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Re: Godels Theorem [Re: xFrockx]
    #15000938 -

yeah id say its a problem of our logical system (only a problem because we want completeness). The logical system just happens to be part of this reality because we are in it...


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Re: Godels Theorem [Re: Noteworthy]
    #15001554 -

If math is the language of the universe, then if you think of this theorem on a universal scale, isn't it just the mathematical way of describing the paradoxes of existence?

Things like, the universe can't contain itself, but it also does contain itself.

And nothingness can't exist because the thought of nothingness makes it something?

Edited by Midnight_Toker (08/30/11 02:49 PM)

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Re: Godels Theorem [Re: Noteworthy]
    #15001845 -

Noteworthy said:
so is this all related to a pipe dream of wanting logic to prove logic? or what


Gödel's result did shatter a pipe dream from the early 20th century, but it's much deeper than that because it tells us something important about the differences between what can be expressed within a formal system and what can be proven within the same formal system, when that formal system is powerful enough to allow self-referential expressions.

And the pipe dream in question (called "Hilbert's program") was actually founded in some serious concerns about how much we can trust mathematical proofs. It took two centuries before mathematicians could prove that the calculus invented by Newton and Leibniz was really dependable, and not just a magic trick that only worked sometimes. This is the calculus that seeded the european enlightenment and the industrial revolution, so mathematicians were a bit frustrated that it was all founded on the concept of dividing something infinitely small by something else that was also infinitely small. After all, infinitely small means zero, right? Anyway, they finally did come up with a rigorous theory of calculus that doesn't involve dividing zero by zero, and after that the mathematicians thought "hey, why not create a theory for all of mathematics, a theory that is provably correct, provably consistent, and in which any possible true mathematical statement can be proven true from a small set of fundamental axioms?"

This last idea turned out to be a pipe dream.

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Re: Godels Theorem [Re: Rhizoid]
    #15004820 -

Hmm nice response. Hilberts program eh?

I guess on this topic im not sympathetic to people with a problem with the infenetissimal and infinity. This might be because I was born into a world where these concepts have already been dealth with for hundreds of years.

It seems clear to me that infinetissimal means less than any finite value, which would have to be 'zero' (if anything) because if it was anything else it would be a finite value.
Also im unsympathetic to mathematicians who try to talk about the 'number of members of an infinite set' because I think its 'stupid' to say that infinity is an equal number to another infinity. Neither are finite, so neither can be 'equal' in the sense we normally speak about. so there arent the same number of members of the 'set of cardinal numbers' verses' the set of even numbers'. There are an infinite number of each, infinite numbers being fairly incomparable.
However there are twice as many cardinals as even numbers... I dont yet understand why people say there are the same number.

They will then say theres an infinite amounts more values on a continuous line than there are cardinal numbers. And yes id say there is, for the same reason there are twice as many cardinals than even numbers


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Re: Godels Theorem [Re: Noteworthy]
    #15005115 -

Infinity is a tricky concept. It isn't an amount or a quantity, because those are measured by numbers and infinity is not a number. My take on infinity is that it is a shorthand word for open-ended collections, i.e. collections about which you can say "the further you look, the more you will see."

Open-ended collections can be compared, by finding if there is any possible one-to-one mapping between them. That's what you do when you figure out the cardinality of a set: you compare it with the set of natural numbers or one of its power sets. But I think it's unfortunate that the same word "cardinality" is used both for finite sets and infinite sets. They are only vaguely similar concepts.

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Re: Godels Theorem [Re: Noteworthy]
    #15006548 -

Noteworthy said:
It seems clear to me that infinetissimal means less than any finite value, which would have to be 'zero' (if anything) because if it was anything else it would be a finite value.


It seems clear to me that this is incorrect.  It is not zero, if it was then what good would calculus be?  An infinitesimal is less than any finite value but more than zero.

