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Godel's second incompleteness theorem: Dr. Akira K
This result of Goedel, like Einstein's relativity theory, is a most misunderstood result of the last century mathematical science. The commonly accepted view of what this theorem means is completely mislead. The result is of relative nature in two ways. 1. The particular formula which is not deducible from the first order arithmetic means the consistency of the first order arithmetic only under the standard interpretation. So, for nonstandard modles of the first order arithmetic it does not mean the consistency. 2. Godel's formula which he showed to be unprovable from the first order arithmetic is depending upon a particular intensional representation of the provability. He did not show that all other possible representation of the consistency of the first order arithmetic are nonprovable from the first order arithmetic. Indeed it was Mostowki who showed in the mid 60's that there is yet another formula, which under the standard interpretation means the consistency of the first order arithmetic, is deducible from the first order arithmetic.
Mike Alexander - Godel's Incompleteness Theorem
One thing always puzzles me in relation to Godel's Theorem. By creating what amounts to Epimenides' Paradox within a formal system we can demonstrate incompleteness, but how can we be sure there are other inconsistencies beyond these relatively trivial and deliberately-constructed paradoxes? One contradiction may prove the point, but if it's a lone exception rather than the manifestation of a more widespread systemic flaw, the significance of the proof is surely less far-reaching.
Jim Russell - Goedel
A really informative and fascinating programme. Although I have been aware of Goedel and his theorem for some time, while listening I believe that I 'got' it at least for the moment. Many years ago was involved in discussions on 'trustworthy' and mathematically proven software and wish now that I had been aware of Goedel then. My personal frustration at the time resulted in a comment that mathematicians seriously discussing 'proof' reminded me of medieval schoolmen discussing how many angels can dance on the head of a pin. My boss's (a mathematician by training)response was, "Typical *** chemist". I am sure that I would have had a constructive response had I been aware of Goedel.
Dryopithecus - Goedel's Theorem
I can't remember what I was doing last Thursday, but somehow I forgot to listen to IOT! Now I'm having trouble downloading an mp3 version. (The "podcast" is has taken over an hour so far and appears to be compiling a bitstream for the QuickTime player, which is not what I want.)IFrancis Roads asks what is the definition of "true" in this context. This is a very good question. A provable statement is one that follows from the set of axioms. Professor Penrose says, in "The Emperor's New Mind", that there are statements that we know to be true but that cannot be proved from the axioms. Risky though it may be to disagree with the eminent professor, I beg to disagree. How we can "know" something is true if we can't prove it? If we had a higher set of axioms under which the statement is true, this would be fine, but it's not clear to me what these axioms could be in the case of arithmetic, for example. D.
Best Godel Story
The Best Godel story I know:Sydney Morgenbesser , as a young graduate student, got the job of getting Godel through immigration into America. Godel had to read a piece of the American Constitution and swear allegiance or something, and he read it, turned to Sydney, and said `There's an inconsistency in here', to which Sydney elbowed him hard in the ribs and said `Not now' or something equivalent.
Rob Nield
There is a seeming paradox in the incompleteness theorem - reflected in Godel's death. He came to believe he was being poisoned - so he stopped eating! Brilliant!The seeming paradox is: how can a rigorous proof be established, within the subject of which we are speaking, to show that the subject itself is not entirely rigorous? The answer, of course, is to use a "meta proof" which lies "outside" the subject in question.It is important to emphasise that Godel's theorem only applies to sufficiently complex mathematical systems - like arithmetic. If the system is sufficiently simple, the incompleteness theorem is NOT valid. Thus, very simple mathematical systems ARE perfectly logically consistent - eg Euclidean geometry.
Rob Nield - Godel's incompleteness theorem
One point which needed real emphasis -though I might have missed it in the programme - is that the key thing about the incompleteness theorem, which is so important, is that it shows that Mathematics cannot be reduced to a single set of consistent axioms - as Russell's programme in The Principia had tried (in vain) to do.
Chris Jeynes: Goedel
Another superb IOT! Thank you. I join other commentators in thinking that this was incomplete: Melvyn, we need a followup! My view is that the ontological argument of Anselm in Proslogion was a fascinating precursor of Goedel in that it was the first (?) sustained discussion of a self-referencing statement. It is clear to me that Kant's and Hegel's strictures on Anselm were unconvincing, but also I do not think that Anselm proved his point (how could he have done?). I struggled for three weeks to understand the argument when I was a young man; how delighted I was when I learned the Incompleteness Theorems (at Bristol, actually)! The other omission that I would like to see explored in a future IOT is the related question of meta arguments. Undecidability is proved by a (formal) metamathematical argument: does this not mean that the validity of metaphysical arguments cannot be ruled out a priori?
