All aboard the Atheist Bus
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@Numeromancer said:
Wow. I thought that a breeze was cool because it removed the ambiant pressure caused by diffusive latency, allowing more of the water to overcome liquid-phase intermolecular forces. But now I know better; it just "receives evaporation". Brillant! You are sooo much smarter than Mr. Zealot.
Phrased rather sloppy, but out of the two approximations, the imaginary guy who says the breeze is heating you is, well, a lot more wrong. My layman's terms describe what is actually happening. And yes, I'm obviously smarter than the strawman Zealot. Are you claiming "his" explanation was more true? It isn't. You can be a dick to me, but everything I said was still completely right :)
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@CDarklock said:
@bstorer said:
You might wanna go reread that theorem; that's not at all what it says.
Yes it is.
"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."
And that would be an "axiom": "a principle that is accepted as true without proof". Since it is true, we clearly accept it as true. Since it is not provable, we clearly do so without proof. Which means we can accurately restate Godel's first incompleteness theorem as follows:
"For any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an axiom."
Unfortunately, the word "axiom" also means other things, so this would be confusing. The original wording is unambiguous, so we retain it. But the fact remains, by saying an axiom always exists, Godel's theorem says we can never eliminate all axioms.
Unless, of course, you want your theory to be inconsistent, ineffectively generated, or informal. And before you pull up "basic arithmetic truths", Stephen Hawking has expressed a belief that this theorem applies to theoretical physics every bit as much as to basic arithmetic. I'll take his word for it; if you want to argue that, go talk to him.
Fail.
The unprovable statements demonstrated by the incompleteness theorem are not axioms themselves, but instead are statements that are true but not provable given the set of axioms that make up the formal theory. One can prove them by adding axioms, but this only introduces more unprovable statements. There are always axioms in any useful formal theory and there certainly was never an attempt to eliminate all axioms -- such a claim on your part shows a very flawed understanding of formal logic. What the incompleteness theorem demonstrates is that any theory that can prove certain arithmetic truths will contain statements that are themselves true but are not provable given the axioms of the theory, thus the axioms themselves are not complete in the sense they can be used to prove the entirety of all true statements that can be expressed in that theory.
Once again, this has nothing to do with proving axioms because the whole fucking point of axioms is that they are assumed to be true without proof. You don't have to prove them. What we now know is that there is no formal theory that can prove arithmetic and all true statements within that theory using only the recursively enumerable axioms that belong to that theory.
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"I can't stand atheists. All they ever do is talk about God".
Heinrich Böll
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@CDarklock said:
And that would be an "axiom": "a principle that is accepted as true without proof".
Yes, axioms are accepted as true. But truth is not sufficient for something to be a proposition. Just because it's true (if it really is) doesn't make it an axiom. To quote the authorities:@Wikipedia said:
In traditional logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject to necessary decision. Therefore, its truth is taken for granted, and serves as a starting point for deducing and inferring other (theory dependent) truths.
The theorem covers more than that (otherwise it would be really trivial and not interesting) . For instance, consider the problem of P vs NP (or Goldbach's conjecture, etc). It may be true but unprovable.
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@CDarklock said:
You misunderstand. The incompleteness theory states that not all sentences in a given theory can be proven to be either true or false, even though they have a fixed truth value. That we cannot derive its truth doesn't make an unprovable sentence into an axiom. We can create a new system wherein the unprovable statement is added as an axiom, but this is not the same thing.@CDarklock said:@bstorer said:
You might wanna go reread that theorem; that's not at all what it says.
Yes it is.
"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."
And that would be an "axiom": "a principle that is accepted as true without proof". Since it is true, we clearly accept it as true. Since it is not provable, we clearly do so without proof.
You are taking a broad definition of axiom. Yes, axiom may mean something else in the dictionary, but we're discussing formal systems. An axiom is a statement assumed to be true in the creation of a formal system. Other statements which are unprovably true aren't axioms because they aren't assumed to be true, they are true, given the axioms are true. A simplification would be to say that axioms are the foundation or starting point, while the Godel sentences are destinations that should be reachable, but aren't.Which means we can accurately restate Godel's first incompleteness theorem as follows:
"For any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an axiom."
Unfortunately, the word "axiom" also means other things, so this would be confusing.
@CDarklock said:
But the fact remains, by saying an axiom always exists, Godel's theorem says we can never eliminate all axioms.
