How would you even define a notion of "follows from the axioms" that didn't presuppose proof in a finite number of steps?
GP is talking about a concept known as logical implication. A set of axioms logically implies a statement if that statement is true in every model satisfying those axioms. A model is a particular interpretation of a set of axioms. Say, if your only axiom is (For all x, For all y (x+y = y+x)) then an abelian group of order two is a model, and so is Z, and so is any mathematical structure where operation + is, in fact, commutative.
"Provable" (or "deducible"), on the other hand, means that there is a finite sequence of formulas which constitutes a formal deduction of a statement from the axioms.
As logicians, we work in the meta-theory, which is widely accepted to be ZFC. A model is just a certain kind of set. Giving a rigorous definition here would be excessive, but we can at least mention that models are "big" objects which include the universe of discourse. So a model for natural numbers omega would, naturally, include the infinite set omega. "A logically implies B" is a statement about how A and B relate in various models. Compare that to "B is deducible from A", a statement about existence of a finite formal deduction, which, in meta-theory, is a different kind of set. A deduction is literally a record of finitely many formulas over a finite alphabet.
Yes and yes, except that your first sentence is a bit unclear. "That you can make a list of all true statements of mathematics" is just saying that the set is countable.
The set of all true statements of arithmetic does exit, but we cannot write a program that would reliably recognize true statements (i.e., always halt with a definite answer). We cannot even have a program that would recognize some set of axioms which we could use to derive all facts of arithmetic.
Godel of course proved that you can never have a complete list of all true statements in mathematics. So if they're just limiting it to "central" theorems, how do they decide what's central and what's not?
Godel proved that you can never have a complete and recursive list of axioms for arithmetic. Regardless, "central" in TFA means well-known, which, IMHO, is fair enough. Letting a computer know proofs of all major results seems to be a necessary step on the way to building a silicon mathematician.
You know what is more mainstream than Futurama? DVDs. Both of you should be ashamed. On your way out, please hand in passwords for your very low Slashdot IDs.
Godel's Incompleteness I is not really about decidability. Here's a nice list of theories, with some examples for (1)+(3)+(4). Some of the most notorious ones are dense linear orders without endpoints and real closed fields. You will find quite a few examples that are both complete and decidable, including the pure identity theory, which has no non-logical axioms.
I like the following elementary presentation I picked up from prof. M. Stanley at SJSU:
A first order axiomatic theory can have any three of the following, but not all four:
(1) Be recursively axiomatizable, i.e. a computer program can decide the set of axioms.
(2) Be expressive enough to capture all the basic facts about arithmetic.
(3) Be consistent, i.e. not allow to derive a contradiction.
(4) Be complete, i.e. for any statement F, prove either F or (not F).
I fail to see how he is not excited about "extinguishing life", and a whole lot of it.
So you are saying that a person (like Feynman) was not excited when he or his colleagues finally found a way to develop the weapon and state that, with high probability, the damage will be such and such and the human casualties will be such and such? According to you, they must have burst into tears of sadness when they put the dots above i's in their report. According to you, Feynman thought at that moment something like: "Dude, I wish that the bomb would instead explode into fireworks at the safe distance from the intended target". You want hogwash? Here it is: "We are exited about completing our work on the bomb, but we are very sad about its uncanny ability to kill people".
What gets me is that people like SoVeryTired entertain this junkie dream about a perfectly impartial scientist who is able to completely detach himself from the object of his research. My position is still the same: people who worked on the bomb were excited about wholesale killing. I never claimed that they did not have other (even conflicting) emotions or aspirations.
He said, "All right, there's a meeting at three o'clock. I'll see you
there."
I said, "It's all right that you told me the secret because I'm not
going to tell anybody, but I'm not going to do it."
So I went back to work on my thesis -- for about three minutes. Then I
began to pace the floor and think about this thing. The Germans had Hitler
and the possibility of developing an atomic bomb was obvious, and the
possibility that they would develop it before we did was very much of a
fright. So I decided to go to the meeting at three o'clock.
OK. What I can see is a man who decided to work on a bomb because he wanted to be able to bomb Hitler before Hitler could bomb US. Not just for the love of physics involved, but mainly for strategic reasons. I sure didn't read the whole thing, but whatever I found supports my position.
the same way that the people working on the Manhattan project were not monsters who wanted to extinguish life.
Huh? Assuming that a person knows that he is developing a bomb which can totally level a metropolitan area, I fail to see how he is not excited about "extinguishing life", and a whole lot of it. And I am not even against killing, be it for fun or for profit. I am just pointing out that you must be dreaming if you think that those researchers were driven solely by the love of Physics.
Please do not tell posters what to do. If you think that a post is offtopic, mod it as such instead of explaining your point of view, which itself is even more offtopic than that of GP!
Slashdot Mirror
Say rather that all mathematics stems from philosophy and you'll be more correct.
More correct could still be a far cry from correct.
Why do star photos have crosses over bigger stars?
Oh my god, it's a word??
I agree, I was just saying that the original sentence was somewhat ambiguous, but may be it's just me.
How would you even define a notion of "follows from the axioms" that didn't presuppose proof in a finite number of steps?
