Register | Sign In


Understanding through Discussion


EvC Forum active members: 65 (9162 total)
4 online now:
Newest Member: popoi
Post Volume: Total: 915,817 Year: 3,074/9,624 Month: 919/1,588 Week: 102/223 Day: 13/17 Hour: 1/1


Thread  Details

Email This Thread
Newer Topic | Older Topic
  
Author Topic:   Gödel, Tarski, & Logic. (for grace2u)
:æ: 
Suspended Member (Idle past 7184 days)
Posts: 423
Joined: 07-23-2003


Message 1 of 22 (67740)
11-19-2003 1:41 PM


grace2u,
I thought it would be a good idea to begin a new thread to focus specifically on your assertions that the universe is ruled by some fundamental "laws of logic." In this post I hope to educate you a bit in the work that has been done in metalogic and the implication it has upon logic in general. (Admins, if you think this thread is inappropriately placed, or should simply be included in the original thread once it is re-opened, feel free to move or delete it since I have saved a copy of its text and can easily re-insert it where/when appopriate)
First, grace2u, I will state your position as I understand it.
You asserted that there exist some "laws of logic" that are universal, absolute and fixed. You also asserted that the fact that these are is evidence of your God, but I will ignore the theistic arguement for a moment since it is really only tangential to this topic. I only intend to show that the things you assert exist in fact do not and provably so.
I take your assertion that the "laws of logic" are universal, absolute and fixed to mean that there exists some single set of fundamental axioms from which all logical systems proceed. You seemed to contest this in your last response to me in the former thread however, when you claimed that your "laws of thought" were different than the axioms of elementary logic. The fact is that your laws of thought state A = A, (A & ~A) = 0, and (A or ~A) = 1. The axioms of elementary logic state A = A, (A & ~A) = 0, and (A or ~A) = 1 (among additional axioms). I hope that if you wish to continue to assert that your "laws of thought" are different than the axioms of elementary logic you will kindly show me where the properties you enumerated as your "laws of thought" do not appear within elementary logic's axioms.
Now, moving along to the focus of this post, I wish to introduce you to some prominent logicians and their proofs -- the first of which is Kurt Gdel.
Gdel is famous for his Completeness and Incompleteness Theorems (theorems of course being statements that are consistently derivable from the axioms of the logical system in which they are formulated). What I intend to focus on here is the Incompleteness Theorem. What it states essentially is that for any set of axioms at least sufficiently complex as to model elementary arithmetic there exist within the system well-formed formulae which are true yet unprovable in the system lest the system suffer inconsistency.
Gdel's statement basically said this:
G(x) = "The well-formed formula G(x) is unprovable in the system."
Once he was able to formulate that statement arithmatically, he was able to show that if the system could prove G(x) true, then it resulted in a contradiction, and in order to prove it false it would have to prove it true resulting also in a contradiction. Therefore the system is incomplete since there exist statements which could be formulated according to its syntactical rules yet were undecideable.
Now, you asserted that the axioms of every logical system proceed from these supposedly universal, absolute, and fixed "laws of logic" implying that they must be able to model elementary arithmetic. If this is the case, then those laws must be either inconsistent or incomplete, and therefore they cannot sufficiently model the entire universe. That is to say, truth is not entirely bound to be logical.
That brings me to my second famous logician: Alfred Tarski. Tarski also has a theorem which basically says that truth cannot be entirely represented in a logical system. In fact, as the page I linked you to states, Tarski's Theorem can be regarded as a corrolary of sorts to Gdel's Theorem, and it states in no uncertain terms that there must be true statements which are not logical theorems. in other words, truth is not only that which is logically provable, and therefore we can conclude that reality is capable of exhibiting behaviors which are not in strict accordance with ANY logical "laws."
Now, I leave you to explain how this can be if we are to suppose that the "laws of logic" are universal, absolute and fixed. If there were such "laws", wouldn't we expect that all true statements must be therefore derivable from them? Wouldn't we expect that all truths would be logically provable?

