80 years ago, Kurt Gödel toppled empires of mathematical philosophy with his famous Incompleteness Theorems.
 Kurt Gödel proved, ironically, that it’s impossible to prove everything. And yes, he proved it. |
I’m every bit as interested in science, philosophy and engineering as I am in business. Gödel’s theorem has profound implications for every branch of knowledge.
The materialist view prevails in secular circles. Materialism states that the laws of physics and the universe we know are all that is. It sees the universe as a giant machine. It assumes that everything we experience is purely the result of blind cause and effect. It scoffs at the idea that there is any such thing as God or metaphysics.
This view was epitomized by “Logical Positivism” which was espoused by a group known as “The Vienna Circle” in Austria, led by Ludwig Wittgenstein. Logical Positivism says that anything that cannot be experimentally verified or mathematically proven is invalid.
The Logical Positivists were confident that very soon, all the loose ends of mathematics would be nailed down by a single unifying theory. The world would finally fully embrace reason and logic and leave the failures of religion behind.
Kurt Gödel was a member of the Vienna Circle and in 1931 proved that a single unifying theory was impossible. He proved that the goal of the Logical Positivists was unachievable. This was a devastating blow.
Godel’s Incompleteness Theorem says that any system that is complex enough to express mathematics cannot prove, by itself, that everything it says is true. It will always rely on something outside the system that you have to assume is true but cannot prove.
You can then step outside the system and complete your proof, but in order to do that you will now have to invoke something else from the outside. So you keep expanding ever outward, invoking still more things that you cannot prove.
This was very disturbing to mathematicians, because mathematicians hate uncertainty.
Many people have raised the question of whether Gödel’s incompleteness theorem applies to the universe itself. If the universe is mathematical, then yes in fact it does.
Stated in Formal Language:
Gödel’s theorem says: “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”
The Church-Turing thesis says that a physical system can express elementary arithmetic just as a human can, and that the arithmetic of a Turing Machine (computer) is not provable within the system and is likewise subject to incompleteness.
Any physical system subjected to measurement is capable of expressing elementary arithmetic. (In other words, children can do math by counting their fingers, water flowing into a bucket does integration, and physical systems always give the right answer.)
Therefore the universe is capable of expressing elementary arithmetic and like both mathematics itself and a Turing machine, is incomplete.
Syllogism:
1. All non-trivial computational systems are incomplete
2. The universe is a non-trivial computational system
3. Therefore the universe is incomplete
Some time ago I posted an article about this: http://www.perrymarshall.com/articles/religion/godels-incompleteness-theorem/ and I was greatly interested in seeing if anyone would be able to poke a hole in my argument. (The article is a much more thorough explanation of Gödel than I am giving you here.)
Nearly everyone agrees that math is incomplete. The idea that the universe is also incomplete apparently makes some people very uncomfortable. If the universe cannot explain itself then there has to be some kind of higher power at work.
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The debate essentially comes down to this:
- If the universe is illogical and inconsistent then it is possible for it to be complete.
- If the universe is logical and consistent then it is incomplete.
- If the universe is incomplete, then it depends on something on the outside.
In other words, if the laws of mathematics and logic apply to the universe, then the universe has to have a metaphysical source. Atheism can only be true if the universe is irrational.
(By the way, my experience from conversing with literally thousands of atheists via email and on my various blogs is this: When you get down to the core emotional center of why they don’t believe in God, it’s often because they feel deep down that the universe is irrational. They’re immensely disappointed that the world is full of evil and suffering. Because of this, they reject the idea of God.)
You cannot prove that the universe is mathematical. But belief that the universe is mathematical is the #1 assumption of modern science. Toss out that assumption and the whole philosophical framework of western civilization crumbles.
In the history of science, you will find that belief in a God who created an orderly mathematical universe was one of the foundations of scientific discovery.
If you visit the world’s largest atheist website, Infidels, on the home page you will find the following statement:
“Naturalism is the hypothesis that the natural world is a closed system, which means that nothing that is not part of the natural world affects it.”
If you know Gödel’s theorem, you know that all logical systems must rely on something outside the system. So according to Gödel’s Incompleteness theorem, the Infidels cannot be correct. If the universe is logical, it has an outside cause.
Thus atheism violates the laws of reason and logic.
The Incompleteness of the universe isn’t formal proof that God exists. But… it IS proof that in order to construct a rational, scientific model of the universe, belief in God is not just 100% logical… it’s necessary.
Practically speaking, all knowledge we have about anything is incomplete. There are always some things you’re certain of, some things you’re somewhat sure of, and some things you cannot prove at all. Human knowledge is always enlarging the circle of what is known, but every question that we answer just provokes more questions. In real human experience, the quest to enlarge that circle never stops.
And I would submit to you that this is the essence of faith, as actually practiced by thinking, reasoning people.
Many people assume that religious faith is some mystical imaginary idea that is embraced purely on the basis of emotion or intuition. That it has nothing to do with facts, reason or logic.
This is completely untrue – at least in in the Judeo-Christian tradition. No one is asking you to believe without evidence or rational reason. Belief in God, in Jesus, and even the afterlife is based on historical events, logical propositions, and reasonable arguments.
Science itself originated from theology. Science assumed then, and assumes now, that the universe is rational. That the universe operates according to fixed, discoverable laws. Even science itself is a very practical outworking of faith in the reliability and consistency of the natural order.
The practice of faith is in many ways living out a hypothesis: That if you follow the teachings and embrace the Spirit, you will have an excellent opportunity to experience success in your work and your family. And that you will be rewarded in your search for meaning and pursuit of the deepest questions.
Perry Marshall
P.S.: If this intrigues you, make sure you read my more extended article, Gödel’s Incompleteness Theorem: The #1 Mathematical Discovery of the 20th Century
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156 Comments on “Gödel’s Incompleteness Theorem: The Universe, Mathematics and God”
1. All non-trivial computational systems are incomplete
2. The universe is a non-trivial computational system
3. Therefore the universe is incomplete
Really? Try this
1. All non-trivial computational systems are incomplete
2. The Universe by definition is complete
3. Therefore the Universe is not a non-trivial computational system.
You could argue I suppose that the known universe is not the whole Universe and that the whole Universe must include something (God?) that makes it complete and allows the known universe to be a non-trivial computational system.
But the whole Universe would still, by definition, be complete and therefore it could not possibly, per the above syllogism, be a non-trivial computational system.
My hunch is that neither the (known) universe nor the Universe (as a whole) are themselves non-trivial computational systems in anything like the sense that would make them subject to Godel’s theorem and also that there is some equivocation in the term “complete.”
Think about it: we may try to understand the universe by constructing axiomatic systems that describe it, but that doesn’t mean the universe itself is an axiomatic system. So, it very well may be true that a complete and consistent axiomatic system that describes the universe is impossible, but it doesn’t follow that therefore the universe itself is incomplete.
Prove your point #2.