Jane - Godel
I am truly amongst the uninitiated this week but I enjoyed the programme and my resultant thinking very much.The pursuit to understand reality (as semantics coin it), both in our terms and its terms, always involves the blowing apart of particular paradigms and their encircling beliefs. A few love it but many hate it - preferring the familiar. It's ongoing whatever the preference. I'm actually responding to Jazspeak's comment about the omission of women on this week's IOT. When I glanced at the names of the three guests, my response was a slight registering of the unprecedented (since I've been listening) lack of women which I automatically assumed to be nothing to do with gender issues but with practicality of some sort. I actually think this programme, whilst being very aware of the implications, is way beyond the somewhat crass confines of political correctness. In my experience, male and female are neck and neck in this show. Best wishes - Jane
Paulpic Axiom's razor
I wonder how much of trouble these more specialized axioms cause for dilettants (not to mention interdisciplinary studies). It is like they topppled the tower of Babel.
Geoff Beard - Godel
Thank you, thank you, thank you. Radio 4 presenters often joke that science as something far beyond their own understanding (and, by implication, not of particular interest?). I can't imagine them daring to profess the same ignorance or lack of interest about arts subjects. Melvyn Bragg is an honourable exception and the Godel programme was a joy from beginning to end. Thank you.
Jennifer Parsons
Thank you for another discussion on mathematics. The subject is a unintelligible to me,yet, paradoxically it sounds interesting. Probably because of the structure of the programme with MB feeding the questions and the lively responses of the specialists. Thanks for the newsletter and the programme annotation which I pass on to a 17year old heading for university to read maths & physics.
Mike Wood - Godel
Competition for the greatest logician since Aristotle is arguable a three way fight between Godel, Frege and Tarski. Its a shame that time wasn't found for a mention of Frege, his Begriffsschrift creates the predicate calculus at one go work which Russell (and Whitehead) in Principia Mathematica continued. It was in connection with Freges work that Russells paradox was first formulated. Hilberts formalist school was in a sense a development of this, particularly introducing the idea of meta mathematics - mathematical proofs are a series of "moves" (the chess analogy) the comsistency, completeness, decidability etc of which can be proved (or not) at the meta level. Godel, as was mentioned exploits this meta idea very considerably.
DavidinBrussels Incompleteness, miracles, translat
Gödel’s theorum may be expressed in a simpler form as : ‘There are things which seem to have consistency, but on being more carefully discussed, are found to be inconsistent.’ The above principle was enunciated not by Gödel or Russell but by a first century Jewish-Christian logician in a document that dates itself as 37 CE. This Aramaic-speaker also added the complement, which modern mathematicians have failed to recognize. ‘There are many things which to some seem inconsistent which yet on a more profound investigation have consistency in them.’ The Aramaic-speaking Jews were extremely sharp in Greek logic and exposed its inherent faults. That logical campaign remained successful until the Platonic revival, when, in the wave of Imperial and ecclesiastical anti-Semitism following the Jewish wars, their contributions in holistic logic were not understood or wilfully rejected. Platonism was advantageous to the official policy of pagan syncretism; Jews were hated, their literature destroyed. The supremacy of Aramaic logicians is perhaps one reason that much of the early Greek literature was first translated into Aramaic and only then outside the Roman Empire into Arabic whence it found its way into Europe during the so-called renaissance of Greek ‘new learning’. (The major force was really the increased unofficial circulation and popular translation of the New Testament and its insistence on truth.) In his argument, the Jewish Christian logician was proving the Messiahship of Jesus and debunking arguments against him. He added: ‘No-one can be proved to be a prophet merely by consistency, for it is possible for many to attain this; but if consistency does not make a prophet much more inconsistency does not.’ He was able to debunk the false ‘miracles’ and Greek logic of his opponent who was a magician by pointing out the inconsistency in both argument and in alleged facts of miracles. Modern physics while consistent(light quanta, relative time and mass, particles smaller than atoms= 'uncuttables' etc) would seem illogical and inconsistent to Greeks but is based on proven experimental facts. Consistent truth will out -- across time, culture or human logical system!