Godel's incompleteness theorem has nothing to do with axiom reduction. You've played fast and loose with words that have clear, specific definitions to arrive at a conclusion that wasn't addressed by Godel because it didn't need to be: it was already well known. Godel's theorem came as a response to work done by Hillbert and others to create a system with the fewest possibile axioms that could prove everything. It was apparent, even before the incompleteness theorem, that some axioms were necessary, but this was because they were trying to create a specific model. A system can be created with no axioms at all, but it doesn't really get you a whole lot if you have nothing to derive from. @CDarklock said:
I don't really see what this has to do with the conversation at hand, other than to attempt to appeal to authority. Stephen Hawking would be just as happy to tell you that you've misunderstand the basic idea.Unless, of course, you want your theory to be inconsistent, ineffectively generated, or informal. And before you pull up "basic arithmetic truths", Stephen Hawking has expressed a belief that this theorem applies to theoretical physics every bit as much as to basic arithmetic. I'll take his word for it; if you want to argue that, go talk to him.
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@dtech said:
Because of this I can't get a accurate image of religous-promoting activities in Texas, but over here there are no people occasionly knocking at your door and asking if they may spread the word of god)
It's no surprise, considering gun laws (and acceptable defence law) there...
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@morbiuswilters said:
What the incompleteness theorem demonstrates is that any theory that can prove certain arithmetic truths will contain statements that are themselves
This statement is true.
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<meta http-equiv="CONTENT-TYPE" content="text/html; charset=utf-8"><title></title><meta name="GENERATOR" content="OpenOffice.org 2.4 (Win32)"><style type="text/css"> </style>It's nice that religion has helped people you know, but I'm afraid what you say simply isn't the case. Just look at the theocracies of this world, look at fundamentalist Christianity, look at the historical role of the Church, look at the Vatican's political affiliations over the past 50 years, think about the nature of faith, unquestioning obedience and belief always in the absence of proof and how one might exploit that to control a nation. A nation where perhaps 50% of people choose not to believe in the Science that is in their faces all day, every day.
Also, personally, I think telling small children that if they don't follow a set of arbitrary rules (and morality does NOT stem from religion), and give unconditional love to a celestial dictator, that they'll burn in a place of unimaginable torture for eternity, is morally reprehensible.
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@some posters before said:
Hunk of characters
Woah! I'm glad I read the thread yesterday before people turned on the overcomplicators.I'll still going to read this stuff when may brain becomes available - after working hours.
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@alegr said:
@morbiuswilters said:
What the incompleteness theorem demonstrates is that any theory that can prove certain arithmetic truths will contain statements that are themselves
This statement is true.
WTF?
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@morbiuswilters said:
@alegr said:
@morbiuswilters said:
What the incompleteness theorem demonstrates is that any theory that can prove certain arithmetic truths will contain statements that are themselves
This statement is true.
WTF?
That's just an example of unprovable statement.
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@vt_mruhlin said:
The real WTF is that atheists waste all that money spamming their beliefs to others, but then whenever you ask them they only talk about how christians are all in-their-face about religion all the time.
The U.S. atheist propaganda juggernaut is either the smartest damn smart-bomb ever, or the christian myth-machine executing it's A game. I hear a lot of christians complaining about atheists trying to push their beliefs on others, but when I open my door it's always a christian trying to convert me - never an atheist, when I watch tv it's always a christian commercial, never an atheist commercial. When I drive my car, it's always a christian billboard or sign, never an atheist billboard or sign. When I see a vanity plate, it's the "In god we trust" one, not the 'There is no god" one. I live in a super-liberal college town, and I see 3+ orders of magnitudes more christian stickers and decals than atheist ones. I've walked past hundreds of firebrand christian preachers in my life, but never a firebrand atheist. I don't understand how anyone could overlook the ubiquitous christian proselytization in the US and complain of atheist spamming unless they were naturally daft, or have burned the love of christ into a bit too much of their brain.
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@obediah said:
when I open my door it's always a christian trying to convert me
Just don't open your door and you can live in a void of nothingness without God forever.@obediah said:
when I watch tv it's always a christian commercial
The TV is trying to save you, man! Can't you see that?
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@GroundZero said:
Also, personally, I think telling small children that if they don't follow a set of arbitrary rules (and morality does NOT stem from religion), and give unconditional love to a celestial dictator, that they'll burn in a place of unimaginable torture for eternity, is morally reprehensible.
Amen to that!
@obediah said:
@vt_mruhlin said:
The real WTF is that atheists waste all that money spamming their beliefs to others, but then whenever you ask them they only talk about how christians are all in-their-face about religion all the time.
The U.S. atheist propaganda juggernaut is either the smartest damn smart-bomb ever, or the christian myth-machine executing it's A game. I hear a lot of christians complaining about atheists trying to push their beliefs on others, but when I open my door it's always a christian trying to convert me - never an atheist, when I watch tv it's always a christian commercial, never an atheist commercial. When I drive my car, it's always a christian billboard or sign, never an atheist billboard or sign. When I see a vanity plate, it's the "In god we trust" one, not the 'There is no god" one. I live in a super-liberal college town, and I see 3+ orders of magnitudes more christian stickers and decals than atheist ones. I've walked past hundreds of firebrand christian preachers in my life, but never a firebrand atheist. I don't understand how anyone could overlook the ubiquitous christian proselytization in the US and complain of atheist spamming unless they were naturally daft, or have burned the love of christ into a bit too much of their brain.
