The limits of Logic

[Dendrek]

If you read carefully, though, the guy you were replying to never said there is an infinity larger than the other. All he was trying to say is that one can grow faster than other, or that we can count one faster than the other, which is technically true.

I did read carefully, and what he said, and what you're saying, is the layman's way of describing the whole number version of infinity. You're referring to the idea that infinity is generally defined to be the idea of "a number that goes on forever" or "counting out to forever." That same idea doesn't really apply to larger versions of infinity.

Instead of thinking of it as counting, think of it as the size of the set of all numbers. In other words, we're thinking of the "size" of infinity, not how fast you reach it. When you think of it in terms of size, the set of integers is twice the size of the set of whole numbers. There may be an infinite number of values in each set, but there are twice as many infinite number of values in integers.

And when you consider integers and not just whole numbers, the idea of counting, or of how fast you are reaching infinity doesn't really make sense. How do you count in both directions? Likewise, the "size" of the infinity on the positive side of the integers is no different than the infinity obtained when counting whole numbers. So if all you're doing is relying on counting, on the speed at which one set approaches infinity compared to the other, there's no way to compare the integer version of infinity to the whole number version of infinity. And if that's all we're doing (talking about counting) then there's no way at all to count the infinitesimally small.

[Wikiy]

I did read carefully, and what he said, and what you're saying, is the layman's way of describing the whole number version of infinity. You're referring to the idea that infinity is generally defined to be the idea of "a number that goes on forever" or "counting out to forever." That same idea doesn't really apply to larger versions of infinity.

Instead of thinking of it as counting, think of it as the size of the set of all numbers. In other words, we're thinking of the "size" of infinity, not how fast you reach it. When you think of it in terms of size, the set of integers is twice the size of the set of whole numbers. There may be an infinite number of values in each set, but there are twice as many infinite number of values in integers.

And when you consider integers and not just whole numbers, the idea of counting, or of how fast you are reaching infinity doesn't really make sense. How do you count in both directions? Likewise, the "size" of the infinity on the positive side of the integers is no different than the infinity obtained when counting whole numbers. So if all you're doing is relying on counting, on the speed at which one set approaches infinity compared to the other, there's no way to compare the integer version of infinity to the whole number version of infinity. And if that's all we're doing (talking about counting) then there's no way at all to count the infinitesimally small.

What's your point though? Sorry, but English isn't my first language so i know very little of the terms when it comes to mathematics. And at the moment, i can't really see a point in all you're saying, it only seems as if you're trying to say that your way of looking at infinity is the right one. But i still don't see how that relates to anything that i said.

[GarGar]

http://video.google.com/videoplay?do...25684649921614
http://video.google.com/videoplay?do...91361786740235

Links to Google Video for being able to watch them fullscreen. The button on the videos don't work for me, and I couldn't find them by searching, so there they are.

I just got done with a computer logic course at school, so it'll be interesting to see some of the higher level logic stuff they do.

[Stir]

*Someone introduces Gödel to a UTM, a machine that is supposed to be a Universal Truth Machine, capable of correctly answering any question at all.
*Gödel asks for the program and the circuit design of the UTM. The program may be complicated, but it can only be finitely long. Call the program P(UTM) for Program of the Universal Truth Machine.
*Smiling a little, Gödel writes out the following sentence: "The machine constructed on the basis of the program P(UTM) will never say that this sentence is true." Call this sentence G for Gödel. Note that G is equivalent to: "UTM will never say G is true."
*Now Gödel laughs his high laugh and asks UTM whether G is true or not.
*If UTM says G is true, then "UTM will never say G is true" is false. If "UTM will never say G is true" is false, then G is false (since G = "UTM will never say G is true"). So if UTM says G is true, then G is in fact false, and UTM has made a false statement. So UTM will never say that G is true, since UTM makes only true statements.
*We have established that UTM will never say G is true. So "UTM will never say G is true" is in fact a true statement. So G is true (since G = "UTM will never say G is true").
*"I know a truth that UTM can never utter," Gödel says. "I know that G is true. UTM is not truly universal."
*Think about it - it grows on you ...

