And there are as many things in the set of all primes as there are in the set of all natural numbers. The reals have a higher cardinality than the set of naturals, not because they "grow to infinity" faster or anything silly like that, but because there are simply more of them there.
I think it's rather silly to split hairs, but that's just me. Anyway, since you seem to be in the business of splitting hairs, I will indulge you.
Let me give you a quick proof that there are not as many things in the set of primes as there are in the set of natural numbers:
1. A proper subset contains fewer things than its superset.
2. The set of all primes is a proper subset of all naturals. ... (Therefore) The set of all primes contains fewer items than the set of all naturals (i.e., the cardinality of the set of all primes is smaller than the cardinality of the set of all naturals).
I suspect the only reason we have this disagreement is because you start from a different set of axioms and/or take a different route to arrive at the knowledge you so want to prove me wrong with. [And, btw, you're the one who brought the word cardinality into this. I was speaking of degrees of infinities which may be thought of as aleph numbers and thus cardinalities --or-- as a more proper degree which respects some other (and more important to some) things that we know about sets].
But, once again, this sort of collision of models is entirely expected given Godel's theorem. The 'problem' here can be looked at in several ways but since you did not understand my geometry illustration and since you do not seem to want to learn anything from me (or to even think that is possible for you to do so, I will not be wasting my time going in to these additional insights).
And, yes, I agree aleph_1 > aleph_0. I don't agree that in my bijection that the primes grow faster, though I will agree that the nth prime will be larger than the n-1th by more than n in general.
Well, if you realize that, then you should be well on your way to understanding the proof I gave above.
But the important thing is that they're not really "growing to an infinity" because infinity isn't a number (even if we do say infinite number, it's more a convenience of ambiguous terminology than anything).
And, even more importantly, the cardinality of a set isn't what it "grows to" as you keep saying, as that doesn't even make sense.
If you allow me a small aside on how that doesn't make sense, it will become clear.
You've got to be kidding me. Of course it is a convenience, that's why I used it. However, it becomes much less convenient when people start splitting hairs like this.
Firstly, for your notion to make sense, we would have to say that |{2,3,4,5}| = 5 since it clearly "grows to" 5 in your words, but we should be able to see that |{2,3,4,5}| = 4.
I was speaking of the density of the set. When you do your neat folding trick where you take all the negatives and you put them in with the positives with zero at the beginning, are you not doubling the number of numbers that come after zero in that set? The geometric analogy I provided would've shown you this if you'd understood it. [And, if you don't believe this statement, I can prove it.]
Cardinality is a measure of how many things are in a set not how large are the things in the set.Another, hopefully illuminating, related bit: take the sets of the first n natural numbers and the first n prime numbers. Say, n=5, so we have N_5 = {1,2,3,4,5} and p_5 = {2,3,5,7,11}. Surely we can both agree that the cardinality of both of these is 5 and the fact that the set of primes "grows faster" doesn't change that. All that we care about is *how many things are in it*.
Ok, now do this. Take the first n positive numbers where n=5: {1, 2, 3, 4, 5}. Now, of tho
I have to admit, you've lost me with this post. To begin with, I can't make much sense of your geometric description or how my (admittedly rushed and slightly handwaved, though there exist far more formal and robust forms of them) proofs hinged on it. The proofs I used hinged on one of the basic principles underlying a very rich and important field, combinatorics. This principle is that the ability to create a bijection between two sets implies equal cardinality and the provable inability to do so implies unequal cardinality. Further, the simple fact that there exists these bijections means that no information is being lost. As long as you have one set and a bijection between it and another set, you have that second set and all the information it contains.
This proof works in such a way as to confer equality on two sets whilst the two sets in question are intuitively (i.e., at finite scales) different in size. I thought the point of aleph numbers was to distinguish between various degrees of infinity. In this case, if both N and the prime numbers have the same aleph number, then the aleph number is not precise enough and we need something which captures the *speed* at which these cardinalities grow to infinity. If aleph numbers don't capture that (or don't capture it at this granularity), then it would be nice to have some augmented aleph numbering system which would capture it).