Replies to this message:
 Message 2 by Brad McFall, posted 11-19-2003 1:59 PM :æ: has not replied
 Message 6 by grace2u, posted 11-19-2003 10:21 PM :æ: has replied
 Message 12 by grace2u, posted 11-20-2003 2:07 AM :æ: has replied

  
Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 2 of 22 (67743)
11-19-2003 1:59 PM
Reply to: Message 1 by :æ:
11-19-2003 1:41 PM


add for another thread
Later tonight or tommarrow I will post responses to both Zephyr and Mark25 on/in the DISTINGUISHING BARAMINOLGY thread in which I will show (by implication or in utility(without bias to a polarization of Creation and Evolution) that logic may be universal(Actually apply when not knowing if the space is only the solar system or some larger extant extent) while the number systems that relay that logic may not necessarily (also by implication) be consistent. The work is not typed up just yet. A sylllogism may be actual while the numbers used to display the same without words may not.

This message is a reply to:
 Message 1 by :æ:, posted 11-19-2003 1:41 PM :æ: has not replied

Replies to this message:
 Message 3 by mark24, posted 11-19-2003 7:17 PM Brad McFall has replied

  
mark24
Member (Idle past 5195 days)
Posts: 3857
From: UK
Joined: 12-01-2001


Message 3 of 22 (67799)
11-19-2003 7:17 PM
Reply to: Message 2 by Brad McFall
11-19-2003 1:59 PM


Re: add for another thread
Whatever you say, Brad, whatever you say...

This message is a reply to:
 Message 2 by Brad McFall, posted 11-19-2003 1:59 PM Brad McFall has replied

Replies to this message:
 Message 4 by Brad McFall, posted 11-19-2003 7:37 PM mark24 has not replied

  
Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 4 of 22 (67802)
11-19-2003 7:37 PM
Reply to: Message 3 by mark24
11-19-2003 7:17 PM


Re: add for another thread
Mark- 24 , I am working on visual basic homework tonight- no time to get you guys up to me today- BEst ....It is not whatever...(I say) there are 5 steps in response to you before we can question a result, the result I am logically involving. This logic is used by doing both creation and evolution only I doubt it will universally be for as long a time as it takes to USE Godel but That will happen after I do some more Mendelism based on hybrids but my prepared repsonse to you will not go logically to this level. That will be for the future. Sorry I have to be this dense today. I have other things to do. I did not think that Grace 2 U remanded that one put out a response and then "leave". That is why I tried to get people to pay more attention to the baraminology thread today- that's all-
A notion of "the entire universe" has something to do with Bertrand Russel's paradox and my guess is a likely rejection in this thread head of Cantor's solution to the ordinal of all ordinals (absolute Infintiy is not acutal). But this relatively may not mean that one may use Ae's reasoning, just havent had time to think it all thru except to know that I am proposing using an incomplete flesh to "program" the metric by rejection of a physcists reality but more later and then back to here....Making one whole or complete or being saved may not be an issue logically with the universe not as it may be but as it is... again- I could be wrong so it is not "what I say". If you stick with it I am fairly sure you if anyone here may be the first to "get me" in all my ornaments. But give it time. AE- I am still going to get back to your post trust me.
[This message has been edited by Brad McFall, 11-19-2003]

This message is a reply to:
 Message 3 by mark24, posted 11-19-2003 7:17 PM mark24 has not replied

Replies to this message:
 Message 5 by :æ:, posted 11-19-2003 8:12 PM Brad McFall has not replied

  
:æ: 
Suspended Member (Idle past 7184 days)
Posts: 423
Joined: 07-23-2003


Message 5 of 22 (67823)
11-19-2003 8:12 PM
Reply to: Message 4 by Brad McFall
11-19-2003 7:37 PM


Re: add for another thread
Brad McFall writes:
A notion of "the entire universe" has something to do with Bertrand Russel's paradox and my guess is a likely rejection in this thread head of Cantor's solution to the ordinal of all ordinals (absolute Infintiy is not acutal).
What I think you're referring to is the self-inclusion or self-reference which characterizes Gdel's statement. Where Russel's paradox involves the "Set of all sets which do not include themselves," I think there is only a similarity to the Gdel sentence, but not necessarily relevance.
Or perhaps you're referring to the well-know antinomy "The set of all sets" which -- according to its definition and Cantor's Power Set Theorem -- must include its power set which results in contradiction. When regading reality, however, it (the real universe) must correspond to the set of all sets despite the contradiction if reality is to be regarded as a whole. That is to say, that which is not included in "the set of all and only that which is real" is irrelevant since whatever it may be, it is NOT REAL. Still, Cantor's theorem supplies the power set (which is obviously real) and which must therefore be included in the set. I think what we can conclude from this is that it is always illogical to apply definitions and theorems writ within the universe about relationships observed among segments of the universe to the universe as a whole.
I'll certainly be interested in what you can offer, though.