The problem with this scenario, as far as I see it at least, is that G is indeed true, but G's meaning is not. It's like a carpenter who builds a table, but insists on it being a chair, and therefore not recognisable as either.

The problem is that to a non-sentience (which I must assume the UTM is), G is true. G exists as a statement, therefore G is true regardless of the command contained within G.

[Dendrek]

What's your point though? Sorry, but English isn't my first language so i know very little of the terms when it comes to mathematics. And at the moment, i can't really see a point in all you're saying, it only seems as if you're trying to say that your way of looking at infinity is the right one. But i still don't see how that relates to anything that i said.

The point I'm making is that "growing" and "counting" don't really work with certain types of infinity. In fact, that way of thinking only really make sense when referring to an infinity that grows in 1 direction only.

A couple of instances where that way of thinking does make sense:
1. When counting whole numbers or positive integers.
2. When talking about going up a slope or hill that goes up forever.
3. When looking at a graph and examining what the graph does with larger and larger input values (x-values).

Since these are all pretty common situations, it makes sense to think of the "growth" of infinity. These are also the first instances where we are taught the concept of infinity, and we often simply accept that this concept of infinity applies to all situations. In reality, it doesn't.

There are situations where thinking about how fast infinity "grows" does not make sense:
1. When thinking about the full value of an irrational number. For example, when you think of pi (3.14159265358989323...), it will occur to you that that number keeps going on forever; there is no last digit to pi. But is it accurate to say that that number is "growing"? That doesn't even make much sense. How is that number growing? It is what it is. We can say that if we used a calculator or computer to calculate the "full" value of pi, then we can observe how it adds more and more accurate values to the decimal, and so we can witness the decimal get larger as more calculations are done. But the number itself is not growing.
2. When thinking about something that "grows" in multiple directions. For example, in the case of integers, where there are positive and negative numbers, and therefore there is growth in two directions, when we talk about "how fast it's growing to infinity," what exactly are we talking about? Are we talking about how fast the numbers grow to positive infinity? Are we talking about the combined effort of going to both positive and negative infinity? Are we talking about the growth of the set of numbers?
3. When thinking about something that can't be counted. It's easy to count whole numbers, but how do you count decimal numbers (positive real numbers)? If you started at 0 and attempted to count all decimals between 0 and 1, you'd never make it to 1. In fact, you'd never make it to 0.1, or even to 0.00000000000000000001. You'd be stuck trying to count numbers that are infinitely small. In fact, once you tried counting past 0, you'd be stuck. "The infinitely small" is an extremely complicated concept that's far more confusing than the infinitely large, and it's also impossible to count. Where do we begin to describe the growth of it?

---------- Post added 2012-08-13 at 03:46 PM ----------

The problem with this scenario, as far as I see it at least, is that G is indeed true, but G's meaning is not. It's like a carpenter who builds a table, but insists on it being a chair, and therefore not recognisable as either.

The problem is that to a non-sentience (which I must assume the UTM is), G is true. G exists as a statement, therefore G is true regardless of the command contained within G.

Yes, G is true. But the issue is that the computer, which is theoretically designed to be able to always output true statements, is unable to output that one. The machine is faced with a paradox that, although true, it is incapable of declaring true.

Basically, the implication is that paradoxes may exist which are true, but which we would never be able to prove true using logic. We can assume it's true (and humans are definitely able to make assumptions that defy logic -- case in point, religion), but we could never actually prove it. Logic is based on the ability to prove; and if there are situations that are true but which logic cannot prove, then it demonstrates that logic is incapable of finding all truths.

As the last line in that statement you quoted suggested, consider the argument. Let it sink it. "It grows on you."

[Stir]

Yes, G is true. But the issue is that the computer, which is theoretically designed to be able to always output true statements, is unable to output that one. The machine is faced with a paradox that, although true, it is incapable of declaring true.

Basically, the implication is that paradoxes may exist which are true, but which we would never be able to prove true using logic. We can assume it's true (and humans are definitely able to make assumptions that defy logic -- case in point, religion), but we could never actually prove it. Logic is based on the ability to prove; and if there are situations that are true but which logic cannot prove, then it demonstrates that logic is incapable of finding all truths.

As the last line in that statement you quoted suggested, consider the argument. Let it sink it. "It grows on you."