You agree that aleph one represents a greater infinity than aleph null, no? Do you also agree that with the bijection you defined between N and the primes, the primes grow faster towards infinity than do the members of N? Therefore, in my mind, the infinity that the primes are growing to is larger than the infinity that the naturals grow to (or alternatively the infinite cardinality of primes is smaller than the infinite cardinality of naturals, i.e., not every natural is a prime).
Also, I don't get your last sentence or in what way I depended on zero....
Zero was of interest because it was the first number of the set: {0, 1, -1, 2, -2, 3, -3} which was used to prove that |Z| == |N| (which is really only interesting as background information on our way to the proof of the equality between the cardinalities of N and the primes). Like I said previously though, I don't think you had to choose zero there. You could've had a set that looked like this for instance:
{18, 19, 17, 20, 16, 21, 15,...} and that would've been analogous to pick 18 to 'pull' from the line as in my geometric description. After having written that, I'm now pretty convinced that the specialness of zero had nothing to do with this proof.
Well, that is an interesting trick (and I've just responded to this in another post as well--maybe we you could join the discussion there)?
Math is fun, but at some point, it starts to lose information which we intuitively know to be true (and thus it's applicability to the real world is limited).
So, cardinality is precise--that wasn't the right word. But, it is somewhat artificial and abstract. To put this in geometric terms, the entire 'proof' hinges on taking a line and turning it into a ray (which you could picture as picking a point on the line and pulling it out perpendicularly until the two rays formed by the picking of the point lie on top of one another). And, that's obviously losing some information (all of the information of one of the original rays, i.e., the negative numbers in your example).
Cardinality is simply an artificial construct and does not entirely capture the real nature of things. It may be that we came to this result by including an improper axiom. I.e., maybe reality is beyond this mode of looking at the world. Maybe (and I entirely suspect this is the case) reality is beyond all modes of looking at the world.
This is the interesting thing about Godel, to me. Models work in isolation; but when they reach a certain critical mass, they start to break down.
In this particular case, maybe we need another definition of cardinality that doesn't lose the intuitive nature of the relationships between the sizes of these sets). Also, of course, the fact that zero is a special number (though even it is a somewhat abstract creation) plays a big role in this, but I think that your 'trick' would work even if you had picked another number besides zero to be the bounded endpoint).
How can there be a 1:1 correspondence? Just take a look at this sieve in action.
As you can see, right off the bat, all of the even numbers are thrown out. So, even if you go no farther than that, there are more integers than primes.
[And, when you include negative numbers, none of which can be prime by a standard definition of prime, then the count of integers is even more!]
Well, what you're missing is that sticking with what's real is just a doomed a process as believing in fantasy. All this means to me is that the universe is greatly mysterious and we cannot know everything, i.e., reality outruns knowledge.
As long as your content with that, fine. But, for me, that realization raises profound philosophical implications about the nature of humanity and our universe and causes me to seek higher knowledge. I'm not saying that I understand the mystery, I'm just embracing it's existence and engaging it.
Yea, this system and the system I want are two different entities. The system in question could keep each set of axioms and theories separate and accept the fact that it is essentially shooting a shotgun out into the universe of knowledge. But that isn't as particularly interesting to me as a system which will collect all of scientific knowledge. I think that the points where we see contradictions (i.e., collisions) between theories will provide very useful insight into the nature of our universe. And, it may even be the case that this is exactly the reason that quantum mechanics and Newtonian physics do not mesh.
I suppose it could keep the different theories separate, but that wouldn't do anything to help us get a holistic view of truth.
And, I disagree with the 'irrelevance' bit. It was my understanding (and this may just be due to my particular philosophy of mathematics slant--there are different schools of thought on it) that truth is absolute and transcendent. I think that all theorems may be translated from one theory to another (providing that the sets of axioms are sufficiently advanced). The 'truth' is beyond each particular perception of it.