This message is a reply to:
 Message 4 by Brad McFall, posted 11-19-2003 7:37 PM Brad McFall has not replied

  
grace2u
Inactive Member


Message 6 of 22 (67861)
11-19-2003 10:21 PM
Reply to: Message 1 by :æ:
11-19-2003 1:41 PM


So much for my 24hr vacation...
I thought it would be a good idea to begin a new thread to focus specifically on your assertions that the universe is ruled by some fundamental "laws of logic."
I do not agree that the universe is RULED by the laws of logic(rather, in a sense they RULE us). I did state however that there exist within the universe these fundamental laws that reflect the nature and character of God known by Him at a minimum and partly by us. They are in a sense a radiant property of this God. We see the radiance, and theism can begin to explain it. Meanwhile, atheists like yourself are forced to deny the realities of these truths.
In our lengthy discussions from inconstancies within atheistic evolution, it was my custom to read and reread your statements in an attempt to understand what you are saying. I gave you the benefit of the doubt on many occasions and always allowed you to clarify anything that was unclear-as any rational individual would. The argument you are posing on the surface smack of straw man, although I grant I need to reread your post and try to understand where "you think" I am coming from before I present a more lengthy response, responding to your statements more directly.
I will make a quick point however and will look forward to your response on it ...
Since the laws of logic(reason) are not absolute or binding in your system of thought(or view of reality), what type of reason are you using now? What is your evidence for or against this system of reason? Do you agree that you are using logic now in this conversation or argument, in an attempt to show me where I am wrong? What are you presupposing in order to do this? Can anyone simply produce their own system of reason(laws of logic) in order to prove their case on this forum? I contend they can not. I contend that laws of logic(reason) must exist in order to even have this conversation. WE ARE BOTH, IN ESSENCE PRESUPPOSING THE SAME SET OF LAWS OF REASON(logic), I freely admit the obvious, while you contend that they do not exist. How can you not see this? Forget that I am a Christian... How can you be so biased such that you would dispute this??? If we did not both presuppose some set of universal truths, either one of us could make any argument we wanted and claim it to be true. While we could, it would be an illogical system. For some reason, you have gone off on a crusade, apparently thinking you have me cornered. In doing this, you are missing my point altogether. I will still respond to your post but please address these questions in as rigorous as terms as you can. Please think about what I am saying. Don't be too quick to respond. I expect you to give account to the questions I ask. The other option is you can contend that I and the forum, plus the universe as you've come to know it do not really exist and that this argument is going on inside the depths of your imagination. I would suggest that this is the only logical argument against my stated position. SO now, please address each of my (?'s) to you.
Oh, one more question.. If we are not both presupposing the same laws of reason(logic)-if you do contend this, how is it possible for either of us to even begin to communicate our point to the other? This is what I mean when I say you might as well argue that you are the only entity in existence, all communication is futile and certainly science is.
Thanks for the interest...
"Christ eleison"

This message is a reply to:
 Message 1 by :æ:, posted 11-19-2003 1:41 PM :æ: has replied

Replies to this message:
 Message 7 by crashfrog, posted 11-19-2003 10:32 PM grace2u has replied
 Message 9 by JustinC, posted 11-19-2003 11:33 PM grace2u has not replied
 Message 11 by :æ:, posted 11-20-2003 2:04 AM grace2u has not replied

  
crashfrog
Member (Idle past 1467 days)
Posts: 19762
From: Silver Spring, MD
Joined: 03-20-2003


Message 7 of 22 (67863)
11-19-2003 10:32 PM
Reply to: Message 6 by grace2u
11-19-2003 10:21 PM


They are in a sense a radiant property of this God.
So, if logic is to be taken as a reflection of the character of God, can we then conclude that, like logic, God is inconsistent and incomplete?
If we are not both presupposing the same laws of reason(logic)-if you do contend this, how is it possible for either of us to even begin to communicate our point to the other?
You've agreed to use the same language. How could you communicate with each other if English isn't a fundamental law of the universe?