Even though I'm a guy, can I have your babies! Wait no, that would not be the christian thing to do. :( I guess I'll have to settle for a "well said".
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@ahnfelt said:
Neither physics nor dieties are facts (in the sense that they are or can be proven). The difference is that there is observable evidence supporting the theories in physics, whereas dieties are pure imagination.
The Judeo-Christian belief system has scientific support through the fields of literature and archaeology.
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@GroundZero said:
It's nice that religion has helped people you know, but I'm afraid what you say simply isn't the case. Just look at the theocracies of this world, look at fundamentalist Christianity, look at the historical role of the Church, look at the Vatican's political affiliations over the past 50 years, think about the nature of faith, unquestioning obedience and belief always in the absence of proof and how one might exploit that to control a nation. A nation where perhaps 50% of people choose not to believe in the Science that is in their faces all day, every day.
Millions have died at the hands of forced atheist governments. Let's face it: this has little to do with religion and much to do with the hatred and greed of man.
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@operagost said:
The Judeo-Christian belief system has scientific support through the fields of literature and archaeology.
What kinds of support? I remember hearing about historical evidence or something (don't remember where, so I can't say if it's true or not) that showed that Jesus was a real person, but that doesn't prove anything at all about Christianity in general.
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@operagost said:
Millions have died at the hands of forced atheist governments. Let's face it: this has little to do with religion and much to do with the hatred and greed of man.
Agreed. I'm sick of people acting like violence and exploitation can be linked easily with one religion. Most human tragedy comes as a result of one group greedily trying to wield power over another group and it takes the forms of communism, fundamentalist Christianity, National Socialism, Islamofascism and others.
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@burntfuse said:
@operagost said:
The Judeo-Christian belief system has scientific support through the fields of literature and archaeology.
What kinds of support? I remember hearing about historical evidence or something (don't remember where, so I can't say if it's true or not) that showed that Jesus was a real person, but that doesn't prove anything at all about Christianity in general.
First you prove that various historical figures existed, usually through archaeology; then you support the validity of the texts through analysis. Textual criticism can be used to verify the faithful transmission of a text. Biblical MS stand up well to this. On the other hand, most secularists summarily dismiss the validity of biblical texts merely because they include supernatural events ("miracles"); this is circular reasoning and has no place in legitimate research. These kinds of preconceived notions are often a part of the higher critcism of a text, which is why theologians are often wary of that kind of analysis. For example, the "Q" or "Quelle" document theory is the result of higher criticism of the Synoptic Gospels. Much damage has been done by overzealous secular researchers who treat "Q" as if it is a proven fact, even though absolutely no archaeological evidence of such a document exists.
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@morbiuswilters said:
I'm sick of people acting like violence and exploitation can be linked easily with one religion. Most human tragedy comes as a result of one group greedily trying to wield power over another group and it takes the forms of communism, fundamentalist Christianity, National Socialism, Islamofascism and others.
One very prominent, recent example being the sort of so-called environmentalism that results in bans of DDT.
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@boomzilla said:
@morbiuswilters said:
I'm sick of people acting like violence and exploitation can be linked easily with one religion. Most human tragedy comes as a result of one group greedily trying to wield power over another group and it takes the forms of communism, fundamentalist Christianity, National Socialism, Islamofascism and others.
One very prominent, recent example being the sort of so-called environmentalism that results in bans of DDT.Yes, the "watermelons": green on the outside, but red on the inside.
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@morbiuswilters said:
The unprovable statements demonstrated by the incompleteness theorem are not axioms themselves
If they're true, and they cannot be proven, how are they not principles "accepted as true without proof"?
@morbiuswilters said:
One can prove them by adding axioms, but this only introduces more unprovable statements.
In other words, adding axioms only adds axioms. This appears self-evident.
Indeed, it is itself an axiom, since you can't prove that adding an axiom adds an axiom. But it's true all the same, and only an idiot would deny it.
(Suggested idiot argument: "Adding the axiom may remove other axioms by allowing them to be proven, making the theory more sound because there are fewer net axioms!")
@morbiuswilters said:
There are always axioms in any useful formal theory
Which is precisely what I said. There are always axioms. You can't eliminate them, and Godel's theorem states this.
@morbiuswilters said:
Once again, this has nothing to do with proving axioms
You're right, it doesn't, and I don't understand your point.
Any formal theory contains axioms. You cannot eliminate all of them.
Therefore, the unprovability of an axiom is not and can never be an acceptable criticism of the theory.
Which means that whether you can prove 1 is in fact 1 is not and can never be an acceptable criticism of mathematical thought.
Do the math.
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@CDarklock said:
@morbiuswilters said:
The unprovable statements demonstrated by the incompleteness theorem are not axioms themselves
If they're true, and they cannot be proven, how are they not principles "accepted as true without proof"?