Perry,
I basically agree with you, just from reading some of your writing; and know with you that God has the answers recorded for honest truth-seekers, by His inspiration, already in Scripture (2 Peter 1:21, 2 Timothy 3:16-17). So here are a few verses for consideration re. consistency and incompleteness in our world (creation). These in effect honor the God who commands all mankind’s attention, even “brilliant” scholars, as giving Him all reservation for holding the unequivocal truth (Proverbs 25:2). Here they are — at least some of them!:
John 1:3, Hebrews 1:3, Colossians 1:16-17, Ephesians 1:10.
These verses tell us plainly that Jesus is actively on the scene, doing just that: holding it all together in a way that His creation alone could not — but now He has the authority to do so — to refute people honestly and with integrity by the fact that He was actually raised from the dead (yet some will foolishly scoff as in the Athens depicted in Acts 17). Nevertheless, from a very practical personable standpoint, the evidence is strong and really incontrovertible, that Jesus was raised from the dead! So He holds a scepter of Authority that no one else does. Also we have the facts that so many attested Him by giving up their very lives (as in Hebrews 11:38 and Revelation 12:11). Not to mention that, Jesus being severely tortured, himself would have denied his divinity and purpose and mission, if he was indeed a liar — but because He didn’t shows He was who He said He was. So God’s consistency in history as well as science stands clearly enough to convince people — yet many will stop their ears and refuse to listen.
There are a lot more verses that could be brought to support but many of the audience members are probably too glossed over to listen (Hebrews 5:11-22), but rather are figuratively stopping their ears and yelling (Acts 7:57). As Jesus warned the crowds, “He who has ears to hear, let him [he had better] hear.” (Matthew 11:15). God warns us all only so many times– beyond which [evidence firmly rejected], people themselves are rejected by God if they continue to choose hypocrisy (claiming they are sinless or good despite the “bad fruit” stinky sin products coming from their lives (Matt. 12:33). Ultimately, they must be shown their breaking of God’s laws and their own misgivings (Romans 3:19-20), in order to lead them to Christ for Him to change them to become acceptable to God — and this is by His grace-gift of salvation alone– a total new birth/ starting over that He provides, forgiving Love to those willing to honestly and openly change (John 3:16).
P.S. KUDOS to you Perry for turning the other cheek (like Jesus did) to people who are not really truth- (God-) seekers anyway — which is the least we can do as creatures made in God’s image to honor Him in repentant, honest humility in the world He has created (per Romans 1, a passage well worth reading to sense how God deals with men with great grace of patience, yet justly holding them accountable for rejecting Him)!
You make some important points. About 10-12 years ago I asked myself:
“Is Romans 1 really true? Is the following statement true?
…what may be known about God is plain to them, because God has made it plain to them. 20 For since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse.”
At the time it wasn’t 100% clear to me that it was true. Maybe it was, maybe it wasn’t. I decided to find out, so I created http://www.cosmicfingerprints.com and http://www.coffeehousetheology.com and I answered emails from well over 10,000 people over a period of 5 years. I played ball with anyone who was willing to engage with me and exercise a little manners.
Eventually I confirmed, yes it is true. The arguments for God are absolutely logical and there is no counter-argument. You can see that clearly here and in the sister article to this one, http://www.perrymarshall.com/godel.
I’ve made some observations:
-Atheists are often quite surprised to find that, in fact, the arguments for God are very robust and yes, basic scientific inference does in fact imply (though not prove) God. They are shocked to find that the arguments they’ve been believing up to this point are flimsy and circular.
-The vast majority of atheists I find are too ashamed or possibly even too cowardly to come out and argue with their real names and identities. If you could wave a magic wand and remove the option for atheists across the internet to be anonymous, 80-90% of the crappy blog comments, YouTube comments and slander would stop overnight.
-Part of the problem here is Christians themselves not being willing to practice real science. Young Earth Creationism, for example, is not scientifically defensible. It requires one to make up or accept all kinds of just-so stories in order to force the data to fit a peculiar Biblical interpretation. Christianity has traditionally been the dog that wags the tail in the history of science – science came from a Christian context after all and most of the early great scientists were Christians. So when Christians abandon science (as many have in the 20th century) and abdicate their responsibility to embrace the truth wherever it may lead them, the atheist side also springs up and becomes a mirror image, also making up just-so stories… in their case, about cells magically appearing in warm ponds once upon a time long long ago, etc. From then on, the two sides depend on each other in order to exist. They thrive on useless controversy because neither is practicing science.
The evidence for God is very robust and it is time for people of faith to stop their mealy-mouthed excuses and foot shuffling, and press forward and engage in the real thrill of scientific discovery.
Even if we assume everything you say about the universe is true, and something must exist outside of the universe to explain its existence, we have no way of knowing what this “something” is. You can’t simply state;
The universe need something to exist out =side of it
Therefore something exists out side of the universe
Therefore JESUS! (lolz,atheists are dumb)
Especially if you are saying it is this way because the universe is logical. The “evidence” for GOD is a book full of non-logic.
Okay…
Did you respond to the right comment? You don’t seem to be addressing my comment at all. Regardless, here is my counter response, line by line. Your words are in quotations.
“There is no empirical evidence of other universes etc. That is pure unbridled conjecture. There are some models that invoke other universes but this is speculation.”
-Irrelevant. There is also no empirical evidence of God. If there were, we wouldn’t be having this conversation.
“Godel’s work was a proof. It applies to any system above a certain level of complexity. it applies to Euclid’s postulates, for example, which are much simpler than the universe.”
-I’m not a mathematician, but I see no problem with this assertion. I will grant you this proposition.
“You are free to reject the notion that reason and logic apply to the universe.”
-Logic and reason only “apply” to the universe in as much as they are tools that allow us to understand the world around us in an increasingly sophisticated manor. These tools were created by us for this purpose.
“Then and only then can you assert that Godel’s theorem doesn’t apply.”
-I assert no such thing.
“In which case you pull the rug out from under science and reject it outright.”
-The scientific method is the best tool we have for understanding the universe. The realization that it may have limits does nothing to diminish this.
“However if reason and logic apply to the universe, and if science is valid, then Godel’s theorems apply to the universe.”
-Even if this is true, you have not begun to prove the existence of God. You have only shown that humans have failed to completely understand how the universe works. I’m fairly certain that this concept is not questioned by any thinking person.
I will be posting this as a reply to my original comment since you did not reply there.
Rusty,
I disagree that we do not have empirical evidence that God exists. I’ve watched deaf people get healed right in front of my eyes after being prayed for (two times now) just to name one thing. We have much empirical evidence for God.
In the specific sense that you are saying, no we do not have physical evidence i.e. God being a physical “thing” that we can put on a scale and weigh. However, if is valid to infer the existence other universes or string theory or other abstractions in physics because it makes a model work (and I do believe such reasoning is in principle valid), then it is equally valid to infer the existence of God, by the exact same process of logic.
No you cannot PROVE God. However science and mathematics emphatically infer God. The only way to overcome this is to do what atheists do and arbitrarily forbid such inferences.
If you are going to invoke other universes that you cannot see, then you have to play fair. You don’t get to have your cake and eat it too.