This message is a reply to:
 Message 6 by grace2u, posted 11-19-2003 10:21 PM grace2u has replied

Replies to this message:
 Message 8 by Brad McFall, posted 11-19-2003 10:57 PM crashfrog has not replied
 Message 10 by grace2u, posted 11-20-2003 12:01 AM crashfrog has replied
 Message 14 by Rrhain, posted 11-20-2003 3:44 AM crashfrog has replied

  
Brad McFall
Member (Idle past 5032 days)
Posts: 3428
From: Ithaca,NY, USA
Joined: 12-20-2001


Message 8 of 22 (67866)
11-19-2003 10:57 PM
Reply to: Message 7 by crashfrog
11-19-2003 10:32 PM


"inconsisent" and "incompelte" apply to the numerology used to express natrualistic magnitudes if. One may choose a theology tht does not think that nature's "numbers" (say Kepler's idea or Plato's) are material but rather as Hume remarked a difference of "spirit" between the math and what the math represents when discussing a physcial void. I do not hold this position. It is out there though. I assume the numbers do not merely bound this vacuum or void.
[This message has been edited by Brad McFall, 11-19-2003]

This message is a reply to:
 Message 7 by crashfrog, posted 11-19-2003 10:32 PM crashfrog has not replied

  
JustinC
Member (Idle past 4844 days)
Posts: 624
From: Pittsburgh, PA, USA
Joined: 07-21-2003


Message 9 of 22 (67874)
11-19-2003 11:33 PM
Reply to: Message 6 by grace2u
11-19-2003 10:21 PM


Re: So much for my 24hr vacation...
quote:
did state however that there exist within the universe these fundamental laws that reflect the nature and character of God known by Him at a minimum and partly by us.
Why do theists always write such obfuscatory statements? Please explain this very vague, possibly meaningless, sentence. Exactly how does God radiate the fact that "John Did Jump" and "John Did Not Jump" is a contradiction? Does he dictate this, can he dictate otherwise? When I say the argument you are using right now is illogical, am I radiating logic? Is that you mean God does?
Does God radiate the fact that "homosexuality is wrong", or are you saying that's how he feels about the subject?
Then explain, if you didn't while answering the questions above, how "John Jumped and John Did Not Jump" is not a contradiction if God doesn't exist.
quote:
They are in a sense a radiant property of this God. We see the radiance, and theism can begin to explain it. Meanwhile, atheists like yourself are forced to deny the realities of these truths.
Why are we forced to deny this? Why can't we just accept them all a priori? How does presupposing a God that explains every pressuposition better than just accepting the presuppositions?
These are serious questions that I would like answered considering I've never gotten an answer when I ask these to other theists (though I don't ask them alot).

This message is a reply to:
 Message 6 by grace2u, posted 11-19-2003 10:21 PM grace2u has not replied

  
grace2u
Inactive Member


Message 10 of 22 (67880)
11-20-2003 12:01 AM
Reply to: Message 7 by crashfrog
11-19-2003 10:32 PM


So, if logic is to be taken as a reflection of the character of God, can we then conclude that, like logic, God is inconsistent and incomplete?
Of course not. I do not follow you're reasoning here. We are finite, God is infinite. How could the standard of an infinite entity be measured by the error prone standard of the finite. Because the man described laws of logic appear to be inconsistent in some cases or incomplete in others(or rather in their aplication to mathematics at times) certainly doesn't imply that the universal laws are similar. As I'm sure you know, you can only measure a system to the degree accuracy of the tool you are using. If the tool(man and his logic) is flawed(or imperfect), then how can the measurements taken be anything but that? We do suppose however, and evidence suggests I might add(I think even most logicians would agree on this) that there is some "absolute-truth" that can be obtained or reached. Else all scientific work would be futile.
You've agreed to use the same language. How could you communicate with each other if English isn't a fundamental law of the universe
Again, you are missing the deeper point being made. We could use any language, however the methods used and reason sought or honored, would be aplicable for both languages. The methods of reason are the same. This reason ,is far more complex than any language or even logical system. Since all logical systems and philosophies are in a sense trying to understand what these concepts are. The fact that they are searching for this truth even suggests it exists. We see glimpses of it in mathematics, man's logic, science and humanity.
Thanks...
"Christe eleison"

This message is a reply to:
 Message 7 by crashfrog, posted 11-19-2003 10:32 PM crashfrog has replied

Replies to this message:
 Message 15 by Rrhain, posted 11-20-2003 4:58 AM grace2u has replied
 Message 17 by crashfrog, posted 11-20-2003 9:17 AM grace2u has not replied

  
:æ: 
Suspended Member (Idle past 7184 days)
Posts: 423
Joined: 07-23-2003


Message 11 of 22 (67885)
11-20-2003 2:04 AM
Reply to: Message 6 by grace2u
11-19-2003 10:21 PM