@morbiuswilters said:
One can prove them by adding axioms, but this only introduces more unprovable statements.
In other words, adding axioms only adds axioms. This appears self-evident.
Indeed, it is itself an axiom, since you can't prove that adding an axiom adds an axiom. But it's true all the same, and only an idiot would deny it.
(Suggested idiot argument: "Adding the axiom may remove other axioms by allowing them to be proven, making the theory more sound because there are fewer net axioms!")
@morbiuswilters said:
There are always axioms in any useful formal theory
Which is precisely what I said. There are always axioms. You can't eliminate them, and Godel's theorem states this.
@morbiuswilters said:
Once again, this has nothing to do with proving axioms
You're right, it doesn't, and I don't understand your point.
Any formal theory contains axioms. You cannot eliminate all of them.
Therefore, the unprovability of an axiom is not and can never be an acceptable criticism of the theory.
Which means that whether you can prove 1 is in fact 1 is not and can never be an acceptable criticism of mathematical thought.
Do the math.
Wow, your understanding of formal systems is so flawed that there is no way I can help without tearing your whole mind down and rebuilding from scratch.
Instead, let's just say the sheer irrationality of your reply actually ripped a whole in the Logicsphere and unleashed a giant Fallacy Dragon that pesto and myself had to fight and kill, much to the delight of the fair maidens who rewarded us with sexual favors.
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@CDarklock said:
If they're true, and they cannot be proven, how are they not principles "accepted as true without proof"?
The obvious point that you're missing is that they aren't "accepted as true without proof." No one knows if they are true based solely on the axioms and operations of the formal system. It may be possible to demonstrate that they are true with additional information or operations (but then you're not staying within the formal system). Or they may be suspected to be true, but since they can't be proven, they aren't accepted as true. Goldbach's conjecture or the Riemann hypothesis may be true but unprovable. All we know for now is that they are currently unproven.
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@CDarklock said:
@morbiuswilters said:
Once again, you cling to an inaccurate definition of what an axiom is. An axiom is not any unprovable true statement within a formal system, it's a presupposition that defines a formal system. In fact, an axiom need not be unprovable; an axiom could be derivable from the other axioms. A statement which is true but unprovable within a given formal system F isn't an axiom. Yes, it is true, but you can't prove it. If you want to simply accept it to be true, you must define a new formal system F' wherein it is added as an axiom. But it still doesn't become an axiom of F. @CDarklock said:The unprovable statements demonstrated by the incompleteness theorem are not axioms themselves
If they're true, and they cannot be proven, how are they not principles "accepted as true without proof"?In other words, adding axioms only adds axioms. This appears self-evident.
This quote is so mind-bendingly moronic, only an idiot would make it.Indeed, it is itself an axiom, since you can't prove that adding an axiom adds an axiom. But it's true all the same, and only an idiot would deny it.
@CDarklock said:
(Suggested idiot argument: "Adding the axiom may remove other axioms by allowing them to be proven, making the theory more sound because there are fewer net axioms!")
You really don't understand formal systems, do you? That an axiom is provable by other axioms doesn't make the axiom "go away". Yes, it means that it is redundant, but it's still an axiom of the system.@CDarklock said:
@morbiuswilters said:
No it doesn't: the definition of a formal system does. Godel's incompleteness theorem doesn't touch on the subject of axiom, except to define classes of axioms for which all formal theories will be incomplete or inconsistent. The reason that axioms must always exist in a useful system is that there's nothing to derive from if you have no presuppositions. This is so basic that I would silly repeating it again, except that you can't seem to understand it.There are always axioms in any useful formal theory
Which is precisely what I said. There are always axioms. You can't eliminate them, and Godel's theorem states this.@CDarklock said:
You're right, it doesn't, and I don't understand your point.
Yes you can, but you don't get anything out of it.Any formal theory contains axioms. You cannot eliminate all of them.
@CDarklock said:
Therefore, the unprovability of an axiom is not and can never be an acceptable criticism of the theory.
This makes so little sense I don't even know how to respond to it. Or perhaps, since you've apparently broken logic beyond the point of repair, it makes perfect unsense, and I don't know how to not respond with anti-fallacies, and of course the mushroom banana hops with the lazy brown dog at twilight. @CDarklock said:
You are not qualified to discuss mathematics or thought, because you don't understand the former, and you can't form a coherent example of the latter.Which means that whether you can prove 1 is in fact 1 is not and can never be an acceptable criticism of mathematical thought.
@CDarklock said:
Do the math.
I did. I stabbed that Fallacy Dragon in the heart, but still it kept coming. It was only the proper and timely use of formal systems on morbiuswilters' part that saved me from certain insanity.
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@bstorer said:
You are taking a broad definition of axiom.