GP is talking about a concept known as logical implication. A set of axioms logically implies a statement if that statement is true in every model satisfying those axioms. A model is a particular interpretation of a set of axioms. Say, if your only axiom is (For all x, For all y (x+y = y+x)) then an abelian group of order two is a model, and so is Z, and so is any mathematical structure where operation + is, in fact, commutative.
"Provable" (or "deducible"), on the other hand, means that there is a finite sequence of formulas which constitutes a formal deduction of a statement from the axioms.
As logicians, we work in the meta-theory, which is widely accepted to be ZFC. A model is just a certain kind of set. Giving a rigorous definition here would be excessive, but we can at least mention that models are "big" objects which include the universe of discourse. So a model for natural numbers omega would, naturally, include the infinite set omega. "A logically implies B" is a statement about how A and B relate in various models. Compare that to "B is deducible from A", a statement about existence of a finite formal deduction, which, in meta-theory, is a different kind of set. A deduction is literally a record of finitely many formulas over a finite alphabet.
Yes and yes, except that your first sentence is a bit unclear. "That you can make a list of all true statements of mathematics" is just saying that the set is countable.
The set of all true statements of arithmetic does exit, but we cannot write a program that would reliably recognize true statements (i.e., always halt with a definite answer). We cannot even have a program that would recognize some set of axioms which we could use to derive all facts of arithmetic.
Godel of course proved that you can never have a complete list of all true statements in mathematics. So if they're just limiting it to "central" theorems, how do they decide what's central and what's not?
Godel proved that you can never have a complete and recursive list of axioms for arithmetic. Regardless, "central" in TFA means well-known, which, IMHO, is fair enough. Letting a computer know proofs of all major results seems to be a necessary step on the way to building a silicon mathematician.
Actually, you got it all wrong. As anyone concerned with personal security, Bruce Schneier has a decoy.
You know what is more mainstream than Futurama? DVDs. Both of you should be ashamed. On your way out, please hand in passwords for your very low Slashdot IDs.
Ha! CmdrTaco sends me $200 every month for reading Slashdot, and I still think that I pay him too much.
sleep `expr $RANDOM % 60 + 1`m ;
echo "Am I, `uname`, realy thinking, or am I just programmed to seem so?" ;
read
Make your UNIX workstation self-aware today!
echo "Am I, `uname`, realy thinking, or am I just programmed to seem so?" ; read
examples that are both complete and decidable, including the pure identity theory
Er, I take that back. But as long as we throw in axioms which determine the size of the model, it will be complete.
Godel's Incompleteness I is not really about decidability. Here's a nice list of theories, with some examples for (1)+(3)+(4). Some of the most notorious ones are dense linear orders without endpoints and real closed fields. You will find quite a few examples that are both complete and decidable, including the pure identity theory, which has no non-logical axioms.
System administrator, eh? You can start by showing your scars.
Idle is really growing on me lately. I never read it, but it is funny how many hateful comments that inane column generates.
I like the following elementary presentation I picked up from prof. M. Stanley at SJSU:
A first order axiomatic theory can have any three of the following, but not all four:
(1) Be recursively axiomatizable, i.e. a computer program can decide the set of axioms.
(2) Be expressive enough to capture all the basic facts about arithmetic.
(3) Be consistent, i.e. not allow to derive a contradiction.
(4) Be complete, i.e. for any statement F, prove either F or (not F).
I wrote:
I fail to see how he is not excited about "extinguishing life", and a whole lot of it.
So you are saying that a person (like Feynman) was not excited when he or his colleagues finally found a way to develop the weapon and state that, with high probability, the damage will be such and such and the human casualties will be such and such? According to you, they must have burst into tears of sadness when they put the dots above i's in their report. According to you, Feynman thought at that moment something like: "Dude, I wish that the bomb would instead explode into fireworks at the safe distance from the intended target". You want hogwash? Here it is: "We are exited about completing our work on the bomb, but we are very sad about its uncanny ability to kill people".
What gets me is that people like SoVeryTired entertain this junkie dream about a perfectly impartial scientist who is able to completely detach himself from the object of his research. My position is still the same: people who worked on the bomb were excited about wholesale killing. I never claimed that they did not have other (even conflicting) emotions or aspirations.
OK. What I can see is a man who decided to work on a bomb because he wanted to be able to bomb Hitler before Hitler could bomb US. Not just for the love of physics involved, but mainly for strategic reasons. I sure didn't read the whole thing, but whatever I found supports my position.
the same way that the people working on the Manhattan project were not monsters who wanted to extinguish life.
Huh? Assuming that a person knows that he is developing a bomb which can totally level a metropolitan area, I fail to see how he is not excited about "extinguishing life", and a whole lot of it. And I am not even against killing, be it for fun or for profit. I am just pointing out that you must be dreaming if you think that those researchers were driven solely by the love of Physics.
Mod it up, boys.
What do you guys use jelly and syrup for?
Please do not reply to your own post.
Oops...
Please do not tell posters what to do. If you think that a post is offtopic, mod it as such instead of explaining your point of view, which itself is even more offtopic than that of GP!