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Re: Godels Theorem [Re: DieCommie]
    #15006619 -

Besides, if "infinitesimal" means zero, why not just say zero? It has fewer syllables, is easier to write, and simplifies communication by using a single term instead of muddying the discussion by adding a redundant word for something that already has a word.


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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Re: Godels Theorem [Re: Diploid]
    #15006659 -

This is kind of like how some people call the universe "God". :rolleyes:


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Well I try my best to be just like I am, but everybody wants you to be just like them. --  Bob Dylan
fireworks_god said:
It's one thing to simply enjoy a style of life that one enjoys, but it's another thing altogether to refer to another person's choice as "wrong" or to rationalize their behavior as being pathological or resulting from some sort of inadequacy or failing so as to create a sense of superiority or separation as yet another projection of a personal fear or control issue.

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Re: Godels Theorem [Re: Poid] * 1
    #15006718 -

I thought they called god the universe?  :doh:

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Re: Godels Theorem [Re: DieCommie]
    #15009427 -

Calculus has always seemed fundamentally flawed to me because it manipulates an integer that cannot ever be calculated completely. I once had someone try to explain it to me, a professor of philosophy who specialized partly in mathematics. He was talking about how to manipulate infinity, and I asked, "How can we use it as a number if it can never be finished?" And he said "Let's just assume that it can be." and I replied "But it is infinity, how can be be finished? By definition, doesn't it mean not finite, not with an end?"

He kept telling me just to assume that it worked. :shrug:

I think it works maybe due to the formalism that can exist and produce calculations, not as a result of actual infinity, but possibly the assumption of just using it as if it is finite? I don't know.

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Re: Godels Theorem [Re: xFrockx]
    #15011037 -

Calculus has always seemed fundamentally flawed

That's because you don't understand it.

because it manipulates an integer that cannot ever be calculated completely

Neither can pi, but circles still exist, and their dimensions can be calculated to arbitrary precision.

By definition, doesn't it mean not finite, not with an end?"

It does, but the symbol pi represents the value EXACTLY not approximately. You don't need infinite pages to write it down because the symbol means the same thing even though it takes up finite space to write. Just like you can think of the ratio of circumference to diameter of a circle without an infinitely large brain to contain the infinite digits of pi.

Consider the following paradox involving "infinitely short distances":

In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.

Your assertion that calculus is "flawed" because of infinities is equivalent to the above absurdity. Clearly, Achilles will win the race, but how can that be given that the above paradox describes a true situation?

The reason Achilles will win even though the paradox is true is explained easily by calculus. It is the naive interpretation of the paradox that is flawed. This is not apparent to the student until he has mastered the nuances of elementary calculus. At that point the student :facepalm: and realizes how naive he was and how much more he still has to learn.

And he said "Let's just assume that it can be."

He told you that because in order to cover the elementary stuff, you have to accept some of his pronouncements at first. Later when you've mastered the elementary stuff, the several semesters of hard study covered by that one simple statement get expounded on in great detail. If not for that simplifying assumption at the beginning of study, it would take years worth of explanation to move onto the next step, just like it took Isaac Newton years to invent calculus the hard way without the benefit of a teacher's initial simplifications. By asking you to accept his word (for now) that it's true, he can answer your original question now instead making you wait for the explanation until you finish undergrad studies.

I replied "But it is infinity, how can be be finished?

To understand the answer requires some knowledge of the field of math called Limits. Without that background, it's not surprising you misunderstand. Here's the ADD version of the class:

Consider the fraction 1/x. If x is 2, the result is one-half. If x is 10, the result is one-tenths. As x grows, the fraction represented becomes smaller. If x is 100, the result is one one-hundredth. See a pattern?

You can't write 1/infinity because infinity is a concept, not a number. But you CAN state that as x APPROACHES infinity, the result of the fraction APPROACHES the limit zero. This is a useful result in the same way pi is a useful result. It not only works, but is absolutely required in order to send a satellite into orbit, or design an airplane, or build a hard drive, or invent MRI scanners, or design cars, and even to architect skyscrapers. If calculus were "fundamentally flawed" as you say, it would not be possible to build any of those things, and we'd still be riding around on horse and buggy.