Therefore, I sort of pictured a system which would collect all of mankind's knowledge in some abstract syntax which could be converted to meaningful contexts (and the conventional 'theories') at will (sort of like the Hilbert approach).
Mmmm... that's a little bit counterintuitive. It seems obvious to me that there are more integers than there are primes. I suppose cardinality is a bit less precise property than I'd imagined.
What I would be interested in knowing is if any of our known theorems contradict one another when translated from one valid axiomatic system to another? This is more directly applicable to Godel's work than undecidability, I think. Based on his work, I picture in fact an infinite number of such contradictions (and this may be why quantum mechanics and Newtonian physics do not mesh).
You should read more philosophy. Materialism isn't the only one out there [and it has serious problems as many thinkers have pointed out]. I would start with JR Lucas if I were you.
Well, in that case, the system will deterministically (although incomprehensibly to the human mind) exclude some valid corners of the vast universe of knowledge. And, it will depend on which axioms it starts with and which theorems are entered first--they will essentially steer it on a course that's doomed to ignore certain valid pieces of knowledge. That's the limitation of Godel. Of course, the person entering one of these excluded theorems will know that his theorem is correct based on the original axioms of his theory so maybe it will be of interest to see where these collide.
Nevertheless, I don't think a system which will actually do the sort of verification and discovery of new theorems which I described is all that far off (just ask Ray Kurzweil).
and check to see that existing theorems are all logically consistent with each other.
I think it's particularly that part that Godel's theorem will make impossible. Consider the case where a theorem from one view of the world (theory) is inconsistent with a second theorem from another theory which has already been entered. Either the system must reject the second or forget the first. Neither is desirable.
And, actually, it doesn't just have to be limited to philosophy and law. But, any claim on the interwebs. We could start from any sentence (or truth claim) and drill down from there (of course, depending on your slant, you may categorize this last suggestion as either philosophy or law).
That's really neat. I would like the same thing to be done with philosophy and law. We should require that our lawmakers abide by the same stringent syntaxes and correctness/consistency requirements as us computer scientists/logicians do.
The system must still be able to exhaustively 'check' each new theorem against all pre-existing theorems (and the original set of axioms). The only difference here between a machine that learns new theorems and one that checks theorems is the source of the input. In this case, the input is coming from humans, but it doesn't necessarily have to be so. Both types of machines though I would imagine will use the same algorithm to 'check' the 'inputs' for contradictions with previously known good theorems.
But, who says that the axioms of set theory are complete or correct (or the only possible set of axioms)? There may be another set of axioms where continuum hypothesis can be proven, no?
... thus, the natural translation from all known theorems of all other theories must be made just to enter them...
And, to answer how a mathematical truth can be known but not proven:
I suspect that it is because our 'minds' include something more than just material--something akin to a soul. We can see on a 'meta' level that mere machines cannot.
Also, almost everything else JR Lucas says about Godel is interesting if you're into that kind of thing (and specifically 'reality outruns knowledge' from the 'Implications of Godel's Theorem' essay).
It's relevant because it is possible and even quite likely that we humans already know contradicting theorems (most likely coming from different theories, number, graph, etc.)
No one has translated all of each of the theories to the others, so we as of the time of this writing do not know if many of the *known* theories will contradict. That's where this system becomes very interesting in the context of Godel. It seems that this system will have to stick to one set of axioms (i.e., one theory) and thus, the natural translation from all known theorems of all other theories must be made just to enter it (and I suspect that this one set of axioms isn't really those of any one 'theory' of math but rather some optimal [from the computer's perspective] set).
So, whichever of two conflicting theorems is entered first will take precedence over the second and the second (which was entirely valid when working within its original context), will not be allowed to be entered.