The machine, however, is a machine. The paradox relies on the machine recognizing a human concept as objective truth. For the machine, the actual meaning of the sentence means nothing. Therefore, there is no paradox as far as the machine is concerned. The paradox is obvious to me, because I have a rich conceptual reality in which I dwell. A machine, however, does not. There is a 'physical' level of truth, which the machine's logic can deduct not by reasoning, but by calculation. A conceptual level of truth as is expected of the machine, however, should not be recognized by the machine at all. It's as if you're expecting your PC to be able to think. Your computer can calculate using predetermined input, but it cannot register.

Basically, what I'm saying is that the machine would automatically assume the truth of G without finding any paradox. It is simply incapable of seeing a paradox in this situation.

By the way, I'm not suggesting that the UTM is a feasible idea. I'm not convinced that it's possible to come up with a series of formulae that compose the answer to everything. I'm merely saying that I feel this thought experiment to be logically fallacious.

[Vasti]

If you read carefully, though, the guy you were replying to never said there is an infinity larger than the other. All he was trying to say is that one can grow faster than other, or that we can count one faster than the other, which is technically true.

I think he knows this, but the guy he was replying to said:

In my opinion, there is no infinity that is "larger" than any other

Even though in math it is accepted that there are multiple 'sizes' of infinity, which he then explained. In terms of countable infinities and uncountable infinities and if I remember correctly there are more distinctions to be made between infinities. Like he mentioned with the integer set is twice as large as the whole number set.
So it is generally accepted that some infinities are larger than others, although I understand where the guy is coming from with the fact that infinity is not something that can be exceeded by anything else (not even other infinities)

On-topic:
All in all this was one of the best reads/watch threads I visited in a while. This stuff really interests me. Now I just hope it doesn't interests me as much as the guys in the video as they usually ended up dead in a tragic way usually preceded by a stay in the asylum.

---------- Post added 2012-08-13 at 05:38 PM ----------

The machine, however, is a machine. The paradox relies on the machine recognizing a human concept as objective truth. For the machine, the actual meaning of the sentence means nothing. Therefore, there is no paradox as far as the machine is concerned. The paradox is obvious to me, because I have a rich conceptual reality in which I dwell. A machine, however, does not. There is a 'physical' level of truth, which the machine's logic can deduct not by reasoning, but by calculation. A conceptual level of truth as is expected of the machine, however, should not be recognized by the machine at all. It's as if you're expecting your PC to be able to think. Your computer can calculate using predetermined input, but it cannot register.

Basically, what I'm saying is that the machine would automatically assume the truth of G without finding any paradox. It is simply incapable of seeing a paradox in this situation.

By the way, I'm not suggesting that the UTM is a feasible idea. I'm not convinced that it's possible to come up with a series of formulae that compose the answer to everything. I'm merely saying that I feel this thought experiment to be logically fallacious.

But how would the UTM be able to know that sentence is true without realizing the implications of it being true? It must know that it can only be true if it can't say it was true. And it will say it is false if it will someday say it is true. So there is no way the machine can either determine it to be true or false. Thus there are truths that the machine will never be able to comprehend/asses/determine/calculate/deduce.

[Wikiy]

Yes, G is true. But the issue is that the computer, which is theoretically designed to be able to always output true statements, is unable to output that one. The machine is faced with a paradox that, although true, it is incapable of declaring true.

Basically, the implication is that paradoxes may exist which are true, but which we would never be able to prove true using logic. We can assume it's true (and humans are definitely able to make assumptions that defy logic -- case in point, religion), but we could never actually prove it. Logic is based on the ability to prove; and if there are situations that are true but which logic cannot prove, then it demonstrates that logic is incapable of finding all truths.

As the last line in that statement you quoted suggested, consider the argument. Let it sink it. "It grows on you."