Re: So much for my 24hr vacation...
grace2u writes:
I do not agree that the universe is RULED by the laws of logic(rather, in a sense they RULE us). I did state however that there exist within the universe these fundamental laws that reflect the nature and character of God known by Him at a minimum and partly by us.
You stated that the "laws of logic" were universal. Are you now retracting that statement? If the universe is not ruled by them, then they're not universal, wouldn't you say? In what sense can they be universal yet not hold everywhere in the universe?
grace2u writes:
Meanwhile, atheists like yourself ...
Where did I say that I was an atheist?
grace2u writes:
...are forced to deny the realities of these truths.
Well I'm not so sure what your claiming is the truth. Are the laws of logic universal or aren't they, according to you?
grace2u writes:
Since the laws of logic(reason) are not absolute or binding in your system of thought(or view of reality), what type of reason are you using now?
Whatever one I want. I just so happens though, that my species has developed a widely accepted abstract structure to govern the constructions of our language. It's not so surprising really since we all share highly similar pathways for experiencing reality. If I wish to be understood in my communications, it is pragmatic that I adhere to a common structure. That doesn't make it anything more than a construction of the mind.
grace2u writes:
WE ARE BOTH, IN ESSENCE PRESUPPOSING THE SAME SET OF LAWS OF REASON(logic), I freely admit the obvious, while you contend that they do not exist.
I never said that they don't exist. I'm saying that they are not universal or absolute. I'm saying that they do not exist as objective features of reality. I'm saying they only exist in our minds.
grace2u writes:
If we did not both presuppose some set of universal truths, either one of us could make any argument we wanted and claim it to be true.
That's right. But if I want someone to agree with me, I should likely construct my argument using a system of thought that is common to other individuals. That we all agree on certain rules does not make them universal or absolute. We could all agree on different rules if we wanted. Heck, we do use different rules when describing different aspects of reality, i.e. elementary logic, fuzzy logic, quantum logic, etc...
grace2u writes:
I expect you to give account to the questions I ask.
Yet apparently I can't expect the same from you. You know what prompted my opening this thread? The fact that I couldn't get you to answer the following question.
And I quote myself thusly, and like so...
quote:
If logic were indeed universal, then we would expect that there would exist no statement X such that X is a well-formed statement and yet we can not prove X true or false logically, do you agree? In other words, assuming that logic is universal, absolute and fixed, we would expect that every conceivable statement is theoretically decideably true or false, right? A simple yes or no will suffice. If your answer is no, please explain how logic can be universal, absolute and fixed and yet there exist well-formed statements which it cannot prove true or false.
So what is it? Yes or no.
grace2u writes:
The other option is you can contend that I and the forum, plus the universe as you've come to know it do not really exist and that this argument is going on inside the depths of your imagination.
This debate has literally nothing to do with solipsism.
grace2u writes:
If we are not both presupposing the same laws of reason(logic)-if you do contend this, how is it possible for either of us to even begin to communicate our point to the other?
In many instances impossible. If you were thinking along the lines of classical logic, a statement in quantum logic would appear absurd. So what? That we do assume certain rules at certain times when communicating does not make those rules universal or absolute. We assume different rules all the time when we speak about different aspects of reality.
grace2u writes:
This is what I mean when I say you might as well argue that you are the only entity in existence, all communication is futile and certainly science is.
Non-sequitur.

This message is a reply to:
 Message 6 by grace2u, posted 11-19-2003 10:21 PM grace2u has not replied

  
grace2u
Inactive Member


Message 12 of 22 (67886)
11-20-2003 2:07 AM
Reply to: Message 1 by :æ:
11-19-2003 1:41 PM