The tools of reason and logic were not created by us. We DISCOVERED them.
The scientific method assumes the universe is logical. if the universe is logical, then Godel’s theorem necessarily applies to the universe, and the universe is incomplete. If the universe is not logical then science itself is doomed.
Fascinating! It seems that Gödel proved what Thomas Aquinas said almost 800 years ago.
‘Any physical system subjected to measurement is capable of expressing elementary arithmetic. (In other words, children can do math by counting their fingers, water flowing into a bucket does integration, and physical systems always give the right answer.)’
‘Therefore the universe is capable of expressing elementary arithmetic and like both mathematics itself and a Turing machine, is incomplete.’
No. It should be ‘Anything within the universe that is capable of expressing elementary arithmetic and like both mathematics itself and a Turing machine, is incomplete.’ You have no image of the Universe. The Universe (defined as everything that exists) is actually complete.
Also, depending on the definition of god it is either:
a) god is defined as The Universe. (god = Universe) nothing to see here.
b) god is defined as something that is Not The Universe. Then god is either 1)incomplete, or 2) not omnipotent. Why call him a god?
‘You cannot prove that the universe is mathematical. But belief that the universe is mathematical is the #1 assumption of modern science. Toss out that assumption and the whole philosophical framework of western civilization crumbles.’
‘If you know Gödel’s theorem, you know that all logical systems must rely on something outside the system. So according to Gödel’s Incompleteness theorem, the Infidels cannot be correct. If the universe is logical, it has an outside cause.
Thus atheism violates the laws of reason and logic.’
No again. A system can be complete by being inconsistent(I’m hinting here that The Universe is complete and inconsistent). So atheism or science do not violate the laws of reason and logic. Belief in god does though(but not belief in general).
Some Uneven Structure lyrics(which i really like):
6. Delusions Of Grandeur
Same patterns repeated.
A constant failure process.
The fables of fortune remain untold.
Delusions of grandeur is defeated
as nothingness takes control.
7. Depression
Crave for the indecision.
Awaiting the ultimate answer.
Restless in infinite tediousness.
Verity uprises from Nihilant.
Womb of all ill shaped self reflections.
Imaging wrongness.
If the universe is complete, then the universe does not obey the laws of reason and logic.
So if science DOES obey logical laws, then science does not describe the universe and is therefore useless.
‘If the universe is complete, then the universe does not obey the laws of reason and logic.’
No. Inconsistency and completeness are part of logic. The Universe is complete AND obeys the laws of logic.
‘So if science DOES obey logical laws, then science does not describe the universe’
Correct. Science does not describe The Universe. Nothing can describe The Universe.
‘and is therefore useless.’
Nope. Science describes successfully anything INSIDE The Universe. So science is the most useful thing anyone can ever have WITHIN The Universe. By definition, everyone IS INSIDE of The Universe.
God, on the other hand, is nowhere to be found anywhere in the picture. Believing in God is illogical.Believing in general isn’t, you must believe that the world exists in order to achieve anything.
You seem to confuse ‘The Universe’ with ‘INSIDE The Universe’.
You do not understand Godel’s theorem at all. If the universe is consistent, it cannot be complete.
Oh, and by the way if you want to discuss further you must use your full name. No anonymous cowards.
‘You do not understand Godel’s theorem at all. If the universe is consistent, it cannot be complete.’
From wikipedia:
Gödel’s incompleteness theorems are two theorems of mathematical logic (…)
I understand Godel’s theorem. Godel’s theorems are part of mathematical logic. It is obvious though that You don’t understand Godel’s theorem.
As i said, The Universe is not consistent, it is complete. Anything trying to ‘explain and predict’ The Universe, is consistent and incomplete. That is the limitation inherent to being INSIDE The Universe, by definition. Being inside IT we only see a finite horizon, and as such we can only conceive an incomplete axiomatic system(we can’t list an infinite amount of axioms).
A complete axiomatic system would contain contradictions. e.g. The sentence: ‘apples are fruit AND apples are NOT fruit’ would be True in a complete system(along with every other possible sentence).
However absurd it might sound, that is exactly the nature of The Universe. That is because of the very definition of The Universe (with capital U). “The Universe is everything that exists” and as such, every possible statement (and mathematics too!) is part of The Universe. (From wikipedia : More customarily, the Universe is defined as everything that exists, (has existed, and will exist))
In our horizon, it is obvious that ‘apples are fruit’ is correct, and ‘apples are NOT fruit’ is incorrect. This horizon is called the observable universe.
For The Universe, however, this is not ‘decided’. The universe is Eternally Indecisive on Any Logical Statement.This makes both sentences True AT THE SAME TIME: the proposition ‘Apples are fruit AND apples are NOT fruit’ is true! (along with any other conceivable proposition)
To quote the lyrics i posted:
Crave for the indecision.
Awaiting the ultimate answer.
Restless in infinite tediousness.
Verity uprises from Nihilant.
That is the nature of The Universe.
So to sum up: The Universe is complete(and inconsistent). Both completeness and consistency are PART OF mathematical logic(and as such part of The Universe).
Science DOES obey logical laws, and is useful, because it describes the finite world we see successfully(science is consistent and incomplete,like the observable universe). We can and will always be able to see only a finite part of The Whole.
There is no thing that can describe The Universe(by definition). There is a way to describe something inside The Universe (the observable universe). It is called science.
Reading about the conversation between Bertrand Russell and Ludwig Wittgenstein might help clear things up for you.
It would be nice for the conversation if you try and say something about what i write, or explain something. Why would you think i don’t understand Godel’s theorem? Why did you say ‘ If the universe is consistent, it cannot be complete.’? I said twice that The Universe is complete.
Let’s be accurate from now on, please use the phrase ‘The Universe’ when you talk about ‘Everything that exists(existed and will exist)’ and the phrase ‘the observable universe’ when it’s about something finite inside of ‘The Universe’. As i said before you seem to confuse the two.
P.s. I don’t see how my full name helps, but here it is. Thomas is actually my last name(strange but true), my fathers name is Jordan Thomas. Sad that you call me a coward, i have good intentions. I don’t believe in God, and i believe God does not exist(in most definitions used by believers). You can define God in such a way that He can exist, he will not be ‘omnipotent’ though, but all mighty compared to us (like a super-intelligent being). And that’s because of Godel’s theorem, he actually proved the opposite.
Mark,
99% of the atheists who come here and post do so anonymously and disappear when I call them out on it. I have found you cannot have an honest conversation of this nature with an anonymous person. Such a person is a coward. Thanks for not being a coward.
You said
“That is because of the very definition of The Universe (with capital U). “The Universe is everything that exists” and as such, every possible statement (and mathematics too!) is part of The Universe. (From wikipedia : More customarily, the Universe is defined as everything that exists, (has existed, and will exist))”
This is an unhelpful definition and using Wikipedia as a reference is not a good idea as most people know.
The universe as I am defining it here (see my article) is what we can see and measure, which came into existence at the time of the Big Bang. What science has access to.
