Memories of Kurt Gödel
rudyrucker.com
rudyrucker.com
"Although other members of the institute found the gloomy logician baffling and unapproachable, Einstein told people that he went to his office 'just to have the privilege of walking home with Kurt Gödel.'"
"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.
1) Smith's Introduction to Godel's Theorems 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 is enlightening in a different way.
I'll also mention a favorite author around here http://www.scottaaronson.com/blog/?p=710 .
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.
"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."
From: https://math.stackexchange.com/questions/617124/peano-arithm...
"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.
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).
[1] https://math.stackexchange.com/questions/106635/why-does-the...
Is he saying that our brains exist over all time simultaneously but they "give" us a sequence of instants from which we perceive the illusion of passage of time ?
We can perceive this moment in time, right now. We can also perceive moments in time which are not right now. To do this, use your imagination and explore your memories or fantasies. Therefore, there are many moments in time. However, it seems like only one moment in time is experienced simultaneously; this is the illusion.
Quantum Mechanics, as a formal system can encode the natural numbers, and thus can form a substrate for self referential Gödel encoding. The same principles would apply to a quantum system, and any other capable of encoding natural numbers.
But models that generate results that can be tested, through observation and experiment? Sure. Maybe even arbitrarily integrated models.
However, there's uncertainty throughout. So models can't be deterministic. Certainly at the "ends", at the quantum level, and at the level of consciousness.
That said, it's interesting to think about a universe with truly infinite rules. Each physical law could have minor exceptions caused by smaller more detailed phenomenon. Each time you would discover some new principle, it would reveal more yet unknown questions, a fractal of infinite knowledge to be refined and science to do. But I think most scientists hope for a finite set of underlying rules for reality.
There is a connexion there between Gödel and Popper.
Except inside a formal system, you can never prove that something is true, only that one explanation is better than other in an endless pursue of better explanations.
I'm not sure there is such thing as not-scientific knowledge, by the way.
I'm fully on-board with the overwhelming, world-changing effectiveness that the scientific method provides for distilling factual, empirical knowledge and truth.
Lately, however, I've been contemplating forms of knowledge and understanding that are more difficult to assess and validate -- things that might be typically described as wisdom or keen insight. Our scientific instruments can't provide observations that let us robustly verify such knowledge, but to me it seems very evident that it exists.
Some examples: What is important to building and maintaining strong relationships? How can one prepare for and handle personal hardship? If one finds themselves in a fortunate position with excess resources, what are good ways to use those resources to help others?
Science can help us with these questions, but humans have useful knowledge to bring to bear in answering those questions that can't be yet described within the framework of science.
Differentiating by quality or truthiness is ridiculously hard in such domains, but I don't think that is a valid reason for dismissing such things altogether.
Even so, one can also apply the scientific method to those sorts of knowledge. One can look at performance. Quality of relationships. Success at dealing with hardship. That's part of psychology. But it hasn't received enough attention, I think.
Anyway, I get that they're similar. But I don't see them as the same, but rather complementary.
I also get what you say about knowledge. Scientific knowledge is what you get by studying external reality. But there is also knowledge that you get through introspection.
"Despite being known for his pioneering work on chaotic unpredictability, the key discovery at the core of meteorologist Ed Lorenz's work is the link between space-time calculus and state-space fractal geometry. Indeed, properties of Lorenz's fractal invariant set relate space-time calculus to deep areas of mathematics such as Gödel's Incompleteness Theorem."
"Consider a point p in the three-dimensional Lorenz state space. Is there an algorithm for determining whether p belongs to IL? There are certainly large parts of state space which don’t contain any part of IL. However, suppose p was a point which ‘looked’ as if it might belong to IL. How would one establish whether this really is the case or not? If we could initialise the Lorenz equations at some point which was known to lie on IL, we could then run (1) forward to see if the trajectory passes through p. If the integration is terminated after any finite time and the trajectory still hasn’t passed through p, we can’t really deduce anything. We can’t be sure that if the integration was continued, it would pass through p at some future stage. The Lorenz attractor provides a geometric illustration of the Gödel/Turing incompleteness theorems: not all problems in mathematics are solvable by algorithm. This linkage has been made rigorous by the following theorem [7]: so-called Halting Sets must have integral Hausdorff dimension. IL has fractional Hausdorff dimension - this is why it is called a fractal. Hence we can say that IL is formally non-computational. To be a bit more concrete, consider one of the classic undecidable problems of computing theory: the Post Correspondence Problem [46]. Dube [16] has shown that this problem is equivalent to asking whether a given line intersects the fractal invariant set of an iterated function system [4]. In general, non-computational problems can all be posed in this fractal geometric way."
Source: Lorenz, Gödel and Penrose: New Perspectives on Determinism and Causality in Fundamental Physics https://arxiv.org/pdf/1309.2396.pdf
A more interesting interpretation of the question (to me) is what if he had access to computers used as mind amplification devices. For example; such as by using Mathmatica or Maple to explore and visualize theorems and things. I'd imagine the benefit of this activity for someone like Gödel would be "not much". Computations and simulation have inherit limitations; precision and rounding errors for scientific computation, and the fact they can only model what we can imagine for another. These people such as Gödel, Neumann, and their ilk, new this, they begat the era of computation we have today, and all the limitations that involved. Neumann in particular was famous for, when presented with your problem, would tell how to solve it.
What's new today that they may not have foreseen is the vast level of internetworking and human communication that arose from ubiquitous presence of computers and networks.
Something that strikes me about the present article is the fact that Gödel, keen to see Rucker before he knew him, was not so keen to converse with him after their first encounter. One might think Gödel was not too impressed with Rucker, maybe found him boring and dull for example.
In an alternate universe, Godel published his proofs in a paper on arXiv in 2002. However it was largely overlooked or dismissed as the work of a crank, so he went back to his day job at MSR.
A casual reference to the paper in a comment on on Math Overflow five years later led to wider discussion and eventually to Scott Aaronson publicising it on his blog. Within a year the two theorems were accepted by the global mathematics community.
Since then, numerous blogs, subreddits, and Facebook posts have challenged the legitimacy of the proofs or claimed prior credit. Speculation persists on some parts of the internet that GCHQ or the NSA knew of the theorems as early as the 1950s.