So which is the more likely to be flawed: calculus or your assertion that calculus is flawed?


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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Re: Godels Theorem [Re: Diploid]
    #15011132 -

"You can't write 1/infinity because infinity is a concept, not a number."

I guess this is what confuses me about it. I like airplanes though don't worry.

"But you CAN state that as x APPROACHES infinity"

I'm not trying to question calc here, just to understand, but how can we ever "approach" infinity? Especially if it is a concept? I do see the formalistic relationship of as one counts up the other counts down, but if infinity is a concept of not finite, how can it ever be approached? Or even be said to be be being approached, if it is infinite?

:confused:

"So which is the more likely to be flawed: calculus or your assertion that calculus is flawed?"

I think it is likely that calculus is not flawed, but maybe that the idea that it manipulates infinity rather than the formalism of numbers might be a misunderstanding or a poor choice of words? I really don't know. Maybe we need to use that concept of infinity to get at the formalism. I have no idea, sorry if I am seeming like I know more than your math teacher, I don't. I just never understood the idea that we can ever get closer to the end of an infinity if there is no end. BUT I do see the formalism of when we do do this. Does that make sense?

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Re: Godels Theorem [Re: xFrockx]
    #15011269 -

but how can we ever "approach" infinity? Especially if it is a concept?

By using abstract thinking. It's one of the defining characteristics that separates us from the rest of the animals.

I can write the symbol pi and mean the ratio of circumference to diameter EXACTLY, to the very last digit, even though it's not possible to write the value down to the last digit. By abstracting that impossible-to-write value to a single symbol, it becomes possible to use it to, for example, calculate the correct size tire for your car.

And while you can't write out infinity either, it's a concept, not a number, you CAN write a symbol that represents it: ∞, and you can use that abstraction to solve myriad real and academic problems.

Think of the words "approaches infinity" as the words "gets bigger and bigger". It's the same concept.


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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Re: Godels Theorem [Re: Diploid]
    #15011287 -

"Think of the words "approaches infinity" as the words "gets bigger and bigger". It's the same concept."

Ah. This is what I meant by that I see the formalism that exists. The "getting bigger and bigger."

As I understand infinity though (and I am definitely not saying I am correct), it's not something that gets bigger and bigger, it is something that is not finite, it has no quantity. Now, I am not saying that the "getting bigger and bigger" is not legit, I just don't think it is using "infinity" in the way that I would use infinity if I were talking about an infinite space.

Does that make sense?

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Re: Godels Theorem [Re: Diploid]
    #15011293 -

By using abstract thinking. It's one of the defining characteristics that separates us from the rest of the animals.

Except for Cromwell the Crow.:wink:


--------------------
"Don't believe everything you think". -Anom.

" All that lives was born to die"-Anom.

With much wisdom comes much sorrow,
The more knowledge, the more grief.
Ecclesiastes circa 350 BC

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Re: Godels Theorem [Re: DieCommie]
    #15011424 -

DieCommie said:
Noteworthy said:
It seems clear to me that infinetissimal means less than any finite value, which would have to be 'zero' (if anything) because if it was anything else it would be a finite value.


It seems clear to me that this is incorrect.  It is not zero, if it was then what good would calculus be?  An infinitesimal is less than any finite value but more than zero.


For some reason I remember my calculus classes using 0 for dx to calculate dy. Of course, it was TAUGHT as if it was a limit that was only approached. But the maths saw not this distinction.

Furthermore,
Quote:
Consider the fraction 1/x. If x is 2, the result is one-half. If x is 10, the result is one-tenths. As x grows, the fraction represented becomes smaller. If x is 100, the result is one one-hundredth. See a pattern?

