Godels incompleteness theorem ends in absurdity meaninglessness

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pam69ur
 
Reply Mon 10 Sep, 2007 06:21 am
All views end in meaninglessness ie self-contradictory As and example of this he presents the case that Godels incompletness theorem ends in meaninglessness. And that his proof was a complete failure
I present this Book which i downloaded of the net. What do you think of this undermining of Godel

Quote:
http://www.earlham.edu/~peters/cours...s/low-skol.htm
Insofar as this is a paradox it is called Skolem's paradox. It is at least a paradox in the ancient sense: an astonishing and implausible result. Is it a paradox in the modern sense, making contradiction apparently unavoidable?



from

Skolem's paradox - Wikipedia, the free encyclopediahttp://www.earlham.edu/~peters/cours...s/low-skol.htm
Insofar as this is a paradox it is called Skolem's paradox. It is at least a paradox in the ancient sense: an astonishing and implausible result. Is it a paradox in the modern sense, making contradiction apparently unavoidable?

Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it

attempted solutions
Bunch notes


"no one has any idea of how to re-construct axiomatic set theory so that this paradox does not occur"



http://www.earlham.edu/~peters/cours...s/low-skol.htm

One reading of LST holds that it proves that the cardinality of the real numbers is the same as the cardinality of the rationals, namely, countable. (The two kinds of number could still differ in other ways, just as the naturals and rationals do despite their equal cardinality.) On this reading, the Skolem paradox would create a serious contradiction

The good news is that this strongly paradoxical reading is optional. The bad news is that the obvious alternatives are very ugly. The most common way to avoid the strongly paradoxical reading is to insist that the real numbers have some elusive, essential property not captured by system S. This view is usually associated with a Platonism that permits its proponents to say that the real numbers have certain properties independently of what we are able to say or prove about them.

The problem with this view is that LST proves that if some new and improved S' had a model, then it too would have a countable model. Hence, no matter what improvements we introduce, either S' has no model or it does not escape the air of paradox created by LST. (S' would at least have its own typographical expression as a model, which is countable.

then the faith solution


Finally, there is the working faith of the working mathematician whose specialization is far from model theory. For most mathematicians, whether they are Platonists or not, the real numbers are unquestionably uncountable and the limitations on formal systems, if any, don't matter very much. When this view is made precise, it probably reduces to the second view above that LST proves an unexpected limitation on formalization. But the point is that for many working mathematicians it need not, and is not, made precise. The Skolem paradox has no sting because it affects a "different branch" of mathematics, even for mathematicians whose daily rounds take them deeply into the real number continuum, or through files and files of bytes, whose intended interpretation is confidently supposed to be univocal at best, and at worst isomorphic with all its fellow interpretations.

ISBN 1876347724
 
Aristoddler
 
Reply Mon 10 Sep, 2007 07:39 am
@pam69ur,
I'd like to see for once...someone actually building on an idea and making corrections with justification, rather than debunking an idea and making it "meaningless".
The Wright Brothers first ideas of flight were meaningless and fruitless. If they had simply debunked the idea of flight, then they would not have made the achievements they did. Instead they chose to build on their idea, make corrections, and achieve a success that was previously unheard of.

Due to lack of time I only scanned the post you made, and skipped right over the theorem, so I can't actually make a proper post or comment regarding that. I just wanted to blow steam regarding the title, and the general attitude portrayed by the author.
 
pam69ur
 
Reply Thu 13 Sep, 2007 07:01 am
@pam69ur,
Quote:


proof of 1)
Godel used Peanos axioms but some argues this system is impredicative ie a vicious circle- many outlaw impredicative definitions as they lead to paradox

Preintuitionism - Wikipedia, the free encyclopedia
Quote:



proof of 2-3
http://www.mrob.com/pub/math/goedel.htm

Quote:
2 A. Whitehead and B. Russell, Principia Mathematica, 2nd edition, Cambridge 1925. In particular, we also reckon among the axioms of PM the axiom of infinity (in the form: there exist denumerably many individuals), and the axioms of reducibility and of choice (for all types).




proof of 4)

Godel used zermelo axioms system
Godel's first Incompleteness Proof at MROB at MROB
Quote:
In the proof of Proposition VI the only properties of the system P employed were the following:

1. The class of axioms and the rules of inference (i.e. the relation "immediate consequence of") are recursively definable (as soon as the basic signs are replaced in any fashion by natural numbers).