Actually, there are things which utterly defy our logic, but are completely true and have been proven. The best case that comes to mind is quantum mechanics (the most proven and predictable theory (as in, able to predict results of experiments) of all scientific theories). If it weren't true, not only would we not have a lot of technology that we do, the whole universe would be a lot different. It says particles have no real size, can teleport, and can interact through information (real, physical information, which sounds bizarre) faster than the speed of light. This all defies everything that might sound logical to us, but it's completely true and has been proven to be true. My point is, our logic (or what should be considered logic) actually adapts to what we know. What we usually consider as "logic" has nothing to do with truth, only with what we have been thought and what makes sense, but science tells us our senses, including the sense of intellectual thinking, are extremely misleading in the physical universe.

[Vasti]

Actually, there are things which utterly defy our logic, but are completely true and have been proven. The best case that comes to mind is quantum mechanics (the most proven and predictable theory (as in, able to predict results of experiments) of all scientific theories). If it weren't true, not only would we not have a lot of technology that we do, the whole universe would be a lot different. It says particles have no real size, can teleport, and can interact through information (real, physical information, which sounds bizarre) faster than the speed of light. This all defies everything that might sound logical to us, but it's completely true and has been proven to be true. My point is, our logic (or what should be considered logic) actually adapts to what we know. What we usually consider as "logic" has nothing to do with truth, only with what we have been thought and what makes sense, but science tells us our senses, including the sense of intellectual thinking, are extremely misleading in the physical universe.

You should never forget that scientific theories (especially those of physics) are just that, theories.
Theories can be adapted to encompass the results of new experiments, phenomenon etc. Just because it seems VERY VERY likely, it does not mean it is as the theory describes it. Evolution theory, Theory of gravity,.....while likely to be true it is so far impossible to know 100% whether they describe the process as they truly are.

But this topic and logic do not deal with theories. This is about certainties and uncertainties. Logic proves things that are certain. There has yet to be found anything that defied logic (not even quantum physics). What the video's now tell us, is that logic will never be able to encompass everything that is certain.

[Dendrek]

Actually, there are things which utterly defy our logic, but are completely true and have been proven. The best case that comes to mind is quantum mechanics (the most proven and predictable theory (as in, able to predict results of experiments) of all scientific theories). If it weren't true, not only would we not have a lot of technology that we do, the whole universe would be a lot different. It says particles have no real size, can teleport, and can interact through information (real, physical information, which sounds bizarre) faster than the speed of light. This all defies everything that might sound logical to us, but it's completely true and has been proven to be true. My point is, our logic (or what should be considered logic) actually adapts to what we know. What we usually consider as "logic" has nothing to do with truth, only with what we have been thought and what makes sense, but science tells us our senses, including the sense of intellectual thinking, are extremely misleading in the physical universe.

Keep in mind that logic is a set of axioms that are used to prove or disprove arguments. Quantum Mechanics expanded the set of axioms we had, thus allowing our logic to expand enough to explain things that it could not before. Given the set of axioms we had before, a lot of questions had arisen about matter that were unanswerable. With our now expanded axioms, we have explanations for those previously unexplainable phenomenon. But with that new set of axioms came even more questions, and theoretically there will be questions raised that current axioms can't explain. Those questions will then require new axioms, which if discovered will bring about more questions, requiring new axioms, producing more questions, etc. ad nasium.

Basically, your argument, taken a little further, directly demonstrates the point I was making.

[Gheld]

http://en.wikipedia.org/wiki/Planck_units

Planck uses applied science on Cantor.
It was very effective.
Cantor becomes irrelevant.

EDIT: That is to say that the infinite and infinitesimal don't apply to any observable natural phenomenon, therefore this whole discussion is absolutely without any practical application.

[Deleted]

I think he knows this, but the guy he was replying to said:
Even though in math it is accepted that there are multiple 'sizes' of infinity, which he then explained. In terms of countable infinities and uncountable infinities and if I remember correctly there are more distinctions to be made between infinities. Like he mentioned with the integer set is twice as large as the whole number set.

The integer set is twice as large as the set of natural numbers, true. It is also exactly as large as the set of natural numbers, as you can order it differently. And if I wanted to be silly, I could say five times as large. That is the essence of the first step in Cantor's theory. there is a "size", cardinality as he called it, of infinite sets which you can count. He called this "size" Aleph0, countable infinite.