That is NOT the same as everything that exists. By definition, whatever else exists, whatever it may be, is not available for us to observe – other universes etc. Including EVERYTHING that *might* exist (including God or whatever else) renders “Universe” with a capital U so conflated with everything else as to render it impossible to even properly ask the questions.
And if you believe in reason and logic, cause and effect, something else necessarily exists. Godel’s theorem gives us a clue as to what it can and cannot be.
Sir, you clearly do not understand Godel’s theorem, despite your claims to the contrary. I know this for a fact because you are trying to tell me that a universe can be both consistent and complete. It cannot. If you think it can, then go back and read the theorem. The only way something can be both consistent and complete at the same time is if it’s infinite and indivisible.
The universe is neither.
Thanks for posting.
‘The universe as I am defining it here (see my article) is what we can see and measure, which came into existence at the time of the Big Bang. What science has access to.’
But this is the observable universe. It is finite, and Godel’s theorem makes a consistent system incomplete when there are INFINITE truths about it, some of which one cannot prove. It is possible to know everything there is when we deal with a finite number of truths! One must show that one of those finite truths is inaccessible, which you didn’t do, and neither Godel’s theorems do. In fact a system with a finite number of truths is ‘weaker’ (mathematically) than the systems Godel deals with in his theorem. There exist ‘weaker’ systems that are both complete and prove their own consistency! Also the mathematical use of ‘weaker systems’ doesn’t make them less useful or illogical.
If we take science as ‘incomplete’, that does not mean what you claim it to mean. Incompleteness in mathematics does not mean that something is ‘unfinished’ or that it ‘misses something’. Incomplete axiomatic systems contain true or false statements that can be expressed in those systems but can’t be proven within them. It doesn’t follow from this that there must be ‘something’ outside of the system. All this is irrelevant to the discussion anyway, since we deal with finite things.
For the observable universe, we can have a consistent theory. But since the observable universe is finite, we can’t have any meaningful discussion about completeness in this sense. How do you know that unprovable truths exist and can be expressed in the observable universe? I saw no proof of this so far. You must prove this is true. Godel’s theorems don’t deal with finite things at all. They deal with a finite number of axioms that can create an infinite number of Truths or Falsities. That’s not the case for us.
I never said the Universe is both consistent and complete. I said 2 times that from the definition “the Universe is everything that exists(existed, and will exist)” follows that the Universe is complete (and inconsistent)(From our point of view). You still haven’t addressed this, but again it doesn’t matter when it comes to Godel’s theorems. In particular, Godel’s theorems do not apply in systems with an infinite number of axioms(and truths) or systems with a finite number of axioms and truths. The first is for the Universe, and the second is for the observable universe.
If anything Godel proved the opposite, that God cannot exist no matter what, unless you define god as the Universe (everything). Then God and Universe are the same, we just use different words for the same thing. In every other case, if God exists, he is not be omniscient or omnipotent, because he would necessarily be incomplete(or ‘weak’ in the mathematical sense). He is only a being super-intelligent compared to us, much like humans compared to ants.
“By definition, whatever else exists, whatever it may be, is not available for us to observe – other universes etc. Including EVERYTHING that *might* exist (including God or whatever else) renders “Universe” with a capital U so conflated with everything else as to render it impossible to even properly ask the questions.”
It’s not useless. First of all we have the definition. We can sometimes deal with questions about the Universe using the definition(e.g. completeness).After this, the important thing to consider is time. In time we might be able to know or see more through science.
‘And if you believe in reason and logic, cause and effect, something else necessarily exists. ‘ This webpage doesn’t prove any of this so far. Godel’s theorems do not deal with this. If anything, the more we learn about the world the more it seems that only ‘nothing’ exists. Fluctuations of this ‘nothingness’ is the reason we are here.
The most plausible thing is that we had this conversation an infinite amount of times in the past and we will do so again in the future. If the eternal recurrence is real, where is god to be found? I just fail to see where there is room for Him, and have never seen any reasonable claims or proof for the opposite.
I’m having a hard time believing you have actually read Godel’s work or my article. Please still yourself and go through the process of understanding what is being said here.
Godel says:
“for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”
If a statement is true and if it is possible for that truth to be known, it necessarily follows that there must something outside the theory that makes it provable. Otherwise none of us would have any grounds for saying “a statement is true.” All of mathematics and science implicitly assumes that some objective truth or reality does exist, after all, and that we are attempting to understand it.
You said
“How do you know that unprovable truths exist and can be expressed in the observable universe? I saw no proof of this so far.”
I refer you to Godel’s theorem itself, and his proof of that theorem.
Many things you say here hang in mid-air, and appear to be pure conjecture or wishful thinking. Prime example being “Fluctuations of this ‘nothingness’ is the reason we are here” and “The most plausible thing is that we had this conversation an infinite amount of times in the past and we will do so again in the future.”
Please back up these statements with empirical evidence.
“for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”
That can only hold for certain if there is an infinite amount of arithmetical statements. If the amount of statements is made finite through the definition of the formal theory then the ‘G’ statement (which is true but not provable) might not belong to the formal theory! This means that the statement is ‘nonsense’ inside this theory.So there is no ‘truth’ left outside the theory. Again as i said this happens in ‘weaker’ formal systems, which are consistent and capable of proving their own consistency.
You must show that for a finite amount of truths (because there is a finite amount of possible configurations in the observable universe), such a self-referential statement exists inside our universe. Where is it? I haven’t seen anyone ‘point’ to it or show it yet. Certainly not here in this webpage.
‘I refer you to Godel’s theorem itself, and his proof of that theorem.’
As i said that deals with a certain part of mathematical logic. You need to prove this holds true for the observable universe. Where is this correspondence proved? So far the opposite is observed. In particular, you can have finite systems that Godel’s theorem cannot deal with, it just becomes meaningless because from the definition of those systems you cannot construct such an unprovable but true sentence.
‘Please back up these statements with empirical evidence.’
I cannot back up infinity with empirical evidence. There are theories which predict such things(e.g. eternal inflation theory).These theories make predictions about both the world around us(and have been tested) and the Universe(which have not YET been tested).
There is empirical evidence of scientific theories such as inflation theory. These theories predict an infinite number of universes. The important thing in the usefulness in science is not only what you observe, but what you can predict using theories. That is why we must give humanity,science and knowledge time. Time is the most important thing. We know more than we ever did, and soon we will know even more. The book “Our mathematical Universe” which i read recently has both empirical evidence, predictions and ‘speculations’.
I shouldn’t have to explain what is happening on the forefront of physics and cosmology. You should aim to educate yourself, instead of asking me for the evidence(i am no physicist so i take their word and data for it,i don’t know nearly as much as them as well). I study mathematics.
When i know something, i speak with certainty. When i don’t i ‘speculate’ but make it clear: “If anything, the more we learn about the world the more IT SEEMS…” I never stated it for certain truth. I just said what it APPEARS to be true with our knowledge of the world so far. “The MOST PLAUSIBLE thing is …” again the same. When it comes to Godel’s theorems on the other hand, i am certain.