Yes, I am. This is because I am not talking about formal mathematics. I am talking about a very large and comprehensive subject of which formal mathematics is a proper (and miniscule) subset.
@bstorer said:
we're discussing formal systems
No, we're not. What we are discussing is whether a FACT requires a PROOF. I assert that a thing is a fact when it is true, and that an axiom - a principle accepted as true without proof - demonstrates that fact without proof is not only possible, but natural and normal. Godel's theorem actively states that it is necessary.
@bstorer said:
I don't really see what this has to do with the conversation at hand, other than to attempt to appeal to authority.
I'm observing that Godel's theorem can be and has been applied outside the limited problem domain explicitly stated by Godel.
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Going waaay back to the original topic, the total is now at £110,600 / $170,932 / €137 461 / SEK 1 376 976 / ZWD 76870802, or 2011% of the original goal.
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@morbiuswilters said:
Instead, let's just say the sheer irrationality of your reply actually ripped a whole in the Logicsphere and unleashed a giant Fallacy Dragon that pesto and myself had to fight and kill, much to the delight of the fair maidens who rewarded us with sexual favors.
Morbius, always with the fair maidens. What a racist.
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@bstorer said:
Once again, you cling to an inaccurate definition of what an axiom is.
I use words to mean things. The word "axiom" does, in fact, mean what I am using it to mean. It means other things, too, but I've stated quite clearly what I mean by it. The existence of another definition doesn't invalidate the one I'm using, and neither does your irrational preference for some other meaning.
@bstorer said:
You really don't understand formal systems, do you?
You really don't understand the concept of an "idiot argument", do you? It's the kind of argument that would be made by an idiot.
@bstorer said:
No it doesn't: the definition of a formal system does.
You are using the wrong definition of "axiom". When I use a word to mean something, and you know what I'm using it to mean, and you use it to mean something else, it's called a fallacy of equivocation.
@bstorer said:
This makes so little sense I don't even know how to respond to it.
Let me explain in much, much simpler terms.
Imagine that I suggest an axiom of a system.
You examine that axiom and find that it is not proven.
This is IRRELEVANT to the system as a whole. Its importance and utility are unaffected.
@bstorer said:
You are not qualified to discuss mathematics or thought
This is called an "ad hominem argument". It is also a fallacy.
@bstorer said:
I stabbed that Fallacy Dragon in the heart
...and ran off with its treasure trove of fallacies?
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@CDarklock said:
@bstorer said:
Ok, then why are you trying to talk about a result from "formal mathematics"?You are taking a broad definition of axiom.
Yes, I am. This is because I am not talking about formal mathematics. I am talking about a very large and comprehensive subject of which formal mathematics is a proper (and miniscule) subset.@CDarklock said:
No, we're not. What we are discussing is whether a FACT requires a PROOF. I assert that a thing is a fact when it is true, and that an axiom - a principle accepted as true without proof - demonstrates that fact without proof is not only possible, but natural and normal. Godel's theorem actively states that it is necessary.
We can all make up our own definitions, but that isn't a very productive use of our time. There is nothing about a fact, in and of itself, that requires proof. But without proof, we can't decide on whether the fact is real or imagined.@CDarklock said:
I'm observing that Godel's theorem can be and has been applied outside the limited problem domain explicitly stated by Godel.
I think that it's fair to make an analogy to systems other than formal. For instance, I'd argue that the existence of god is something that is not proveable by science. It may be either true or false, and belief one way or the other does not rely on rational thought or proof.Nevertheless, you're making up your own definitions of axioms as they apply to math. And your random babbling about them just makes you look dumb, because you obviously don't understand the topic.
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@CDarklock said:
I use words to mean things. The word "axiom" does, in fact, mean what I am using it to mean. It means other things, too, but I've stated quite clearly what I mean by it. The existence of another definition doesn't invalidate the one I'm using, and neither does your irrational preference for some other meaning.
Thanks for clearing that up, Mr Dumpty....
When I use a word to mean something, and you know what I'm using it to mean, and you use it to mean something else, it's called a fallacy of equivocation.
@CDarklock said:
Let me explain in much, much simpler terms.
No one has argued with this assertion. And if you'd stuck to this, you'd be in fine territory. However, you cannot apparently distinguish between an axiom and something else that is true but unproveable, which is what Godel's theorem deals with.Imagine that I suggest an axiom of a system.
You examine that axiom and find that it is not proven.
This is IRRELEVANT to the system as a whole. Its importance and utility are unaffected.
@CDarklock said:
@bstorer said:
No, it was an observation, based upon your posts. And unlike your arguments, it becomes less unprovable with each post.You are not qualified to discuss mathematics or thought
This is called an "ad hominem argument". It is also a fallacy.
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@CDarklock said:
I use words to mean things. The word "axiom" does, in fact, mean what I am using it to mean. It means other things, too, but I've stated quite clearly what I mean by it. The existence of another definition doesn't invalidate the one I'm using, and neither does your irrational preference for some other meaning.