I suppose that is the one. Unfortunately, it seems, the right's judges have a bit more of a tendency to forget how they got there than the left's do. Sandra Day O'Connor also comes to mind.
Actually, your citations of other finds do nothing to refute my point about the one particular find where scientists worldwide got all giddy over a pig's tooth. If I'm not mistaken, it was before the other finds you cite (if they are for real this time.... once bitten, twice shy you know).
And, explaining how the guy could've made the mistake, although noble, hardly erases the mistake.
Reposting as non-AC so you'll be more likely to see this:
Well, for one, you mentioned it. (In number 2).
And, it wasn't so much Godel's conclusion that is of interest, but his method, which was actually only an extension of the method of Hilbert (and to a lesser degree Russell).
Are you saying that if machines had existed at that time that they could not run the codes? To disprove that assertion if that is what you are saying, then one could simply take the Hilbert system (or the Godel numbering system) and do the computations today starting from known axioms (and in fact, I bet that is exactly what the software in question is doing or going to do). If the numbering system is not identical, then it certainly must be equivalent.
The abstractions are useful because they provide succinct methods to discuss complex subjects, but they aren't useful because they obfuscate the meaning of what they are trying to discuss.
Or..., maybe the concepts really are abstract. How can you obfuscate the meaning of an abstract entity? It is, by definition, abstract. They have to choose something to represent such concepts and they typically choose symbols from a few well-known and ancient languages.
While the proof did contain only one such statement which caused the problem, one was enough. Godel did not even attempt to find more because the proof implies that (assuming an infinite amount of knowledge exists) there are infinitely many such statements (obviously one of these infinities has a greater degree than the other in the same way that there are both an infinite number of integers and an [smaller] infinite number of primes).
In my mind, it's kind of like dividing by zero (everything breaks down at that point). I would imagine it's a little like yeast working through dough.
Slashdot Mirror
And there are as many things in the set of all primes as there are in the set of all natural numbers. The reals have a higher cardinality than the set of naturals, not because they "grow to infinity" faster or anything silly like that, but because there are simply more of them there.
I think it's rather silly to split hairs, but that's just me. Anyway, since you seem to be in the business of splitting hairs, I will indulge you.
... (Therefore) The set of all primes contains fewer items than the set of all naturals (i.e., the cardinality of the set of all primes is smaller than the cardinality of the set of all naturals).
Let me give you a quick proof that there are not as many things in the set of primes as there are in the set of natural numbers:
1. A proper subset contains fewer things than its superset.
2. The set of all primes is a proper subset of all naturals.
I suspect the only reason we have this disagreement is because you start from a different set of axioms and/or take a different route to arrive at the knowledge you so want to prove me wrong with. [And, btw, you're the one who brought the word cardinality into this. I was speaking of degrees of infinities which may be thought of as aleph numbers and thus cardinalities --or-- as a more proper degree which respects some other (and more important to some) things that we know about sets].
But, once again, this sort of collision of models is entirely expected given Godel's theorem. The 'problem' here can be looked at in several ways but since you did not understand my geometry illustration and since you do not seem to want to learn anything from me (or to even think that is possible for you to do so, I will not be wasting my time going in to these additional insights).
And, yes, I agree aleph_1 > aleph_0. I don't agree that in my bijection that the primes grow faster, though I will agree that the nth prime will be larger than the n-1th by more than n in general.
Well, if you realize that, then you should be well on your way to understanding the proof I gave above.
But the important thing is that they're not really "growing to an infinity" because infinity isn't a number (even if we do say infinite number, it's more a convenience of ambiguous terminology than anything).
And, even more importantly, the cardinality of a set isn't what it "grows to" as you keep saying, as that doesn't even make sense.
If you allow me a small aside on how that doesn't make sense, it will become clear.
You've got to be kidding me. Of course it is a convenience, that's why I used it. However, it becomes much less convenient when people start splitting hairs like this.