I still remember when i encountered the theorem for the first time, i was around 19 years old. I immediately realised it was proof that omnipotence and omniscience cannot exist. Unless you speak about ‘everything that exists, existed and will exist’.
Religion has its own importance. In the ancient human societies religion brought order and stability. This explains why people of our time are still religious. They need religion for their inner well-being. Animism was very important for the evolution of human societies, and all religions come from animism. But that is as far as it gets us.
“There is empirical evidence of scientific theories such as inflation theory.”
A theory is not evidence. A theory is a theory. Do you not know the difference?
Where is your empirical evidence?
Show me where in Godel’s theorems it has been proven that “…That can only hold for certain if there is an infinite amount of arithmetical statements.”
“A theory is not evidence. A theory is a theory.”
You have a theory, then you try to find evidence that agree with the theory’s predictions. If your theory holds, then it describes a part of the world successfully. It is useful. This is how science works.There is no absolute knowledge. If through experiments we are 99.9999% certain that inflation theory is true, that is good enough for us. You can get even closer by successive experiments. We have EVIDENCE of inflation theory’s predictions,there is similarity in background radiation from distant points in the observable universe. As for particular evidence, just google ‘inflation theory evidence’, i am not a physicist.
“Show me where in Godel’s theorems it has been proven that “…That can only hold for certain if there is an infinite amount of arithmetical statements.””
Who said anything about a proof? this is a LIMITATION of Godel’s theorems. It is an inherent limitation because of the DEFINITION of mathematical systems that Godel’s theorems talk about. Godel’s theorems do not apply to every single mathematical construct. As i said, it has been PROVEN that there are certain systems Godel’s theorems CANNOT DEAL with. There are ‘weak’ mathematical systems that are both consistent and can prove their own consistency. There are other mathematical systems that escape Godel’s theorems as well. For some systems, it is possible to be consistent AND prove their own consistency.
There are certain hypotheses Godel’s theorems have to satisfy in order to apply to an axiomatic system. So i ask again, where is the correspondence between the hypotheses necessary for Godel’s theorem and the observable universe? I see nothing of the sort in this website.
There is no empirical evidence of other universes etc. That is pure unbridled conjecture. There are some models that invoke other universes but this is speculation.
Godel’s work was a proof. It applies to any system above a certain level of complexity. it applies to Euclid’s postulates, for example, which are much simpler than the universe.
You are free to reject the notion that reason and logic apply to the universe. Then and only then can you assert that Godel’s theorem doesn’t apply. In which case you pull the rug out from under science and reject it outright. However if reason and logic apply to the universe, and if science is valid, then Godel’s theorems apply to the universe.
Perry
Geometry is not the issue in your posted essay. I’m just saying you have been continually playing “bait and switch!” You are applying Godel’s concepts to the universe. You refuse to understand him, misapply him and then go off on Geometry’s issues with its axioms. This has nothing to do with Godel’s work.
All of your comments in your latest post go to using Geometry’s issues with its axioms as a divergence while you continue to ignore the core points I’ve made about GODEL! Go back to my last post and firstly answer the issues I’ve raised. Then maybe we can have a fair and reasonable discussion.
However if you want to go to Geometry then firstly take down your flawed essay based on Godel and write an essay based on Geometry alone. So far it’s all been “smoke & mirrors” in your posts while you mixed in Geometry and ignored my critique on your misuse of GODEL.
I am not the person who is in a loop. You are. I have answered your objections about Godel vis a vis the universe. If you choose to ignore me, that is your decision.
You say geometry is complete but you can’t prove its axioms.
If you can’t prove its axioms, it’s incomplete.
I have not ignored you, I have placed the burden of proof on you. If you want to continue to insist that geometry is consistent AND complete, prove the fifth postulate.
Perry you wrote:
Yes I am reading an infinity assumption into the 2nd postulate. (This is not the first time I have brought this up with you either. But you still have not addressed it.) That is because infinity is implicit in the word “indefinitely” in the 2nd postulate definition:
“2. Any straight line segment can be extended indefinitely in a straight line.”
Perry, I have answered your issue about the 2nd postulate. Read my response again. Geometry is not dependent on the universe being infinite. We do not have to prove whether the universe is finite or infinite for Geometry to proceed from its beginning points or axioms. The 2nd axiom simply means one can extend a straight line continuously/indefinitely and of course this can only be done within space/time. It does not matter whether the latter is finite or infinite. The issue is outside the range of geometry.
You claim that you are addressing my issues while I am dodging your questions but I have to say that is very funny and strange. You simply ignore whatever I bring up and go on. Here I am again, responding to your points but read your own post. You do not even mention a single issue I raised even from my prior to last post.
Perry several times I have explained the reasons why you are confused. Here it is again. You are mixing up issues from Geometry and Gödel. Gödel means something very specific when he says a formal system is “incomplete” or “complete.” He does NOT mean what you do. Here are two unanswered issues/comments from my prior to last post. You have thus far ignored them and these contain the reasons for your continued confusion:
1. You wrote: “The parallel postulate is a perfect example of an undecidable result that is the necessary outcome of an incomplete system.” NO … the parallel line postulate is not a result or the product of a calculation. It is an axiom or postulate and these are the beginning points of formal systems from which processes stem to create results. Geometry does not create undecidable results like the incomplete formal systems that Gödel illuminated. This is a reason why geometry is not incomplete in the Gödel sense. (Please READ the last two sentences again).
2. You wrote: “Gödel does not require all incomplete systems to be consistent. It requires all consistent systems to be incomplete.” Perry you are again showing off your misunderstanding of Gödel. You are wrong on both scores! “Consistency” is one of the three necessary features of incomplete formal systems; a finite list of axioms and that the systems must include the power of full arithmetic are the other two. However there are consistent and complete formal systems like all forms of geometry, the smaller arithmetic called Presburger Arithmetic and others. (Please NOTICE that you are stuck on geometry due to mixing up concepts; while you ignore other consistent and complete formal systems such as Presburger Arithmetic).
You suggested I mean the following:
1) Euclidean geometry is complete (Yes it is, in the Gödel sense. )
2) Euclidean geometry is consistent (Yes, it is consistent as per its logic stemming from its axioms/postulates. It also does not produce “undecidable” results in the Gödel sense. The 5th postulate has been ruled “inconsistent” but this is only when compared to the other four postulates).
3) Gödel’s theorem doesn’t apply to Euclidean geometry in the first place because of Tarski. (NO … Tarski’s work is a footnote to and/or additional to Gödel’s work. Euclidean Geometry is NOT an incomplete formal system because it does NOT include the power of full arithmetic. Gödel’s theorems rule out all such formal systems from being “incomplete.”
Perry you wrote: “Geometry also assumes that space is independent of time, an assumption which is at odds with everything we currently know about the universe. Geometry, contrary to your statement above, is performed without consideration of time.”
And so what? The concept of space/time was not developed until after Einstein’s Special Theory. His former teacher Minkowski firstly elucidated it. Time is also not explicit in arithmetic, algebra etc.