It's not an irrational preference; Godel's theory is very formally defined. If you choose to use definitions for the terms contained therein which differ, then you are the one being irrational. This is like making all roads a single lane and only letting heterosexuals drive in order to shorten commutes because "the shortest distance between two points is a straight line." And even if somebody tried that strategy (no doubt there is a member of the Virginia Department of Transportation stroking his chin thoughtfully right now) and it worked, it doesn't mean that the theory applies to the sexual preference of motorists. I am giving you an absurd analogy, but only because it creates a larger, and more obvious, gap in the logic. Your mistake is much more subtle, but that doesn't make it any better.@CDarklock said:
@bstorer said:
I think you've more than demonstrated it in this thread, thank you.You really don't understand formal systems, do you?
You really don't understand the concept of an "idiot argument", do you? It's the kind of argument that would be made by an idiot.@CDarklock said:
You are using the wrong definition of "axiom". When I use a word to mean something, and you know what I'm using it to mean, and you use it to mean something else, it's called a fallacy of equivocation.
You're the one making the fallacy. You're ignoring the formal definitions on which a theorem is based. You're trying to shoehorn other contexts into the theorem to arrive at new results, but it's garbage in, garbage out.@CDarklock said:
This is called an "ad hominem argument". It is also a fallacy.
It's an observation based upon the evidence at hand.
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@CDarklock said:
No we aren't, we're discussing your complete lack of understanding of Godel's incompleteness theorem. And, by the way, if we're discussing Godel's incompleteness theorem, we're discussing formal systems. That you don't get that point is at the heart of your problems.@bstorer said:
we're discussing formal systems
No, we're not. What we are discussing is whether a FACT requires a PROOF.
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@bstorer said:
This is like making all roads a single lane and only letting heterosexuals drive in order to shorten commutes because "the shortest distance between two points is a straight line." And even if somebody tried that strategy (no doubt there is a member of the Virginia Department of Transportation stroking his chin thoughtfully right now) and it worked, it doesn't mean that the theory applies to the sexual preference of motorists.
Do you mean to tell me that this isn't the point of the new HOT lanes on the beltway?
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@CDarklock said:
Let me explain in much, much simpler terms.
Imagine that I suggest an axiom of a system.
You examine that axiom and find that it is not proven.
This is IRRELEVANT to the system as a whole. Its importance and utility are unaffected.
So very, very, very wrong... Axioms are the base of the formal system. They are accepted as true without proof. They define the "rules" of the formal system. The incompleteness theory shows that no set of axioms exist which can prove all statements that are true, given that set of axioms. It is not about defining or finding axioms. All unprovable statements that are true are not necessarily axioms. In fact, that is the very essence of Godel's theory; proving how little you actually understand it if you can get it completely backwards. If all unprovable statements that are true could just be labeled "axioms" as you suggest, then incompleteness theory is wrong. That's because there would exist no true statements in the formal system that were unprovable and not axioms. However, if you simply label all unprovable-yet-true statements as "axioms" you actually create new unprovable-yet-true statements that are not axioms. See, that is how incompleteness theory works: only because they are not axioms are the unprovable-yet-true statements a thorn in our side. This is what Godel demonstrated. This is what you do not understand. Not all unprovable-yet-true statements are axioms.
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@boomzilla said:
Do you mean to tell me that this isn't the point of the new HOT lanes on the beltway?
Yah, those are affirmative action. You should see the I-66 HOV / I-495 HOT interchanges taking shape; believe me, they're anything but straight!
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@morbiuswilters said:
I'll go one further: axioms aren't just accepted as true, they're defined as true. (Probably more or less what you meant anyway.)[Axioms] are accepted as true without proof. They define the "rules" of the formal system.
@morbiuswilters said:
See, that is how incompleteness theory works: only because they are not axioms are the unprovable-yet-true statements a thorn in our side. This is what Godel demonstrated. This is what you do not understand. Not all unprovable-yet-true statements are axioms.
Again, to go one further:Theorem: in a formal system F, all unprovable-yet-true sentences S are not axioms in F.
Proof (by contradiction): suppose otherwise, that some s in S is an axiom in F. But then, s would be trivially provable by invoking axiom s. But then s is no longer unprovable! Contradiction.
CDarklock: if you want to use Godel's theorem's by analogy in fields other than formal systems, go ahead and have fun with your philosophical exercise. But that's all it is, a philosophical exercise. It might help you creatively come up with the real solution that you're looking for, but the analogy on its own doesn't prove anything.
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@AssimilatedByBorg said:
@morbiuswilters said:
I'll go one further: axioms aren't just accepted as true, they're defined as true. (Probably more or less what you meant anyway.)[Axioms] are accepted as true without proof. They define the "rules" of the formal system.