Nice try...
Ok .. :ae:.. you are getting a little better but still missing the mark I'm afraid.
I could argue this point from two perspectives.
I could state that this is irrelevent since you have misunderstood my statements or ...
I could demonstrate how you misaplied Godel to the context of our discussion(that is an eternal data set, or at a minimum, one that can jump systems).
Just to clear my previous comments:
grace2u writes:
I am not arguing that the AXIOMS contained within the various logical(or nonlogical) systems are universal and invariant, rather that there are in existence a set of universal and aboslute laws(reflected by laws of thought and laws of logic(reason)) that make these laws perceivable and useful to us
I must first begin by copying over the quote that I used to summarize my final point from inconsistencies within atheistic evolution.... I did say that these universal absolutes are REFLECTED by the laws of thought and laws of logic(reason).
Now, in your statement:
:ae: writes:
I take your assertion that the "laws of logic" are universal, absolute and fixed to mean that there exists some single set of fundamental axioms from which all logical systems proceed
I contend that it is possible that there is one single set, but not known by man. How could man ever know that this is the case. Even if a theist said, here they are... I would still maintain that we do not know if that is truly complete. Again, we can only measure a system to the degree accuracy of our tools. Unless one postulates they are god-and therefore error free, they could never be for certain.
Now, for the sake of discussion and to demonstrate you are wrong yet again, even though I really am not obliged to, I will demonstrate where your strict system does not in fact represent reality and where Godels infamous incompleteness theorem does not hold(this by his own definition). BTW it is not merely I that contend it doesn't hold in reality, but many philosophers will agree in the following(references provided if asked-I'm being lazy).
:ae: writes:
What it states essentially is that for any set of axioms at least sufficiently complex as to model elementary arithmetic there exist within the system well-formed formulae which are true yet unprovable in the system lest the system suffer inconsistency
I agree with your summary of the theorem...
Please note the emphasis on --ANY SET OF AXIOMS AT LEAST SUFFICIENTLY COMPLEX(btw I'm not shouting here, simply emphasizing)--and then it continues.
So,
1)Godel states that the proposed SET of axioms must be sufficiently complex. Now as I'm sure you know, the incompleteness theorem requires a system in which to operate in, else it falls apart.
2)That is to say, if the data set is infinte, the theorem does not hold. I am sure we could argue all day and night whether or not the universe and the logic that describes it is infinite or not. I hope we could agree that it is. So, if the set of axioms is infinite(or can in essence cross systems), the theorem is invalid. This theorem is more a problem for a mathematician than a theist philosopher arguing that that an infinite God contains infinite and absolute logic that is reflected towards His creation.
BTW, I am not arguing against Godel, merely that you have misapplied his theorem in this case.
Nice try..
I will have to address your other point tomorrow, its getting late.
Take care and I still do appreciate your comments ,
"Christe eleison"

This message is a reply to:
 Message 1 by :æ:, posted 11-19-2003 1:41 PM :æ: has replied

Replies to this message:
 Message 13 by :æ:, posted 11-20-2003 3:15 AM grace2u has not replied

  
:æ: 
Suspended Member (Idle past 7184 days)
Posts: 423
Joined: 07-23-2003


Message 13 of 22 (67895)
11-20-2003 3:15 AM
Reply to: Message 12 by grace2u
11-20-2003 2:07 AM


Re: Nice try...
grace2u writes:
I could state that this is irrelevent since you have misunderstood my statements or ...
Of course I don't understand your statements. In one thread you asserted that the "laws of logic" were universal. In this, you've said that they're not. Which is it?
grace2u writes:
I could demonstrate how you misaplied Godel to the context of our discussion(that is an eternal data set, or at a minimum, one that can jump systems).
You'll have to be more specific with regard to the relevance of an "eternal data set" and how it supposedly "can jump systems." What do you mean by these statements?
grace2u writes:
I contend that it is possible that there is one single set, but not known by man. How could man ever know that this is the case. Even if a theist said, here they are... I would still maintain that we do not know if that is truly complete.
So you're asserting that they exist, yet you don't know that they do, and in fact they are unknowable? Why should I believe you when you assert their existence then? Would you believe me if I told you that there is an invisible pink unicorn in my living room, but I just don't know it, and in fact neither you nor I can ever know it?
grace2u writes:
Again, we can only measure a system to the degree accuracy of our tools. Unless one postulates they are god-and therefore error free, they could never be for certain.
So what? While it may seem unfortunate to you, if it is indeed the case that we can never reach absolute certainty, it will simply be a fact we'll have to live with. That is no reason to go around postulating the existence of unknowable beings in order to cope with aspects of life that make us uncomfortable. (And honestly, it doesn't make me uncomfortable at all)
grace2u writes:
Godel states that the proposed SET of axioms must be sufficiently complex. Now as I'm sure you know, the incompleteness theorem requires a system in which to operate in, else it falls apart.
Right, and from your assertion that the "laws of logic" were universal and absolute, it necessarily follows that the set of these "laws of logic" be able to model every logical system. Otherwise they would not be universal and absolute as you've claimed (and then retracted, it seems).
grace2u writes:
That is to say, if the data set is infinte, the theorem does not hold.
Why? Because you say so?
grace2u writes:
So, if the set of axioms is infinite(or can in essence cross systems), the theorem is invalid.
If the set of axioms is capable of modeling elementary arithmetic, then it must be either inconsistent or incomplete. That's what the theorem states, and you've agreed that it does. If this set of axioms is universal and absolute as you've asserted, then it follows that all logical systems must proceed from it. Elementary arithmetic is a logical system, and it is incomplete. Therefore this set of axioms can not be absolute and universal since at least one logical system which proceeds from it is incomplete.
[This message has been edited by ::, 11-20-2003]