2. Every recursive relation is definable in the system P (in the sense of Proposition V).

Hence in every formal system that satisfies assumptions 1 and 2 and is ω-consistent, undecidable propositions exist of the form (x) F(x), where F is a recursively defined property of natural numbers, and so too in every extension of such

[191]a system made by adding a recursively definable ω-consistent class of axioms. As can be easily confirmed, the systems which satisfy assumptions 1 and 2 include the Zermelo-Fraenkel and the v. Neumann axiom systems of set theory,47
 
boagie
 
Reply Thu 13 Sep, 2007 01:13 pm
@pam69ur,
Hi All,Smile

All of reality is a fiction, it is the subjects ability to abstract meaning which is true only to itself, which is its reality. Even ones identity is a fiction, which could be peeled away like the layers of an oinion. The distrubing findings of neurology will reveal much about the fictions of ourselves and the abstract abode we have constructed for ourselves from gossamer thread produce by fairy. Why should it be surprizeing that a theory would end in meaninglessness, one might consider this the only truth available to the mind of man.

If anything it should make ones beliefs all the more spiritual, for the intangiable nature of ourselves and our reality leave little choice. It is the wonder and the awe which is spirituality, and perhaps meaningless, accept for the vitality of the awe, would that be meaning or just the rapture of being alive? This is not a scientific approach to the problem of meaninglessness, but I do think it puts it into a certain perspective, one where meaninglessness is not an unreasonable expectation.That which remains unprecived is forever meaningless,objective reality is meaningless, only relative too a subject does objective reality become meaningful----------this does make sense does not guys? Meaninglessness, realities greatest abundance. I am going back to my room now!
 
NeitherExtreme
 
Reply Tue 30 Oct, 2007 07:17 pm
@boagie,
boagie wrote:
Hi All,Smile
All of reality is a fiction, it is the subjects ability to abstract meaning which is true only to itself, which is its reality. Even ones identity is a fiction, which could be peeled away like the layers of an oinion. The distrubing findings of neurology will reveal much about the fictions of ourselves and the abstract abode we have constructed for ourselves from gossamer thread produce by fairy. Why should it be surprizeing that a theory would end in meaninglessness, one might consider this the only truth available to the mind of man.

If anything it should make ones beliefs all the more spiritual, for the intangiable nature of ourselves and our reality leave little choice. It is the wonder and the awe which is spirituality, and perhaps meaningless, accept for the vitality of the awe, would that be meaning or just the rapture of being alive? This is not a scientific approach to the problem of meaninglessness, but I do think it puts it into a certain perspective, one where meaninglessness is not an unreasonable expectation.That which remains unprecived is forever meaningless,objective reality is meaningless, only relative too a subject does objective reality become meaningful----------this does make sense does not guys? Meaninglessness, realities greatest abundance. I am going back to my room now!


I think we come from kind of opposite directions, but I totally agree that the meaninglessness (whew!) we find shows our need for spirituality.Smile

I would just tweak your ideas a little though, and say that although all apears to be fiction, that does not prove that it is! It means that it is either fiction OR that we can't understand it. From our perspective it appears that we define the universe, but that does not mean that there is no other perspective. Within that realization that it could or could not be defined by me, I also now have the freedom to choose under which belief I will live.

(As a side note, doesn't it seem weird that if I would start from myself as what gives meaning to the universe, I would end up having no inherant meaning myself? The only hope I have of meaning is that there is something else that is inherantly meaningful that can give me meaning.)
 
PoPpAScience
 
Reply Tue 30 Oct, 2007 07:34 pm
@NeitherExtreme,
NeitherExtreme wrote:
(As a side note, doesn't it seem weird that if I would start from myself as what gives meaning to the universe, I would end up having no inherant meaning myself? The only hope I have of meaning is that there is something else that is inherantly meaningful that can give me meaning.)


I just had to remark, that I love the way you phrased that.
 
NeitherExtreme
 
Reply Tue 30 Oct, 2007 07:42 pm
@PoPpAScience,
PoPpAScience wrote:
I just had to remark, that I love the way you phrased that.

Well, to be honest... Now that I think about it that wasn't entirely my own.Surprised I recently read a part of a book by Francis Scheaffer that said something similar, though not exactly the same. The part of the book was basically his historical analysis of that idea- following it through Greek, Roman, Renesance, & Reformation philosophy. The biggest problem with the book was that he tried to say too much too quickly to make any solid arguments, but that one point stuck with me, and I believed it because I've experienced it. Smile
 
 

 
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