@morbiuswilters said:
See, that is how incompleteness theory works: only because they are not axioms are the unprovable-yet-true statements a thorn in our side. This is what Godel demonstrated. This is what you do not understand. Not all unprovable-yet-true statements are axioms.
Again, to go one further:Theorem: in a formal system F, all unprovable-yet-true sentences S are not axioms in F.
Proof (by contradiction): suppose otherwise, that some s in S is an axiom in F. But then, s would be trivially provable by invoking axiom s. But then s is no longer unprovable! Contradiction.
CDarklock: if you want to use Godel's theorem's by analogy in fields other than formal systems, go ahead and have fun with your philosophical exercise. But that's all it is, a philosophical exercise. It might help you creatively come up with the real solution that you're looking for, but the analogy on its own doesn't prove anything.
I was trying to explain it as simply and clearly as possible so CDarklock would understand and no longer propagate misinformation.
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@operagost said:
Millions have died at the hands of forced atheist governments. Let's face it: this has little to do with religion and much to do with the hatred and greed of man.
You underestimate the power of religion to influence behaviour. Granted, warlike nations no longer send out missionaries with their armies, but religion still remains an excellent way of creating divisions between people.
Once people think of themselves as being in separate and incompatible groups, they can use these to kill one another, as happens in places such as Iraq and India.
I agree that many different belief systems can lead people to dedicate their lives to helping others or to murdering them. Religion is merely one of the more popular ones.
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@_moz said:
I agree that many different belief systems can lead people to dedicate their lives to helping others or to murdering them. Religion is merely one of the more popular ones.
It's not justoutright murdering, either. Witness the modern church of Climate Change. It's determined to ruin economies, lower standards of living, and generally dictate the way people live, though some of the most extreme do advocate for reducing human population by pretty much any means you can think of.
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@boomzilla said:
One very prominent, recent example being the sort of so-called environmentalism that results in bans of DDT.
What? The Dynamic Debugging Technique for the PDP is banned?
http://en.wikipedia.org/wiki/Dynamic_debugging_technique
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@bstorer said:
No we aren't, we're discussing your complete lack of understanding of Godel's incompleteness theorem.
Straw man.
I've used Godel's theorem to illustrate an overall principle. That principle is that lack of proof does not alter the truth of a statement, and such a lack must be accepted in some cases. This is a valid application of the theorem, albeit outside the original problem domain Godel's theorem covered. Godel's theorem can be and has been used outside that domain.
I've also used the word "axiom" to mean something which is, indeed, a valid and accepted definition of the word "axiom". You don't like that! You think I should use the word "axiom" to mean something else which is applicable only within a specific problem domain - the same domain in which Godel's theorem was originally stated. But since I'm not working in that problem domain, that cannot possibly be what I mean. The word would become useless, and I would then need to use a different word. But there isn't one; the correct word for the concept I am expressing is "axiom". Its applicability to another concept is immaterial.
So rather than address the fundamental question of truth without proof, you've decided to address the question of whether I have any business discussing formal mathematical systems and the axioms from which they proceed. This is irrelevant, because it is not what I am discussing.
@bstorer said:
if we're discussing Godel's incompleteness theorem, we're discussing formal systems
Failure of imagination. How sad.
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@morbiuswilters said:
All unprovable statements that are true are not necessarily axioms.
That depends on your definition of "axiom".
Definition A: "a principle which is accepted as true without proof"
Definition B: "a proposition that is assumed without proof for the sake of studying the consequences"
An unprovable statement which is true is necessarily an axiom by definition A, but is not necessarily an axiom by definition B. This can conceivably be a problem, because the unqualified term "axiom" is ambiguous until the intended definition is specified.
Except I've already qualified it and specified my intended definition. The alternative would be irrational and pointless, as you've demonstrated over and over again. Trouble is, anyone that smart should be smart enough to figure out that when you use the specified definition, it's not irrational and pointless. It's just much, much harder to engage in mathurbation over it.
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@CDarklock said:
@bstorer said:
if we're discussing Godel's incompleteness theorem, we're discussing formal systems
Failure of imagination. How sad.
It's not a failure of immagination, it's that Goedel's incompleteness thereom only applies to formal systems. You cannot use it as an analogy in a religious conversation, as it is the analogue of nothing. And within formal systems, words like "axiom" have specific meanings and you can't throw them around carelessly. If we accept your argument that you weren't really discussing formal systems, then your point is moot. You cannot use a fairly abstract and theoretical branch of formal logic and arithmetic as a metaphor for your wacky religious or philosophical beliefs.
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@morbiuswilters said:
Goedel's incompleteness thereom only applies to formal systems
I do not believe this is true. I believe Godel's theorem is a limited statement of a fundamental truth of the universe: that within any empirically observable problem domain, a complete definition of the problem space cannot be made without reference to entities outside that domain.
