their 700 pages on this is nonsense, a putting of the prior after what should come latter. And maybe Godel's ideas have the same problem: too much application of logic to math
I told you twice what Godel's theorem is:
Quoting TonesInDeepFreeze
Instead of recognizing that, you bring up a different matter.
You will not make any progress here if you can't pay attention.
I told you what I thought of it. It does not mean anything mathematically because it refers back on itself, a move of logic, not math.
Quoting Gregory
What “tripped up” Frege was Russell's paradox, not the barber paradox.
Russel said he took the squiggly part of the barber paradox and used it with sets
[quote=Russell]You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of (the?) words is just noise without meaning.[/quote]
The source is “The Philosophy of Logical Atomism”.
Do you know about Gödel numbering? If so, you should be able to understand why the Gödel statement is a mathematical statement.
Godel and Russell both had many ideas that were mathematical but had an element of the science of logic in how they move. What Russell said in your quote is what I was saying. Sets that contain themselves are not objects of mathematics
The Gödel statement is not a set, it's a statement.
It references itself, but unlike other statements that can be classified as meaningless, like the liar statement, it must either be true or false, because it's a mathematical statement.
Here you have a more detailed explanation
What you think about it is one matter. (What you think about it is based on a collection of confusions and misunderstandings you have.)
But you said that I have difficulty stating Godel's theorem. Yet I stated it without difficulty. So I'd like to know whether you recognize that you were incorrect to claim that I have difficulty stating the theorem. Your recognition of that would help to show that you are not entirely irrational and rhetorically irresponsible.
Russell's paradox is interesting philosophically, but I showed how this paradox can give two answers (both "in a sense"). Everyone has different explanation on how Gödel's arguments are supposed to work, probably from the nature of the case. If you disagree on logic's relation to math, then start with what you think are the logical tools of Gödel's theorem
You provided a statement and have not spoken yet of the internal logic that makes it a proof
Quoting Gregory
You only need these tools: knowledge about Gödel numbering, knowledge about formal systems, and the law of the excluded middle.
Did you even bother reading the Wikipedia article? It's not that long considering the difficulty of the subject.
Anyway, here is just one part of it:
[quote=Wikipedia (Gödel numbering)]Gödel noted that statements within a system can be represented by natural numbers. The significance of this was that properties of statements – such as their truth and falsehood – would be equivalent to determining whether their Gödel numbers had certain properties. The numbers involved might be very long indeed (in terms of number of digits), but this is not a barrier; all that matters is that we can show such numbers can be constructed.
In simple terms, we devise a method by which every formula or statement that can be formulated in our system gets a unique number, in such a way that we can mechanically convert back and forth between formulas and Gödel numbers. Clearly there are many ways this can be done. Given any statement, the number it is converted to is known as its Gödel number.[/quote]
If the Gödel statement were meaningless, then we would not be able to construct it through Gödel numbers.
The Gödel statement is a meaningful statement since its corresponding Gödel numbers can be constructed. Therefore, it's either true or false, in the same way as a statement such as “10001^26278283 is prime” is either true or false, but not meaningless:
[quote=Wikipedia (Gödel incompleteness theorems)]The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in F. However, the sequence of steps is such that the constructed sentence turns out to be GF itself. In this way, the Gödel sentence GF indirectly states its own unprovability within F.
To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate on Gödel numbers of sentences of the system. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete.
Thus, although the Gödel sentence refers indirectly to sentences of the system F, when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by a primitive recursive relation. As such, the Gödel sentence can be written in the language of arithmetic with a simple syntactic form(...)[/quote]
That's a (very) brief summary of it.
If you want to know more, look it up.
I already have read that. The numbers are random and don't form real equation. I was asking for Gödel's theorems stated as verbal paradoxes like Russell's paradox. That way I can explain like I can do with Russell's. Maybe Gödel proves something but it's only about human cognition. The point of my thread was that higher species know things in better ways and spirituality can lead to thinking beyond human thought. How this works with Gödel's theorems is what I was wanting to talk about
You said that I have difficulty stating the theorem. Stating a proof of the theorem is more than stating the theorem.
I stated the theorem without difficulty.
As I said, I'd like to know whether you recognize that you were incorrect to claim that I have difficulty stating the theorem. Your recognition of that would help to show that you are not entirely irrational and rhetorically irresponsible.
The incompleteness proof is intuitionistically valid and does not require excluded middle.
Hmm, ok. I meant that you may need the excluded middle (as it applies to mathematical statements) to show that the Gödel sentence was not meaningless despite being self-referential, like the sentence “this sentence is false”.
I thought the law of the excluded middle was also needed for mathematical proofs by contradiction, like Euclid's proof that there are infinitely many primes.
That doesn't sound right to me. The properties that are checked are syntactical. What semantical properties does Wikipedia claim are checked?
Godel's numbering might not apply to the real world. The real world is mathematical but there might be a theory of everything in term of physics. If there are things we can't prove in mathematics, we at least knows math is true for us. For some Christians God gives us actual grace for our free will to renovate itself and return to it's former pure state and the merits of Jesus make us innocent before the justice of God. So man becomes somewhat divine Lutherans from Jacob Boehme to Hegel emphasized that God was "all in all", or in a more exact sense, was "everything in everything". All religions describe a union with the divine and if our thoughts can be raised up and the experience of the divine and is intellectual in a sense, it's possible our thoughts can be moved where everything is seen as a total unity of truth. This was my concern with Godel. Perhaps Godel helps us gain this vision, idn
No, there are two kinds of proof involving contradiction:
assume ~P
derive contraction
conclude P
assume P
derive contradiction
conclude ~P
The first one requires excluded middle (or double negation, or whatever intuitionistically invalid rule).
The second one does not require excluded middle (or double negation or any intuitionistically invalid rule).
I see, that does make sense I guess.
Paradox is just a logic hiccup. The Universe still gets on with doing what it does, despite human musings about paradox and/or infinities.
I am always surprised that an 'intelligent' person can ever satisfy their personal search for the T.O.E,
with the god fable. Especially when it clearly is just a filler story (or god of the gaps) for phenomena that humans just can't explain yet. As has already been suggested, it is lazy thinking.
For me, despite the barber's paradox, everyone that wants a shave, seems to be able to get one, despite any protest from, or hiccup in, propositional logic.
Paradox and infinities just indicate that we have not answered all the questions yet.
I hope we never do, as I am not sure what we would be for after that.
If we do ever answer every question, it will perhaps turn out that the Universe will become 'self-aware' and all sentient life in the Universe, can as a totality of thought, declare itself God.
Because at that point, the totality of thought from all life in the Universe would effectively satisfy the three qualifiers for godhood,
Omnipotence (no more questions to answer so this must have been achieved),
Omnipresence (no part of the Universe would exist, which is not affected by the totality of life within it), and Omniscience (ditto with omnipotence)
If all questions have been answered then God exists as a totality of life in the Universe.
Universeness 28/01/2022 TPF
ha ha...... :naughty:
"What have I got in my pocket?" :wink:
Bad hobbit, Bad!