Paul Bassett on Godel's Theorem
It's too bad your program ended where it did, as the guest mathemeticians both seem to believe that computers are necessarily limited by Godel's Incompleteness, where as smart people are not. This is a comforting myth, but a myth nonetheless. When computers are allowed to learn from essentially random inputs from the outside, they are no more limited in what they can do than are people; conversely people are just as limited by Godel as machines. Space does not permit me to outline the proof, but let me finish by saying that were I wrong, then all the time and money being spent on research into artififcial intelligence would be wasted, sort of like the fruitless quest for a perpetual motion machine.
Jazspeak - GÖDEL'S INCOMPLETENESS THEOREMS
This was a most interesting discussion and to my mind demonstrated that there will always be room for Art and Philosophy in interpreting and expressing an increasingly mathematical view of the Universe.The use of Euclidean concepts is quite good for visualising or imagining the concept of infinity. However, the use of points on a line is not, to me, as easily pictured as the use of circles.Taking the Euclidian definition of a circle as an infinite number of points all equidistant from a single central point, the definition of infinity would be the total number of points that define an infinite number of circles.Not only does this method clarify, for me, the meaning of infinity but also distinguishes infinity from merely infinite.All in all this was one of the best IOT that I have heard in a long time and I thought that the guest experts tackled the very difficult concepts with exceptional clarity. However, I do have to question the lack of women on the panel and hope that there were very good reasons for the omission.
Derek Greenacre Incompletness Theorum
It seems to me that mathematical systems cannot expect to be complete from a set of axioms alone.There also needs to be "rules of things which are illegal moves" In other words there are things which are allowed and those which are not.This is not as radical as what it might seem as for example division in an equation is valid but division by zero within an equation is not .By encompassing the things which are not valid operations into the system makes its consistency more ( but not totally) universal and as such more practical
Rita Kingham: Godel's Incompleteness Theorem
This week's topic was totally absorbing; forty minutes never passed so quickly. Roger Penrose, in the "Emperor's New Mind", gives a lot of background detail, especially in the chapter headed 'Truth, Proof and Insight'. We need at least another couple of programmes on this subject.
james lewis mathematics foundations
to tell everybody about the uncertainty in mathematics vindicates their dislike of the subject. Move away from numbers as counted to those as points on a line and you're in trouble with philosophy.
GÖDEL'S INCOMPLETENESS THEOREMS
GÖDEL'S INCOMPLETENESS THEOREMS‘The statement on the other side of this card is True’ turns to reveal…. ‘The statement on the other side of this card is False’. Seems a bit strange at first but after flipping the card over a couple of times it all makes perfect sense.Phew! Lucky I picked the card up the right way otherwise it would not have worked properly at all.
Graham Roberts on Barber on Island connundrum
The barber was a woman.
Robert Gore; Maths.
Did they convince you, Melvin? That there are different infinities I mean. Take counting numbers, imagine them on a line, this could be an infinite direction, and never enclosed in a set. Unless one can have sets that expand infinitely to contain their elements? Try the question, 'can there be any counting number, that cannot be increased by one?' Seems obvious to me, but then I have not been programed by mathematicians! Does it have any meaning to compare different ways of proceeding infinitely, whether with games like Pi, or simply by whole number counting. Any way, more please; but you are not alone in being glad they started with barbers!
Greg Michaelson: Godel's Incompleteness Theorem
This otherwise highly engaging programme claimed that Godel's work shows that computers have limitations that are not shared by humans. First of all, humans are finite machines so any limitations that Godel's results place on computers also apply to humans. Secondly, in principle a computer can prove arbitrary results given the same rules as a mathemtician and enough time. In particular, in 1993 Ammon reported a computer generated proof of Godel's incompleteness theorem.
Andrew Wyld—Indeterminately unprovable state
Given that you can encode statements as numbers, surely you can make statements with those numbers and encode those as well. In other words, there must be statements that you can't *prove* are unprovable—statements whose Gödel coding can be put into other, unprovable, statements. Since you can't prove consistency either, if you come across something you think *might* be unprovable, isn't it rather likely that you'll never be able to tell (unless you can prove or disprove it, that is?)
David Derrington - Set theory
Wonderful programme, especially for those of us who come to maths more indirectly via philosophy. Explaining the Barber Paradox and its implications purely in words on air takes some doing. Not sure who it was, but even more clarity could have been given by a few examples of “sets that are members of themselves” eg the set of big things, which is itself big, the set of non-tables, which is itself a non-table, the set of boring things…
Francis Roads
Goedel says that certain statements may be true but unprovable. What is the definition of "true" in this context?
The_Taffinch
My head hurts.
probability
Great show! I'm still scratching my head about the three doors,goat, and a car!Every one of the shows are exceptional..thanks a lot!Lawrence
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