I know this post is aimed at laymen but this sentence is not correct.
"A bit more precisely, the Incompleteness Theorem shows that human beings can never formulate a correct and complete description of the set of natural numbers, {0, 1, 2, 3, . . .}."
The second order Peano Axioms are categorical and thus, up to isomorphism, the only model for this axiom system are the Natural numbers {0, 1, 2, 3, ...}. This is a complete system. We can't happen is a recursively enumerable axiomatic description of the Natural numbers that is complete.
Another way to get a complete description of the Natural numbers is to take the collection of all true statements of the Natural numbers and make that our axiomatic system. It's just not a useful axiomatic system but it is a complete description of the Natural numbers.
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 :)
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 Theoremshttp://www.logicmatters.net/igt/ is a great book, with all the mathematics but willing to go into the philosophy.
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. :)
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).
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.
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.
I'm out of my depth but I found this interesting. Is it accurate?
"But Z2 is usually studied with first-order semantics, and in that context it is an effective theory of arithmetic subject to the incompleteness theorems. In particular, Z2 includes every axiom of PA, and it does include the second-order induction axiom, and it is still incomplete.
"Therefore, the well-known categoricity proof must not rely solely on the second-order induction axiom. It also relies on a change to an entirely different semantics, apart from the choice of axioms. It is only in the context of these special 'full' semantics that PA with the second-order induction axiom becomes categorical."
Yes. Mummert is far more qualified than I am on these matters. Here is a line from his reply that is important:
"So, even though Z2 with full second-order semantics is categorical, for any sound effective deductive system there are still true formulas of Z2 that are neither provable nor disprovable in that system."
The key is sound and effective deductive system. Think computable or mechanistic process for deduction. The second order Peano axioms with second order semantics are not and effective deduction system.
In all proofs you have to start with (or end up with depending on the direction your proofs go) axioms. In the second order Peano axioms with second order semantics you can end up in a situation where you don't know if a given statement is an axiom! Making that determination can be quite hard.
Fascinating! How does the situation arise where you don't know if something is an axiom? Aren't axioms either members of a small agreed-on set, or based on an axiom schema?
Well let's say I take as my axioms all true statements about the Natural numbers. Let's take the Goldbach conjecture. Is it an axiom? No one knows because no one knows if it is true or not.
When you say "small agreed upon set" you are in essence talking about a recursively enumerable set of axioms. A collection of axioms that is "small" enough so that one could easily determine if a statement is an axiom.
If you have time for another question, I'm still confused about how it applies to second-order arithmetic, though, since the Peano axioms are well-known and easily listed on a single piece of paper. What difficulty is there be in determining whether a statement is a second-order Peano axiom?
It seems particularly strange since there are apparently fewer axioms than in first-order Peano arithmetic (by replacing an axiom schema with a single induction axiom).
"A bit more precisely, the Incompleteness Theorem shows that human beings can never formulate a correct and complete description of the set of natural numbers, {0, 1, 2, 3, . . .}."
The second order Peano Axioms are categorical and thus, up to isomorphism, the only model for this axiom system are the Natural numbers {0, 1, 2, 3, ...}. This is a complete system. We can't happen is a recursively enumerable axiomatic description of the Natural numbers that is complete.
Another way to get a complete description of the Natural numbers is to take the collection of all true statements of the Natural numbers and make that our axiomatic system. It's just not a useful axiomatic system but it is a complete description of the Natural numbers.