You wrote: “Since you say that you can do geometry in finite space, then SHOW me how you can still do geometry when there is an upper limit to how long a line can be.”
If space/time is finite then Geometry and other things are confined within its boundaries/limits. This means that straight lines, Geometry and other things only have meaning within space/time. However in principle, Geometry could be done within a finite or infinite space/time.
You have continued to mix up issues and meanings as applied to geometry with those in Gödel. I have proven these are NOT THE SAME as Gödel’s meaning is very specialized and concise. You either do not want to admit it or you do not understand it; but it is important as your essay applies Gödel’s ideas to the universe. It is your turn to answer to this very specific criticism. If you continue to dodge this issue then we wasting time riding on your merry-go-round of nonsense. It’s your turn … “I am not letting you off the hook!”
You said: “We do not have to prove whether the universe is finite or infinite for Geometry to proceed from its beginning points or axioms.”
Yes you are absolutely right. But my point is that those same beginning points or axioms are not formally provable. They are assumed.
In the theoretical constructs of Euclidean geometry, we POSTULATE that a line can be extended indefinitely. Everyone accepts this as being self evident. However no one has proven it. Which is my point.
If you can’t prove it, it’s incomplete.
You said: “Geometry does not create undecidable results like the incomplete formal systems that Gödel illuminated.”
Until the type of coordinate system is specified, the truth of the parallel postulate cannot be verified and thus the 5th postulate is undecidable within Euclid’s original formulations.
You said: “Consistency” is one of the three necessary features of incomplete formal systems.”
Mathematics is littered with incomplete formal systems that were later proven to actually be wrong, i.e. inconsistent.
You said: “Geometry does not create undecidable results like the incomplete formal systems that Gödel illuminated. This is a reason why geometry is not incomplete in the Gödel sense.”
The dependency of the 5th postulate on coordinate system is an explicit counterexample of what you have just said. Which is why geometry is incomplete in the Godel sense.
You said: “You suggested I mean the following: 1) Euclidean geometry is complete (Yes it is, in the Gödel sense.)”
If it’s complete, then prove Euclid’s 5th postulate.
You said: “The 5th postulate has been ruled “inconsistent” but this is only when compared to the other four postulates).”
Please show me where someone has proved the 5th postulate is inconsistent.
Even though Euclidean geometry does not do full arithmetic, it is STILL incomplete. Which tells me that even some systems that do not meet Godel’s criteria for sophistication, still have unprovable axioms.
You said: “If space/time is finite then Geometry and other things are confined within its boundaries/limits. This means that straight lines, Geometry and other things only have meaning within space/time. However in principle, Geometry could be done within a finite or infinite space/time.”
The above is only true if we assume Geometry only has meaning within space/time. Jarmo, if that were true then Geometry would be necessarily false. The fact that is is consistent within its own framework means that logic itself is independent of space and time.
I assert that geometry does have meaning outside of space/time; that logic does have meaning outside of space/time; and the space and time are subject to the laws of logic. I cannot prove this, but I believe that logic is higher than space, time and the material universe. All those things are subject to logic. The profession of science implicitly assumes this to be true.
With that assumption I can practice geometry without contradiction. Within what appears to be your own materialist worldview, geometry is empirically FALSE. Why? Because you cannot practice geometry without the 2nd postulate. If truth can only exist within space/time then the 2nd postulate is false and therefore Geometry only exists in the world of unicorns and fantasies.
Within a hard materialist worldview, much of mathematics is nonsense. Which is why I am not a materialist.
Again you have claimed geometry is complete. So prove the 5th postulate.
Perry, there is a pattern developing in your responses. Why is it that you expect me to answer your criticism and you ignore mine? If we are having a fair discussion/debate then it has to go both ways. I would take time to reiterate my criticism from my last post but you can read it again. I also expect responses, just as you do. If I continually ignored your responses then based on how you play we would not be exchanging. Therefore I ask you to please answer my criticism and the points I raise; not just simply go on to add more of your own.
Geometry isn’t about whether space/time is finite or infinite. That issue is outside its range and it merely accepts that it can only be done within space/time. You are reading an infinity of space assumption into the 2nd postulate but it does not really say that; as a line can only be extended in space. The reasonable realization is that in the special case of space time ending then so would geometry and therefore it’s not an issue. In other words geometry can only be done within space/time and this is a-priori to geometry and really to everything else within space/time. However, in principle geometry can be done in either a finite or infinite space/time.
I am doing my best to keep this conversation on the original points. I believe I have answered your criticisms and I will certainly get to any new ones. But let’s take this a step at a time.
Yes I am reading an infinity assumption into the 2nd postulate. (This is not the first time I have brought this up with you either. But you still have not addressed it.) That is because infinity is implicit in the word “indefinitely” in the 2nd postulate definition:
“2. Any straight line segment can be extended indefinitely in a straight line.”
Please correct me if I’m wrong but your arguments appear to be trying to have your cake and eat it too. Thus far you have been telling me:
1) Euclidean geometry is complete
2) Euclidean geometry is consistent
3) Godel’s theorem doesn’t apply to Euclidean geometry in the first place because of Tarski.
But Tarski’s formulations are a special case. One need not look far to find 100 authoritative mathematical sources that attest that Euclidean geometry is incomplete (= only provable if you assume without proof that the axioms are true.) What you have really been saying is that you get to assume Euclidean geometry is consistent because no contradictions have been found. But that is not the definition of consistent. The standard for consistent is mathematical proof, not inference from lack of failure.
No one has proven the 5th postulate. So you don’t get to claim absolute consistency. You can only postulate consistency.
No one has proven the 2nd postulate either. It rests on assumptions which you take for granted but have no proof for and which are in fact in conflict with what we actually know about the physical universe.
I am addressing your questions. You are dodging mine.
If you want to claim that geometry is both complete and consistent, you will have to prove the 2nd and 5th postulates. Geometry DOES assume that space is infinite. Since you say that you can do geometry in finite space, then SHOW me how you can still do geometry when there is an upper limit to how long a line can be.
Geometry also assumes that space is independent of time, an assumption which is at odds with everything we currently know about the universe. Geometry, contrary to your statement above, is performed without consideration of time. (Which is yet another implicit, unproven assumption which you have not stopped to question let alone prove.)
You don’t get to claim consistency and completeness without supplying proof. I am not letting you off the hook.
Perry all formal systems proceed from their axioms/postulates. There isn’t a necessity or implication/assumption for space to be infinite in the axioms of Euclidean Geometry. Infinite space does not have to be assumed as Euclidean and the other geometries are testable even in finite space. All that would happen in any extended test is it would halt where space ends. However we do not really know if space/time is finite so your comment “It certainly is not in the real universe” is assumptive. In any case, this is off the topic of Gödel and the universe.
You wrote: “The parallel postulate is a perfect example of an undecidable result that is the necessary outcome of an incomplete system.” NO … the parallel line postulate is not a result or the product of a calculation. It is an axiom or postulate and these are the beginning points of formal systems from which processes stem to create results. Geometry does not create undecidable results like the incomplete formal systems that Gödel illuminated. This is a reason why geometry is not incomplete in the Gödel sense.