Firstly, for your notion to make sense, we would have to say that |{2,3,4,5}| = 5 since it clearly "grows to" 5 in your words, but we should be able to see that |{2,3,4,5}| = 4.
I was speaking of the density of the set. When you do your neat folding trick where you take all the negatives and you put them in with the positives with zero at the beginning, are you not doubling the number of numbers that come after zero in that set? The geometric analogy I provided would've shown you this if you'd understood it. [And, if you don't believe this statement, I can prove it.]
Cardinality is a measure of how many things are in a set not how large are the things in the set.Another, hopefully illuminating, related bit: take the sets of the first n natural numbers and the first n prime numbers. Say, n=5, so we have N_5 = {1,2,3,4,5} and p_5 = {2,3,5,7,11}. Surely we can both agree that the cardinality of both of these is 5 and the fact that the set of primes "grows faster" doesn't change that. All that we care about is *how many things are in it*.
Ok, now do this. Take the first n positive numbers where n=5: {1, 2, 3, 4, 5}. Now, of tho
I have to admit, you've lost me with this post. To begin with, I can't make much sense of your geometric description or how my (admittedly rushed and slightly handwaved, though there exist far more formal and robust forms of them) proofs hinged on it. The proofs I used hinged on one of the basic principles underlying a very rich and important field, combinatorics. This principle is that the ability to create a bijection between two sets implies equal cardinality and the provable inability to do so implies unequal cardinality. Further, the simple fact that there exists these bijections means that no information is being lost. As long as you have one set and a bijection between it and another set, you have that second set and all the information it contains.
This proof works in such a way as to confer equality on two sets whilst the two sets in question are intuitively (i.e., at finite scales) different in size. I thought the point of aleph numbers was to distinguish between various degrees of infinity. In this case, if both N and the prime numbers have the same aleph number, then the aleph number is not precise enough and we need something which captures the *speed* at which these cardinalities grow to infinity. If aleph numbers don't capture that (or don't capture it at this granularity), then it would be nice to have some augmented aleph numbering system which would capture it).
You agree that aleph one represents a greater infinity than aleph null, no? Do you also agree that with the bijection you defined between N and the primes, the primes grow faster towards infinity than do the members of N? Therefore, in my mind, the infinity that the primes are growing to is larger than the infinity that the naturals grow to (or alternatively the infinite cardinality of primes is smaller than the infinite cardinality of naturals, i.e., not every natural is a prime).
Also, I don't get your last sentence or in what way I depended on zero....
Zero was of interest because it was the first number of the set: {0, 1, -1, 2, -2, 3, -3} which was used to prove that |Z| == |N| (which is really only interesting as background information on our way to the proof of the equality between the cardinalities of N and the primes). Like I said previously though, I don't think you had to choose zero there. You could've had a set that looked like this for instance: ...} and that would've been analogous to pick 18 to 'pull' from the line as in my geometric description. After having written that, I'm now pretty convinced that the specialness of zero had nothing to do with this proof.
{18, 19, 17, 20, 16, 21, 15,
Well, that is an interesting trick (and I've just responded to this in another post as well--maybe we you could join the discussion there)?
Math is fun, but at some point, it starts to lose information which we intuitively know to be true (and thus it's applicability to the real world is limited).
So, cardinality is precise--that wasn't the right word. But, it is somewhat artificial and abstract. To put this in geometric terms, the entire 'proof' hinges on taking a line and turning it into a ray (which you could picture as picking a point on the line and pulling it out perpendicularly until the two rays formed by the picking of the point lie on top of one another). And, that's obviously losing some information (all of the information of one of the original rays, i.e., the negative numbers in your example).
Cardinality is simply an artificial construct and does not entirely capture the real nature of things. It may be that we came to this result by including an improper axiom. I.e., maybe reality is beyond this mode of looking at the world. Maybe (and I entirely suspect this is the case) reality is beyond all modes of looking at the world.