This message is a reply to:
 Message 12 by grace2u, posted 11-20-2003 2:07 AM grace2u has not replied

  
Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 14 of 22 (67898)
11-20-2003 3:44 AM
Reply to: Message 7 by crashfrog
11-19-2003 10:32 PM


crashfrog writes:
quote:
can we then conclude that, like logic, God is inconsistent and incomplete?
OR, crash. God is inconsistent OR incomplete. The two are just different ways of saying the same thing.
------------------
Rrhain
WWJD? JWRTFM!

This message is a reply to:
 Message 7 by crashfrog, posted 11-19-2003 10:32 PM crashfrog has replied

Replies to this message:
 Message 16 by crashfrog, posted 11-20-2003 9:08 AM Rrhain has replied

  
Rrhain
Member
Posts: 6351
From: San Diego, CA, USA
Joined: 05-03-2003


Message 15 of 22 (67906)
11-20-2003 4:58 AM
Reply to: Message 10 by grace2u
11-20-2003 12:01 AM


grace2u writes:
quote:
How could the standard of an infinite entity be measured by the error prone standard of the finite.
Because logic applies to infinite things, too. In fact, the Incompleteness Theorems were famously and primarily applied specifically to show that a question about infinities cannot be resolved using current axioms of set theory.
What is the size of the continuum?
Well, that's a hard question. "It's infinite" isn't an answer because some infinite things are bigger than others.
Let's start with just the integers. There are an infinite number of them, but let's get a handle on how many there are.
Suppose you had a hotel room with an infinite number of rooms, all numbered: 1, 2, 3, .... Plus, suppose that every single room is occupied. Someone shows up. Can you get a room for the new guest?
Yep. Simply tell everybody in the hotel to move down one room. The guest in Room #1 moves to Room #2, the guest in Room #2 moves to Room #3, and so on. This leaves Room #1 empty and the new guest can move in.
This process works for any finite number of people who show up. Ten people show up? Just have everybody move down ten rooms. This leaves Rooms #1-10 open and they move in. A hundred? A quintillion? No problem.
But suppose you had an infinite number of people show up? Could you fit them in?
Yep. Simply tell everybody in the hotel to move to the room number that is twice as large as the one currently occupied. The guest in Room #1 moves to Room #2, Room #2 moves to Room #4, Room #3 goes to Room #6, and so on.
Everybody that is currently in the hotel has a room since for every guest you can name, I can give you a room number to find that guest.
But look what we did: We just opened up every single odd numbered room. This means that an infinite number of guests is equivalent to an infinite number of guests only occupying even-numbered rooms. Since there are just as many odd numbers as even numbers, that must mean that the new guests can simply move into the odd-numbered rooms and everybody is satisfied.
Now, that's just integers (and positive integers at that). From this, we can tell that one way to see if something is the same size as the integers is to pair them up and see if they're the same size. If you can get this one-to-one correspondance, then they're the same size.
It shouldn't be too difficult to see how to add the negative integers into this (you can easily put the negative integers into one-to-one correspondance with the positive.) But for fractions (and by "fractions," I really mean the "rationals" which means any number that can be written as a fraction of two integers), things get weird.
Without getting too technical, let's just say that Cantor developed an ingenious proof that showed that the set of fractions is the exact same size as the integers (in short, it involves writing down fractions in a matrix method, which can guarantee that we get all the fractions and then, by traversing it diagonally, you can show that it is in one-to-one correspondance to the set of integers, thus the fractions are the same size as the integers.)
But what about the real numbers? Real numbers include numbers that cannot be expressed as a fraction such as pi, e, and the square root of 2 (as well as all the rationals). Because these numbers form a continuous spectrum of numbers between any two you care to name, it is often called "the continuum." What about them? If we include all these other numbers, can we put them into one-to-one correspondance with the set of integers?
It turns out we can't.
First, some prep work. I need to show that a terminating decimal has an equivalent infinite expansion. That is, I need to show that 0.999... = 1.
Let x = 0.999...
Then 10x = 9.999...
Subtracting:
10x - x = 9.999... - 0.999...
Reducing:
9x = 9
Therefore, x = 1. But we started with the statement that x = 0.999.... Therefore 1 = 0.999....