You wrote: “Gödel does not require all incomplete systems to be consistent. It requires all consistent systems to be incomplete.” Perry you are again showing off your misunderstanding of Gödel. You are wrong on both scores! “Consistency” is one of the three necessary features of incomplete formal systems; a finite list of axioms and that the systems must include the power of full arithmetic are the other two. However there are consistent and complete formal systems like all forms of geometry, the smaller arithmetic called Presburger Arithmetic and others.
Perry, even though mathematical systems are imperfect it does not mean the universe is incomplete. We use math as applied to physics to create the best possible approximations of elements in the universe we study. We call the later theories or models in physics but none are considered untouchable. Your entire treatise is ill conceived.
Second postulate:
“2. Any straight line segment can be extended indefinitely in a straight line.”
You said:
“Infinite space does not have to be assumed as Euclidean and the other geometries are testable even in finite space.”
Tell me how you propose to confirm Euclid’s 2nd postulate in finite space.
Perry I erred in my second to last paragraph in my last post. The first & second sentence should read:
“Again you are wrong. Gödel says undecidable results are the necessary outcome of incomplete systems.”
I had incorrectly written “the necessary outcome of INCONSISTENT systems.” This is wrong as Godel requires all incomplete systems to be consistent. I was rushed and hence erred.
“even Euclidean Geometry does not assume anything that is outside itself. It includes the parallel line postulate even though we lack other proofs for it; it is still within the system. Also non-Euclidean geometries do not even include the parallel line postulate.”
Euclidean geometry assumes an infinite amount of available space. No one has proven that this is a valid assumption. It certainly is not in the real physical universe.
The parallel postulate is a perfect example of an undecidable result that is the necessary outcome of an incomplete system.
Minor point: Godel does not require all incomplete systems to be consistent. It requires all consistent systems to be incomplete.
Perry once again, you are misdirecting our discussion about Gödel by reverted to discussing geometry and the parallel line postulate. In any case, you have to realize the article speaks to three forms of non-Euclidean geometry where the parallel line postulate does not hold. “Absolute geometry” is one of these three but it is a special “independent” class where the 5th postulate is ignored. Citing this line about a form of non-Euclidean geometry has nothing to do with our discussion about Euclidean geometry. (The article you cited also includes an error in citing “special relativity” when she/he should have said “general relativity”; but this is beside the point).
You are also mixing up ideas from Gödel’s Incompleteness studies and misapplying them to geometry. All forms of geometry do NOT fall within the class of formal systems that he addresses. You are wrong to say, “a consistent system always assumes something outside the system that you cannot prove.” This is not so; even Euclidean Geometry does not assume anything that is outside itself. It includes the parallel line postulate even though we lack other proofs for it; it is still within the system. Also non-Euclidean geometries do not even include the parallel line postulate.
You also wrote: “Gödel does not simply say that systems will always produce an undecidable result. It also says that consistent systems necessarily refer to something outside of themselves that you cannot prove.”
Again you are wrong. Gödel says undecidable results are the necessary outcome of inconsistent systems. When one appears and even if you add an axiom, another will always appear. This does not happen in any form of geometry since it does not include the power of full arithmetic.
Perry, I am sorry to say that you are blowing smoke. What you have written is nonsense. Over the course of many responses you simply continue to mix up and misapply concepts from Gödel and geometry. Perhaps you are fooling yourself due to your need to prove your god?
Perry I am bewildered by your continual confusion. Not only is my statement correct but you failed to show specific reason for your criticism. Further you went on to claim I do not understand Gödel; but it only shows you do not understand him. Gödel did NOT prove that geometry is incomplete but it is the exact opposite. You simply keep misunderstanding what Gödel means. Gödel proved that any formal system that includes his three features will always produce a result that cannot be determined or decided from its axioms/postulates. This is why he called these systems “incomplete” and/or “undecidable.” Geometry does not include all the power contained in full arithmetic and therefore cannot be “incomplete” in the Gödel sense.
Further Gödel is NOT speaking about the truth of the axioms of formal systems. In fact he showed that it does not matter even if one adds an axiom that renders an unexpected result “decidable” as another “undecidable” result will eventually appear. This goes on and on … adding axioms can’t “complete” the formal system.
The criticism about unproven starting points goes to the 5th postulate in geometry and not to Gödel. This is a different subject; but it’s one that you keep inserting into a discussion that is supposed to be about applying Gödel to the universe. You went on to claim I am only right if I prove Euclid’s 5th postulate. Perry that is bizarre as it does not have anything to do with what I am saying!
There are other imperfections in mathematics. I can cite the irrational numbers as a case of inherent imprecision but we still use them to do math. My point is NOT that Gödel’s incompleteness is wrong; nor is it in any way based on me or anyone else having to solve the 5th postulate problem. I am saying that even though these imperfections exist you still cannot apply Gödel’s Incompleteness Theorems or their results to the universe and then arbitrarily claim one has to include “god” to complete the universe! None of this follows from Gödel and you cannot come to the same theological conclusion arguing from geometry.
In physics we study results and from these we infer laws, mechanisms/principles. In the process we use different forms of math depending on what is being studied. At best we come to very good approximations but scientific theories can always be improved on and even replaced due to more precise detections and better interpretations.
Perry, Gödel’s Incompleteness Theorems only show that some formal systems i.e. mathematics will always include an undecidable result and it does not matter how many axioms we insert. It is in this sense that such formal systems are incomplete. It does not mean the universe is incomplete or that this in any way warrants anyone to cite their idea of “god”!
Quoting http://www2.mtsd.k12.wi.us/Homestead/users/ordinans/Euclidean.htm:
“Absolute geometry assumes that the parallel postulate and all its alternatives do not exist; in other words, absolute geometry ignores the fifth postulate. This allows all the theorems of absolute geometry to hold true in all other forms of geometry. However, absolute geometry is incomplete because many situations are left unsolved. For example, the statement “The sum of all the angles in a triangle equals 180 degrees” cannot be confirmed in absolute geometry. In Euclidean geometry, the statement is true because of the parallel postulate (using the sum of interior angles is two right angles), but in absolute geometry the statement is left unanswered since the fifth postulate is used to prove that previous statement.”
You can easily find similar statements all across mathematical literature over many decades.
Euclid’s unproven 5th postulate is an axiom that enables you to decide otherwise undecidable statements in a specific context. But since it’s not proven, Euclidean geometry is still incomplete.
You are exactly right. Adding axioms can’t “complete” the formal system. That is exactly what I have been saying all along. A consistent system always assumes something outside the system that you cannot prove.
Godel does not simply say that systems will always produce an undecidable result. It also says that consistent systems necessarily refer to something outside of themselves that you cannot prove.
Once we arrive at a proper understanding that yes indeed Godel’s results apply to Euclidean geometry we can then turn our attention to the universe.
Perry the essay I have been commenting on is the one you posted and incorrectly cited Gödel’s Incomplete Theorems as if to prove the universe is incomplete without your god. This is nonsense as I have amply proven but you have retreated and repeatedly tried to redirect the discussion by citing the 5th postulate problem in geometry. This doesn’t change your essay on Gödel.