This is the interesting thing about Godel, to me. Models work in isolation; but when they reach a certain critical mass, they start to break down.
In this particular case, maybe we need another definition of cardinality that doesn't lose the intuitive nature of the relationships between the sizes of these sets). Also, of course, the fact that zero is a special number (though even it is a somewhat abstract creation) plays a big role in this, but I think that your 'trick' would work even if you had picked another number besides zero to be the bounded endpoint).
Yes, but correspondence is rather common. A few examples come to mind:
algebra & geometry (and various forms of each),
number theory, graph theory & set theory
Different 'answers' are only meaningless if the two systems do not correspond. If I'm not mistaken, that would be the exception rather than the rule.
How can there be a 1:1 correspondence? Just take a look at this sieve in action.
As you can see, right off the bat, all of the even numbers are thrown out. So, even if you go no farther than that, there are more integers than primes.
[And, when you include negative numbers, none of which can be prime by a standard definition of prime, then the count of integers is even more!]
Well, what you're missing is that sticking with what's real is just a doomed a process as believing in fantasy. All this means to me is that the universe is greatly mysterious and we cannot know everything, i.e., reality outruns knowledge.
As long as your content with that, fine. But, for me, that realization raises profound philosophical implications about the nature of humanity and our universe and causes me to seek higher knowledge. I'm not saying that I understand the mystery, I'm just embracing it's existence and engaging it.
Yea, this system and the system I want are two different entities. The system in question could keep each set of axioms and theories separate and accept the fact that it is essentially shooting a shotgun out into the universe of knowledge. But that isn't as particularly interesting to me as a system which will collect all of scientific knowledge. I think that the points where we see contradictions (i.e., collisions) between theories will provide very useful insight into the nature of our universe. And, it may even be the case that this is exactly the reason that quantum mechanics and Newtonian physics do not mesh.
I suppose it could keep the different theories separate, but that wouldn't do anything to help us get a holistic view of truth.
And, I disagree with the 'irrelevance' bit. It was my understanding (and this may just be due to my particular philosophy of mathematics slant--there are different schools of thought on it) that truth is absolute and transcendent. I think that all theorems may be translated from one theory to another (providing that the sets of axioms are sufficiently advanced). The 'truth' is beyond each particular perception of it.
Therefore, I sort of pictured a system which would collect all of mankind's knowledge in some abstract syntax which could be converted to meaningful contexts (and the conventional 'theories') at will (sort of like the Hilbert approach).
Mmmm... that's a little bit counterintuitive. It seems obvious to me that there are more integers than there are primes. I suppose cardinality is a bit less precise property than I'd imagined.
What I would be interested in knowing is if any of our known theorems contradict one another when translated from one valid axiomatic system to another? This is more directly applicable to Godel's work than undecidability, I think. Based on his work, I picture in fact an infinite number of such contradictions (and this may be why quantum mechanics and Newtonian physics do not mesh).
You should read more philosophy. Materialism isn't the only one out there [and it has serious problems as many thinkers have pointed out]. I would start with JR Lucas if I were you.
Well, in that case, the system will deterministically (although incomprehensibly to the human mind) exclude some valid corners of the vast universe of knowledge. And, it will depend on which axioms it starts with and which theorems are entered first--they will essentially steer it on a course that's doomed to ignore certain valid pieces of knowledge. That's the limitation of Godel. Of course, the person entering one of these excluded theorems will know that his theorem is correct based on the original axioms of his theory so maybe it will be of interest to see where these collide.
Nevertheless, I don't think a system which will actually do the sort of verification and discovery of new theorems which I described is all that far off (just ask Ray Kurzweil).
and check to see that existing theorems are all logically consistent with each other.
I think it's particularly that part that Godel's theorem will make impossible. Consider the case where a theorem from one view of the world (theory) is inconsistent with a second theorem from another theory which has already been entered. Either the system must reject the second or forget the first. Neither is desirable.