Now to the proof. Suppose you had a list of all possible decimal expansions between 0 and 1. To avoid duplicates, we will make each expression unique by making it the infinite repeating version. That is, instead of 0.5, we will use 0.49999... which is the same thing.
Now, this means we have a list of decimal expansions:
x1 = 0.a1b1c1d1...
x2 = 0.a2b2c2d2...
x3 = 0.a3b3c3d3...
Let us now create a new number, p:
p = 0.a1b2c3d4...
However, we shall alter the numbers of p as follows:
If pi = 3, then new-pi = 2
If pi != 3, then new-pi = 3
Thus, we have a number that is necessarily between 0 and 1 and yet differs from every single number in our list at the ith decimal place. No matter what our list is, we can construct a number that isn't on the list and should be.
Therefore, the real numbers cannot be put into one-to-one correspondance with the integers. While the size of the integers is infinite, the size of the reals is larger.
[And if you really want something that will bake your noodle, consider this: The rationals (all numbers that can be expressed as a fraction of two integers) are "dense" in the reals. That is, between any two real numbers you care to name, no matter how close together they are, there is at least one rational number. Similarly, the reals are "dense" in the rationals...between any two rationals, there is at least one real. And yet, as we just showed, there are more reals than rationals. Where do we find the space to fit them all?]
It turns out that there are a whole hierarchy of infinities. Cantor showed that one can construct a power set of a set by taking every single subset of a set. This power set, he proved, is always larger than the original set. In short, if a set has size n, then the power set has size 2n. And for infinite sets, that means that the power set is larger still.
He assigned the size of the integers the symbol aleph-null. Thus, the power set of the integers would be of size 2aleph-null and he called this new number aleph-one. The power set of aleph-one is aleph-two and so one. Each aleph is bigger than the one below it, even though a set of that size is infinite.
So the question is now: We know that the size of the reals is larger than aleph-null, but what is it? Is it aleph-one? Aleph-two? Or perhaps there is some other sort infinity between the alephs just as there are numbers between the integers.
It turns out that while we know that the size of the continuum (often called "c") cannot be greater than aleph-one, we don't know if it actually is aleph-one. And furthermore, we have come down to this extremely frustrating set of events:
If we assume the c is aleph-one, we do not get a contradiction. For a long time, people thought this proved the point. No contradiction? Then it must be.
Ah, but that isn't good enough. It turns out, after other research was done, that if we assume that c is not aleph-one, we also do not get a contradiction.
And as it turns out, this particular question can never be answered given the current axioms of set theory. It's a very basic question, but given the foundations of our mathematics, we can never answer it.
quote:
Because the man described laws of logic appear to be inconsistent in some cases or incomplete in others(or rather in their aplication to mathematics at times) certainly doesn't imply that the universal laws are similar.
Ah, you're a Platonist. A Platonist would look at the problem of the Continuum Hypothesis and say that the real numbers do have an actual size...we just don't know what it is (and alas, can never know).
However, this doesn't really apply to god. Since when is god an axiomatic set theory sufficiently complex to model simple arithmetic?
quote:
We do suppose however, and evidence suggests I might add(I think even most logicians would agree on this) that there is some "absolute-truth" that can be obtained or reached.
Eh, not really. More accurately, mathematicians claim that some things, given certain assumptions, can be declared to be true but that a lot of things, if the system is sufficiently complex, can never be known.
Some things simply cannot be discovered no matter how perfect your logic is.
quote:
Else all scientific work would be futile.
Not at all! You seem to be dissatisfied with "accurate enough for all known examples." Why is that?
------------------
Rrhain
WWJD? JWRTFM!

This message is a reply to:
 Message 10 by grace2u, posted 11-20-2003 12:01 AM grace2u has replied

Replies to this message:
 Message 18 by grace2u, posted 11-20-2003 9:22 AM Rrhain has replied

  
Newer Topic | Older Topic
Jump to:


Copyright 2001-2023 by EvC Forum, All Rights Reserved

™ Version 4.2
Innovative software from Qwixotic © 2024