You can't write 1/infinity because infinity is a concept, not a number


Actually 1/infinity does = 0 and 1/0 = infinity, its a basic notion that I would like you to disprove.

furthermore, 0, for all practical mathematical purposes, = infinetissimal.

infinetissimal* any number = infinetissimal

infinetissimal+ any number = that number

infinetissimal has all of the same properties as 0 except for certain abstract, possibly ontological or metaphysical properties, which go beyond the mathematics.

Of course you might disagree, but as an answer I would like something other than 'IT JUST ISNT' because people have always said that about uncomfortable ideas, but here I would like a mathematical explanation.

note that 0*infinity could equal anything.

Also note that we talk of limits when describing functions on finite numbers because there is no point on a graph that represents an infinite number. You could say that the axis represents an infinetissimal number though. In fact, any point represents an infinetissimal range of values, which incidentally contains only one finite value in it.

IT is also worth noting that through this system, each value in fact represents the finite value as well as an infinetissimal range around it, hence why 0=infinetissimal=-infinetissimal, and 1=1+infinetissimal=1-infinetissimal.

Might be a bit hard to get your head around, but I would appreciate a refutation


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Re: Godels Theorem [Re: Noteworthy]
    #15011453 -

"note that 0*infinity could equal anything."

0 times anything is 0. If you have no sets of two apples, then you have no apples. 0x2=0

1 times anything is whatever the anything is. If you have 7 sets of one apple, you have 7 apples. 7x1=7

No?

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Re: Godels Theorem [Re: xFrockx]
    #15011507 -

0 times any finite value = 0. 0*infinity could equal anything. It doesnt make sense if you treat 0 as 'nothing' but this system is one which is very close to the notion that nothing is in fact something, just without relevant features.

So, 0 equals 'no finite value' rather than 'nothing'.

Thus, no finite value * any other finite value will result in no finite value.

But infinetissimal is not a finite value either, so it is not excluded from being the result of 0 times something.

The notion that infinetissimal * infinity = anything is perhaps the more unique notion of this system, I have not explored its significance but to show how it is derived:

1/infinity = infinetissimal

2/infinity= 2*1/infinity = 2*infinetissimal = infinetissimal


1/infinetissimal = infinity
2/infinetissimal = 2*1/infinetissimal = 2*infinity = infinity

1/2 = 1/infity * infinty/2 = infinetissimal * infinity

any fraction will also = infinetissimal * infinity this way

hence any finite value = infinetissimal * infinity = 0 * infinity = 0/1 * 1/0


By finite number here I mean REAL number. there is a sense in which pi is not 'finite' but thats not the sense I am using here.


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Re: Godels Theorem [Re: Noteworthy]
    #15011528 -

"It doesnt make sense if you treat 0 as 'nothing' but this system is one which is very close to the notion that nothing is in fact something, just without relevant features. "

So nothing is not nothing then?

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Re: Godels Theorem [Re: Noteworthy]
    #15011561 -

1/0 =

Suppose I have 6 apples and 3 people to divide them over. How many apples does each person get? 6 / 3 = 2, easy.

Now suppose I have 6 apples and zero people to divide them over. How many apples does each... uh... each... wait, there aren't any entities to finish the question with. There aren't any "things" or people or anything else. Do I rephrase as "How many apples does each non-existing entity get"?

I don't think a non-existing entity can get any apples or anything else for that matter. It's a meaningless question because division by zero is meaningless.

The best you can say is that as the number of people approaches zero, the number of apples each gets approaches infinity. But RIGHT AT ZERO, the question is rendered meaningless.

I would like something other than 'IT JUST ISNT'

The answer you want DOES exist, and it is deep and satisfying, but without the technical background, it won't make sense. That's why in the beginning, the teacher says "just trust me for now, it just isn't".


--------------------
Republican Values:

1) You can't get married to your spouse who is the same sex as you.
2) You can't have an abortion no matter how much you don't want a child.
3) You can't have a certain plant in your possession or you'll get locked up with a rapist and a murderer.

4) We need a smaller, less-intrusive government.

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