I have proven that you have taken Gödel out-of-context; and misapplied his principles to try and show the universe is incomplete. Further adding your “god” into the fray could not possibly “complete” the universe in the sense of Gödel’s Incompleteness Theorems. It means you are wrong from the outset and you are wrong in your conclusion. It also doesn’t matter whether you add the 5th postulate issue from geometry into the discussion. It cannot change any that I have proven about your misapplication of Gödel.
Nevertheless, my comments about Euclidean Geometry are correct as its outcomes have been shown to be consistent in a flat space/time applications. This is what it is designed for. You claim that no one has proven the 5th postulate but axioms/postulates are starting points and accepted as such. The logic and processes of formal systems work from its axioms onward. As I have vainly stated several times “consistency” in the Gödel sense only goes to the flow of logic from axioms. It is in this sense that formal systems that have all the features he outlined are inconsistent. Again, this has nothing to do with the inconsistency criticism of the 5th postulate which is strictly comparative to its other four. However there has never been a single “inconsistency” shown in any outcome within the range of Euclidean Geometry.
However and again, all forms of geometry are complete in the sense that Gödel meant. If you want to keep ignoring this it simply shows your inability to admit to error.
Your last comment is completely haywire. I was simply following your imaginative ideas; and my conclusion is completely correct. Even if geometry were one day be shown to have logically inconsistent outcomes (which it never has within its range) it would not make a hill of beans to its consistency in the Gödel sense. The terms used in both systems are the same; but they do NOT mean the same thing. Perry wake up!
You said:
“my comments about Euclidean Geometry are correct as its outcomes have been shown to be consistent in a flat space/time applications. This is what it is designed for. You claim that no one has proven the 5th postulate but axioms/postulates are starting points and accepted as such. The logic and processes of formal systems work from its axioms onward. As I have vainly stated several times “consistency” in the Gödel sense only goes to the flow of logic from axioms.”
The above statement is incorrect. You still do not understand Godel. Godel’s whole point is that axioms/postulates are UNPROVEN starting points and must be accepted as such until proven wrong. Until proven right the systems are incomplete.
You can protest all you want, and you can falsely claim to have countered my argument if you so choose. But until you have proven Euclid’s 5th postulate, Euclidean geometry remains consistent but incomplete, just as Godel’s theorem predicts.
Prove Euclid’s 5th postulate. Then and only then can you support your claim that you are right.
Perry when I post a response I can no longer see it in the pending moderation mode. Yet you claim I should still see it. Again, I only see it after you post a response. This is a very recent change. Has anyone else noticed this and commented?
Perry you seem to want to continue discussing Euclidean Geometry that is not even included in your essay on Godel and the universe. I suspect that you know that I have proven that as far as applying Godel to the universe is a complete NO go!
The inconsistency issue that is raised against Euclidean Geometry goes to comparing the first four postulates/axioms to the fifth. The later was judged to be lacking in rigor and hence not consistent with the first four. Some went as far as to suggest the fifth postulate was redundant as it could be derived from the first four. However this turned out to be a blind alley. Euclidean Geometry requires the fifth postulate, as even you have pointed out.
However there has never been a contradiction shown in the carrying out of Euclidean processes that are within the range of its function. Therefore it is judged to be consistent and logical in this sense. You simply have to shore up on understanding how the same terms are used in differing contexts/systems.
In Godel “consistency,” as one of the three features of an “incomplete” formal system, means its processes and conclusions are without contradictions. Since all forms of geometry do NOT incorporate all the functions of full arithmetic (a second feature of an incomplete formal system) these are complete in the Godel sense.
Lastly let me play along with your fictional scenario. If any form of geometry was shown to create a contradictory result it still could not make it “incomplete” in the Godel. All geometries cannot be so judged, as these are outside the range of Godel’s Incompleteness Theorems.
This fictional contradictory result would only imply a limitation for its range of operation. It would not not undermine the validity of its proven results. The scenario could point to the development of a broader geometry in much the same way that Newton’s limited but-still-successful theory of gravitation was incorporated by Einstein’s General Theory. The later of course went beyond Newton’s range.
My original, larger Godel essay is at http://www.perrymarshall.com/godel, which does reference Euclidean Geometry. I link to it in this above article.
No one has proven Euclid’s 5th postulate. Thus it is an unproven axiom.
You said:
“there has never been a contradiction shown in the carrying out of Euclidean processes that are within the range of its function. Therefore it is judged to be consistent and logical in this sense.”
The above is wrong! Consistency does NOT mean “never proven to be inconsistent” (!) Consistency means PROVEN to be true. With formal mathematical logic. With a PROOF.
Euclid’s 5th postulate is not proven. Therefore it is an axiom, something which we assume to be true but have been unable to prove. We infer that it is true because of our inability to prove it wrong.
You said:
“Since all forms of geometry do NOT incorporate all the functions of full arithmetic (a second feature of an incomplete formal system) these are complete in the Godel sense.”
Because Euclidean geometry does not incorporate all the functions of full arithmetic, it is not sufficiently sophisticated to qualify as a “effectively generated formal theory that proves certain basic arithmetic truths.”
You said:
“ If any form of geometry was shown to create a contradictory result it still could not make it “incomplete” in the Godel…”
That is right. It would be inconsistent. It might still be complete (thought it might not be complete.) A system could be inconsistent AND incomplete at the same time.
Perry,
I have wondered why my posts no longer appear in the to be moderated mode? Now I only see them after you respond. Have you changed the settings?
Yes,I brought up Non-Euclidean Geometry as an example of a complete formal system. It disproves your claim i.e. “I’m saying that if any system is consistent, it’s incomplete.”
Both Euclidean & Non-Euclidean Geometry are complete in the Godel sense. Euclidean Geometry has be critiqued as “inconsistent” (mainly the 5th postulate compared to the first four)but it is “complete” via the lens of Godel. A point that I have raised several times but you ignore it and keep falsely claiming “Euclidean Geometry is incomplete.” You should be saying it is “inconsistent” but this has a different meaning than Godel’s intention.
Axioms/postulates are part of formal systems for their intuitive truth; some can be proven by other lines of thought and some do not have the additional support. However this is the meaning of inconsistency.
Further axioms/postulates are not “outside the system” as you falsely claim.
I agree that all five postulates are essential for Euclidean Geometry. It is used for many scientific/technological purposes including engineering, architecture, surveying and parallax. Astronomers use the principle of parallax to measure distances to the closer stars.
I am still saying you are leading the discussion off topic. Your essay is about Godel as applied to the universe. Shortly I’ll show other instances where you erred in your essay.
Your posts are always moderated. However you yourself may still be able to see them.
Euclidean geometry does not appear to be inconsistent; no one has found a contradiction. Perhaps someone will, in which case it can be judged inconsistent. What we do know is that no one has proven it to be true. If and when it is proven to be true, it will likely require an outside assumption, making it incomplete.