And, actually, it doesn't just have to be limited to philosophy and law. But, any claim on the interwebs. We could start from any sentence (or truth claim) and drill down from there (of course, depending on your slant, you may categorize this last suggestion as either philosophy or law).
That's really neat. I would like the same thing to be done with philosophy and law. We should require that our lawmakers abide by the same stringent syntaxes and correctness/consistency requirements as us computer scientists/logicians do.
The system must still be able to exhaustively 'check' each new theorem against all pre-existing theorems (and the original set of axioms). The only difference here between a machine that learns new theorems and one that checks theorems is the source of the input. In this case, the input is coming from humans, but it doesn't necessarily have to be so. Both types of machines though I would imagine will use the same algorithm to 'check' the 'inputs' for contradictions with previously known good theorems.
But, who says that the axioms of set theory are complete or correct (or the only possible set of axioms)? There may be another set of axioms where continuum hypothesis can be proven, no?
Correction:
... thus, the natural translation from all known theorems of all other theories must be made just to enter them ...
And, to answer how a mathematical truth can be known but not proven:
I suspect that it is because our 'minds' include something more than just material--something akin to a soul. We can see on a 'meta' level that mere machines cannot.
See Minds, Machines & Godel
Also, almost everything else JR Lucas says about Godel is interesting if you're into that kind of thing (and specifically 'reality outruns knowledge' from the 'Implications of Godel's Theorem' essay).
It's relevant because it is possible and even quite likely that we humans already know contradicting theorems (most likely coming from different theories, number, graph, etc.)
No one has translated all of each of the theories to the others, so we as of the time of this writing do not know if many of the *known* theories will contradict. That's where this system becomes very interesting in the context of Godel. It seems that this system will have to stick to one set of axioms (i.e., one theory) and thus, the natural translation from all known theorems of all other theories must be made just to enter it (and I suspect that this one set of axioms isn't really those of any one 'theory' of math but rather some optimal [from the computer's perspective] set).
So, whichever of two conflicting theorems is entered first will take precedence over the second and the second (which was entirely valid when working within its original context), will not be allowed to be entered.
I suppose that is the one. Unfortunately, it seems, the right's judges have a bit more of a tendency to forget how they got there than the left's do. Sandra Day O'Connor also comes to mind.
Actually, your citations of other finds do nothing to refute my point about the one particular find where scientists worldwide got all giddy over a pig's tooth. If I'm not mistaken, it was before the other finds you cite (if they are for real this time.... once bitten, twice shy you know).
And, explaining how the guy could've made the mistake, although noble, hardly erases the mistake.
Reposting as non-AC so you'll be more likely to see this: Well, for one, you mentioned it. (In number 2). And, it wasn't so much Godel's conclusion that is of interest, but his method, which was actually only an extension of the method of Hilbert (and to a lesser degree Russell). Are you saying that if machines had existed at that time that they could not run the codes? To disprove that assertion if that is what you are saying, then one could simply take the Hilbert system (or the Godel numbering system) and do the computations today starting from known axioms (and in fact, I bet that is exactly what the software in question is doing or going to do). If the numbering system is not identical, then it certainly must be equivalent.
The abstractions are useful because they provide succinct methods to discuss complex subjects, but they aren't useful because they obfuscate the meaning of what they are trying to discuss.
Or..., maybe the concepts really are abstract. How can you obfuscate the meaning of an abstract entity? It is, by definition, abstract. They have to choose something to represent such concepts and they typically choose symbols from a few well-known and ancient languages.
While the proof did contain only one such statement which caused the problem, one was enough. Godel did not even attempt to find more because the proof implies that (assuming an infinite amount of knowledge exists) there are infinitely many such statements (obviously one of these infinities has a greater degree than the other in the same way that there are both an infinite number of integers and an [smaller] infinite number of primes).
In my mind, it's kind of like dividing by zero (everything breaks down at that point). I would imagine it's a little like yeast working through dough.