@morbiuswilters said:
And within formal systems, words like "axiom" have specific meanings
And that's the failure. In your world, the word "axiom" must have the definition you prefer, because you cannot imagine Godel's theorem applying outside the domain of formal systems.
@morbiuswilters said:
You cannot use a fairly abstract and theoretical branch of formal logic and arithmetic as a metaphor
I've actually been speaking allegorically. Perhaps you're not qualified to discuss discussion.
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@CDarklock said:
But it is true.@morbiuswilters said:
Goedel's incompleteness thereom only applies to formal systems
I do not believe this is true.
@CDarklock said:
I believe Godel's theorem is a limited statement of a fundamental truth of the universe: that within any empirically observable problem domain, a complete definition of the problem space cannot be made without reference to entities outside that domain.
While the sentence after the colon is (trivially) correct, it has nothing to do with Gödels incompleteness theorem.
@CDarklock said:
@morbiuswilters said:
And within formal systems, words like "axiom" have specific meanings
And that's the failure. In your world, the word "axiom" must have the definition you prefer, because you cannot imagine Godel's theorem applying outside the domain of formal systems.
Since it is a theorem about formal systems, proved in a formal system, it cannot apply to anything but formal systems.
@CDarklock said:
@morbiuswilters said:
You cannot use a fairly abstract and theoretical branch of formal logic and arithmetic as a metaphor
I've actually been speaking allegorically. Perhaps you're not qualified to discuss discussion.
Speaking allegorically about formal sytems? If that isn't a WTF, then what is?
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@CDarklock said:
@morbiuswilters said:
All unprovable statements that are true are not necessarily axioms.
That depends on your definition of "axiom".
Definition A: "a principle which is accepted as true without proof"
Definition B: "a proposition that is assumed without proof for the sake of studying the consequences"
An unprovable statement which is true is necessarily an axiom by definition A,<snip>
Only under the assumption that everything which is true is accepted as true. This isn't the case in the original domain of Gödel's theorem, and far less so if you want to extend it to religious belief.
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@CDarklock said:
@morbiuswilters said:
Goedel's incompleteness thereom only applies to formal systems
I do not believe this is true. I believe Godel's theorem is a limited statement of a fundamental truth of the universe: that within any empirically observable problem domain, a complete definition of the problem space cannot be made without reference to entities outside that domain.
<snip>
Most of this thread is based around a mis-statement of Gödel's theorem. It doesn't say that any axiomatic system is incomplete: rather that any consistent axiomatic system which includes Peano's axioms is incomplete. I can set up an axiomatic system in which every true statement is provable, but it will be self-contradictory.
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@CDarklock said:
Straw man.
Apparently your grasp of the straw man fallacy is as strong as your grasp of the incompleteness theorem. You completely misrepresented the incompleteness theorem. I don't care if you're in the domain of formal systems or not, Godel's theorem does not say anything about the provability of axioms or the ability to remove them. You were wrong, you were called on it. Deal with it, learn what the theorem states, and move on.@CDarklock said:
I've used Godel's theorem to illustrate an overall principle. That principle is that lack of proof does not alter the truth of a statement, and such a lack must be accepted in some cases.
That's not from Godel's incompleteness theorem, though. The big "Holy Shit!" revelation of the theorem is that any attempt to define a consistent system which can be used to derive everything is fundamentally misguided because there will always an unprovable true statement. It doesn't say anything on the matter of what you can or should do with such statements.@CDarklock said:This is a valid application of the theorem, albeit outside the original problem domain Godel's theorem covered. Godel's theorem can be and has been used outside that domain.
Again, no it isn't, because it isn't what the statement says.@CDarklock said:I've also used the word "axiom" to mean something which is, indeed, a valid and accepted definition of the word "axiom". You don't like that!
I don't like it because you use a definition which is incompatibile with the theorem. As I demonstrated before, the word straight has two meanings, but that doesn't make geometric principles valid to sexual orientation.@CDarklock said:You think I should use the word "axiom" to mean something else which is applicable only within a specific problem domain - the same domain in which Godel's theorem was originally stated. But since I'm not working in that problem domain, that cannot possibly be what I mean.
But the problem is you don't use a word which properly translate from the original problem space to your new one. Had you used the word "presupposition" (or even the same meaning of "axiom") you'd be in fine shape. But you didn't, opting instead for a broader, vaguer meaning of the word. Any conclusions you come to from that vaguer meaning are garbage, because you're ignoring what the theorem really says.@CDarklock said:
So rather than address the fundamental question of truth without proof, you've decided to address the question of whether I have any business discussing formal mathematical systems and the axioms from which they proceed. This is irrelevant, because it is not what I am discussing.
No, I question whether you have any business discussing Godel's theorem, because you continue to blatantly misunderstand it. That you don't understand the formal systems on which it was based is a side issue, but a rather telling one.@CDarklock said:Failure of imagination. How sad.
Failure of understanding. How tragic.
