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Thanks for pointing this out. Two questions: (1) Are there any resources you'd recommend reading to learn more about this? (2) If you are familiar, could you re

by monet 9y ago

Thanks for pointing this out. Two questions: (1) Are there any resources you'd recommend reading to learn more about this? (2) If you are familiar, could you restate the Incompleteness theorem in your own words? Just looking to learn more here :)

gfody 9y ago
there's a tiny book called Godel's Proof that is both short and easy to read. I picked it up in my quest to read everything Hofstadter (he wrote the foreword) and thought it made it pretty straight forward. any axiomatic system that's flexible/powerful enough to make some sort of statement about itself will succumb to this method for constructing a paradox - and so all such systems are either inconsistent (paradox is possible) or incomplete (lacks sufficient flexibility to make a statement about itself).
- posterboy 9y ago
  so by contraction, self-reference is paradox? That's certainly true for the liar's paradox. But that shouldn't be mistaken for the actual statement, that some unprovable propositions follow from incomplete axioms and that the axioms can't be completed.
nicky0 9y ago
You're on a dangerous road. I asked these questions once, got a couple of books. Decided I needed to enrol on a couple of courses... 7 years later I had completed a math degree. :)
- smallhands 9y ago
  really i will love love to hear your story.i have feeling that there is more to it ! wow , i am impressed.please what is your email?
jimhefferon 9y ago
Lots of reading are available to technical people who are willing. People elsewhere in this thread have mentioned Nagel and Newman, and also Smullyan. Here are two more: 1) Smith's Introduction to Godel's Theorems http://www.logicmatters.net/igt/ http://www.logicmatters.net/igt/ is a great book, with all the mathematics but willing to go into the philosophy. 2) Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse http://www.ams.org/notices/200703/rev-raatikainen.pdf http://www.ams.org/notices/200703/rev-raatikainen.pdf is enlightening in a different way. I'll also mention a favorite author around here http://www.scottaaronson.com/blog/?p=710 http://www.scottaaronson.com/blog/?p=710 .
yequalsx 9y ago
At the beginning of the 20th century is when abstraction in mathematics really took off. Things started getting abstract quickly. There was also Russell's paradox and the work of Cantor that dealt with infinities. It became clear that math needed to be put on a rigorous, firm foundation. Set theories were developed and logic formalized. One goal was to come up with a collection of axioms for the Natural numbers (capital letter to denote the Natural numbers we all know and love). What was wanted was a collection of axioms that are recursively enumerable. Think computable. The goal was to find a mechanistic process to check theorems and to prove new theorems. In some sense too alleviate the field from human error. In modern language we'd say to find a way to have a computer check/discover theorems in number theory. There are two axiom systems for the Natural numbers. Both are Peano axioms and one system is first order and recursively enumerable. The other system is second order and not recursively enumerable. There are infinitely many models of the first order Peano axioms. For each such model we call them natural numbers (lower case) to signify they are a model of the first order axioms. The Incompleteness Theorem: There are statements that are true in the Natural numbers (upper case!) that are not true in all models of the natural numbers. Furthermore, this will always be the case no matter what system of recursively enumerable axioms you have for describing the Natural numbers. Consequence: The Natural numbers can not fully be described by a nice set of axioms. Whatever recursively enumerable system of axioms you have to describe the Natural numbers will be insufficient to prove all true statements of the Natural numbers. Hence, such systems of axioms are incomplete. One can always find a complete system of axioms by taking as the collection of axioms the collection of all true statements. This isn't helpful because there would no effective way to determining whether or not a statement is an axiom just by looking at it or comparing it to a finite set of axiom schema. Such a collection of axioms is wholly impractical and not useful. But it is wrong to say that a complete axiom system can not be found. Some people falsely claim that the Incompleteness theorem says that there are statements of the Natural numbers that are neither provable or disprovable. What the theorem says is that for a given recursively enumerable set of axioms that the Natural numbers are a model of there will be statements of the Natural numbers that are true but not provable in that system. All true statements of the Natural numbers are provable in the second order system of axioms but the things get dicey from a logic point of view when working with the second order Peano axioms.