You can take twice of it, 2*Aleph0=Aleph0. Aleph0*Aleph0=Aleph0, still the same size. Only 2^Aleph0 is a new larger inifnite number, called Aleph1 or its power set. Also, there is the Continuum number, the size of the set of real numbers. It is the amount of points on a line, of a square, volume, whatever. This is also uncountable infinite, larger than Aleph0. So the question is, is 2^Aleph0=Continuum? We can't say if it is true, false, true but unprovable, or indecidable.

[Dendrek]

http://en.wikipedia.org/wiki/Planck_units

Planck uses applied science on Cantor.
It was very effective.
Cantor becomes irrelevant.

EDIT: That is to say that the infinite and infinitesimal don't apply to any observable natural phenomenon, therefore this whole discussion is absolutely without any practical application.

Not entirely true. There are a lot of theories and properties of math that can't be directly applied to the real world, but those theories are also often the foundations of mathematical concepts that do apply to the real world. An example would be the circle. There is no such thing as a circle in the real world. It's a mathematical concept. But that idealized mathematical concept obviously has real-world applications.

I'm not an expert in that field, so I can't say if there are real-world applications of Cantor's theorems. But I think to dismiss a math concept because it's purely theoretical is short sighted.

[GarGar]

http://en.wikipedia.org/wiki/Planck_units

Planck uses applied science on Cantor.
It was very effective.
Cantor becomes irrelevant.

EDIT: That is to say that the infinite and infinitesimal don't apply to any observable natural phenomenon, therefore this whole discussion is absolutely without any practical application.

This is a mostly mathematical discussion, not a physical one, and their models are very different. Thinking about Galileo's circle problem, you can mathematically model a circle as an infinitely-sided polygon. If you look at it on a computer, you can keep zooming in and zooming in to infinity and it would still look round. But if you tried to build such a model in the real world, obviously you wouldn't be able to make a side smaller than the plank length.

But just because it's physically impossible doesn't mean it's irrelevant or uninteresting. And it doesn't mean it won't lead to a practical application somewhere down the line.

[Deleted]

http://en.wikipedia.org/wiki/Planck_units

Planck uses applied science on Cantor.
It was very effective.
Cantor becomes irrelevant.

EDIT: That is to say that the infinite and infinitesimal don't apply to any observable natural phenomenon, therefore this whole discussion is absolutely without any practical application.

/shoo. This is mathematics, not physics. To physicists calculating with infinities, although not on the level of Cantor's set theory, is still a valuable tool. And infinitesimal calculus is just too simple and useful to be ignored. Whether space can be seen a discrete lattice is still specualtion, Planck length is far to small to be experimented with energies we have access too.

[Wikiy]

You should never forget that scientific theories (especially those of physics) are just that, theories.
Theories can be adapted to encompass the results of new experiments, phenomenon etc. Just because it seems VERY VERY likely, it does not mean it is as the theory describes it. Evolution theory, Theory of gravity,.....while likely to be true it is so far impossible to know 100% whether they describe the process as they truly are.

Scientific theories aren't just theories. They're theories supported by facts. It may be impossible to know 100% that they are true, but if there's a 99.9999% chance they are, then it's practical to assume and operate as if they are as long as the evidence is that much in their favor. And it's also unpractical to point out there's a 0.00001% chance they aren't true.

[Stir]

But how would the UTM be able to know that sentence is true without realizing the implications of it being true? It must know that it can only be true if it can't say it was true. And it will say it is false if it will someday say it is true. So there is no way the machine can either determine it to be true or false. Thus there are truths that the machine will never be able to comprehend/asses/determine/calculate/deduce.

My point is: The machine DOESN'T KNOW that the meaning of the sentence is either true or false. It merely knows that the sentence exists, and, therefore logically, the sentence itself is true regardless of the seemingly conflicting statement it holds. The statement [G] is: '[G] is false.' Basically. Upon being asked whether or not [G] is true, the machine must respond with: 'Yes.' Yes, [G] is true in that [G] is false. This is a Truth; it has been inserted as a Truth, and the machine does not have the ability to discriminate Truths based on comprehension. The machine is merely required to answer. Not to think.

Besides, if you really give it some thorough thought, you might discover that '(The statement of) [G] is true in that (the statement of) [G] is false.' is not a logical paradox. One describes the nature, whereas the other describes the intent.