by monet 9y ago

gfody 9y ago

there's a tiny book called Godel's Proof that is both short and easy to read. I picked it up in my quest to read everything Hofstadter (he wrote the foreword) and thought it made it pretty straight forward. any axiomatic system that's flexible/powerful enough to make some sort of statement about itself will succumb to this method for constructing a paradox - and so all such systems are either inconsistent (paradox is possible) or incomplete (lacks sufficient flexibility to make a statement about itself).

posterboy 9y ago
so by contraction, self-reference is paradox? That's certainly true for the liar's paradox. But that shouldn't be mistaken for the actual statement, that some unprovable propositions follow from incomplete axioms and that the axioms can't be completed.

nicky0 9y ago

You're on a dangerous road. I asked these questions once, got a couple of books. Decided I needed to enrol on a couple of courses... 7 years later I had completed a math degree. :)

smallhands 9y ago
really i will love love to hear your story.i have feeling that there is more to it ! wow , i am impressed.please what is your email?

jimhefferon 9y ago

Lots of reading are available to technical people who are willing. People elsewhere in this thread have mentioned Nagel and Newman, and also Smullyan. Here are two more: 1) Smith's Introduction to Godel's Theorems http://www.logicmatters.net/igt/ http://www.logicmatters.net/igt/ is a great book, with all the mathematics but willing to go into the philosophy. 2) Franzen's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse http://www.ams.org/notices/200703/rev-raatikainen.pdf http://www.ams.org/notices/200703/rev-raatikainen.pdf is enlightening in a different way. I'll also mention a favorite author around here http://www.scottaaronson.com/blog/?p=710 http://www.scottaaronson.com/blog/?p=710 .

yequalsx 9y ago

At the beginning of the 20th century is when abstraction in mathematics really took off. Things started getting abstract quickly. There was also Russell's paradox and the work of Cantor that dealt with infinities. It became clear that math needed to be put on a rigorous, firm foundation. Set theories were developed and logic formalized. One goal was to come up with a collection of axioms for the Natural numbers (capital letter to denote the Natural numbers we all know and love). What was wanted was a collection of axioms that are recursively enumerable. Think computable. The goal was to find a mechanistic process to check theorems and to prove new theorems. In some sense too alleviate the field from human error. In modern language we'd say to find a way to have a computer check/discover theorems in number theory. There are two axiom systems for the Natural numbers. Both are Peano axioms and one system is first order and recursively enumerable. The other system is second order and not recursively enumerable. There are infinitely many models of the first order Peano axioms. For each such model we call them natural numbers (lower case) to signify they are a model of the first order axioms. The Incompleteness Theorem: There are statements that are true in the Natural numbers (upper case!) that are not true in all models of the natural numbers. Furthermore, this will always be the case no matter what system of recursively enumerable axioms you have for describing the Natural numbers. Consequence: The Natural numbers can not fully be described by a nice set of axioms. Whatever recursively enumerable system of axioms you have to describe the Natural numbers will be insufficient to prove all true statements of the Natural numbers. Hence, such systems of axioms are incomplete. One can always find a complete system of axioms by taking as the collection of axioms the collection of all true statements. This isn't helpful because there would no effective way to determining whether or not a statement is an axiom just by looking at it or comparing it to a finite set of axiom schema. Such a collection of axioms is wholly impractical and not useful. But it is wrong to say that a complete axiom system can not be found. Some people falsely claim that the Incompleteness theorem says that there are statements of the Natural numbers that are neither provable or disprovable. What the theorem says is that for a given recursively enumerable set of axioms that the Natural numbers are a model of there will be statements of the Natural numbers that are true but not provable in that system. All true statements of the Natural numbers are provable in the second order system of axioms but the things get dicey from a logic point of view when working with the second order Peano axioms.