Thread: On Infinity

The concept of infinity is a conundrum. Normal logic lends itself to contradictions when dealing with infinity. An example describes this:

Take two lines AB and CD. Obviously, the number of points on them is equal to infinity.

If we were to add the number of points together, we get the seemingly ridiculous answer of infinity + infinity = infinity, even though the number of points on new line AD must be greater than those on AB and CD.

Wouldn't this mean that we must apply new kinds of rules to understand infinity better? That is to say, a new system of mathematics or axcioms that applies only to infinity.

Assuming this were so, what kind of axioms do you think could exist?

In your line question I think one should think of infinity simply as the idea of being uncountable. I can take any line segment and break it up into a smaller one and I will always have an uncountable number of point in between the two ends.

Another way to think about infinity is to think about the rate at which some thing goes to infinity or the rate at which a given infinity is increasing.
Example:

Both and approach infinity but the ratio of the two approaches zero! (Not zero factorial which is 1, humor for the mathematically inclined.)
Line example:
If line AB is always twice as long as line CD and we then make CD longer while keeping the relative size of AB then the ratio of the lengths of AB to CD will remain 2 even though the infinite number of points in each line is getting bigger.

Another idea is one can aways map infinity to some place more manageable. The mapping maps infinity to zero and zero to infinity. Why would one do this? There have been times where I what to to a Taylor series expansion about infinity but as you can image that is kind of hard to do with out a trick. So I replace then do an expansion around zero which is real easy.

Another vizulazarion is called the Riemann sphere also here is a video of a Riemann projection of Riemann.

Finally philosophy types likes to get confused and dazzled by infinity where math and physics types just use the concept to solve real problems.

Originally Posted by c186282

Finally philosophy types likes to get confused and dazzled by infinity where math and physics types just use the concept to solve real problems.

Heh!

And about transcendental numbers and so on. We're allowed, I believe, because we're trying to think about the meanings of things, not just about how they're used (though of course Wittgenstein would say that the meaning lies in the use).

In any case, surely mathematicians themselves took many hundreds of years to come to grips with the concept of infinity and it wasn't formalised until Cantor?

Sorry sunshinewarrior, it is true it took along time and a lot of philosophical thought for people to wrap their heads around the idea of infinity. However, I do see it now as a completed task.

Originally Posted by c186282

In your line question I think one should think of infinity simply as the idea of being uncountable.

Be careful of your terminology here! In mathematics, the term “uncountable” has a very precise technical definition. Every uncountable set is infinite, but not every infinite set is uncountable!

infinity is less a defined 'unlimited' number and more the number we haven't reached yet if you take the biggest current number (probably 9.9X109999989 or something) infinity is simply one more than that


However by giving a set length you nulify the existance of infinity, AB and CD cannot have an inifinite number of points since AB is a predefined length with a finite number of points it and CD arn't infinity and therefore cannot add up to infinity


you can also view it from a 'how long is a piece of string' idea, you cannot make infinity by multiplying or addition. 1 less than inifinity add 1 will not make infinity since it obtains a numerical value. AB and CD have no lengths however they are twice as long are half their lengths so 1/2AB + 1/2AB makes AB. meaning AB and CD = 1/2AB+1/2A+1/2CD+1/2CD which is not infinity

AB and CD cannot have an inifinite number of points since AB is a predefined length with a finite number of points it and CD arn't infinity and therefore cannot add up to infinity

I must argue with you on this. Remember that the area of points is zero; at least, that is the accepted mathematical take on it. It stands to reason that the area multiplied by the number of points gives us the length of the line.

However, this becomes



Where x is the number of points, and l is the length.

If we let x be a finite number, this means that 0 = l; a contradiction. We may infer from this that the number of points must be equal to infinity.

The problem arises when we consider the number of points in the addition of two lines.

Further, even if we assume that that a line consists of a finite number of points, it is easy to show that the area of a single point is impossible to determine. This can be done as so:

Take a line AB. Assume that a finite number of points of some defined area exists.

Now, remember that a line is, by definition, consisting only of one dimension : length.

The sum of the area of the points = the length.

These points cannot exceed the dimension of the line, as they would then no longer form a line. Therefore, a point must have a definite length. However, this implies the existence of further points on the point's line, which in turn can be expressed as lines, and so on. This results in an infinite process, that never ceases. From this, we may infer that actually determining the length of a point is impossible, as in order to do so, we must first define the lengths of the other points.

Therefore, this implies that determining the area of a point is impossible, so we may, for our purposes, replace this "not defined' with infinity.

Originally Posted by JaneBennet

Be careful of your terminology here! In mathematics, the term “uncountable” has a very precise technical definition. Every uncountable set is infinite, but not every infinite set is uncountable!

Thank you for the clarification.

Makes me think of the analogy between a physicist's tool set and a mathematician's tools set.

The mathematician's tools are all neatly hung up in order and only used properly. They also have the best tools.

The physicist's tools are scatter all over the floor and are rarely used correctly. Screw driver to pry, back side of a wrench to hammer. But hey we get the job done.

Originally Posted by Booms

AB and CD cannot have an inifinite number of points since AB is a predefined length with a finite number of points it and CD arn't infinity and therefore cannot add up to infinity

That is wrong. You appear to be confusing “having infinite number of points” with “having infinite length”. A line segment does not have infinite length, but it does have an infnite number of points.

Originally Posted by Booms

infinity is less a defined 'unlimited' number and more the number we haven't reached yet if you take the biggest current number (probably 9.9X109999989 or something) infinity is simply one more than that

That is also wrong. The technical definition of an infinite set in mathematics is a set for which there is a bijection (one-to-one correspondence) between and a proper set subset of , For example, the set of positive integers is infinite. There is a one-to-one correspondence between and the proper subset via etc. A set is then said to be finite iff it is not infinite.

The above definition is a bit abstract. Fortunately there is a more intuitive definition. It turns out that a set is finite if and only if there is a bijection betwen and the set for some natural number – i.e. the elements of can be listed as . If you like, you can take this as a definition of a finite set: A set is finite iff for some . Under this definition, then, a set is infinite iff it is not finite.

The two definitions above are equivalent.

Originally Posted by c186282

Finally philosophy types likes to get confused and dazzled by infinity where math and physics types just use the concept to solve real problems.

I hope this post is relevant to the thread.
You appear to be saying here, and also in a later post, that we have an excellent grasp of infinity and that the job of understanding the concept is a "completed task".
In another thread, in a different sub forum, a poster (william pinn?) stated it would not be possible to reach the present given a universe that has always existed in some form. In this universe time would stretch back to infinity.
Answering his post mitchell mckain dismissed this point of view as a "typical Zeno paradox type argument" and went on to say "the only limitation here is your imagination and possibly your skills in mathematics".
I find it hard to accept, but I am willing to be convinced, that a neat, technical, mathematical proof, which would shatter this "Zeno paradox type argument", exists.
This is what mitchell mckain seems to be suggesting.

I hope this post is relevant to the thread.
You appear to be saying here, and also in a later post, that we have an excellent grasp of infinity and that the job of understanding the concept is a "completed task".
In another thread, in a different sub forum, a poster (william pinn?) stated it would not be possible to reach the present given a universe that has always existed in some form. In this universe time would stretch back to infinity.
Answering his post mitchell mckain dismissed this point of view as a "typical Zeno paradox type argument" and went on to say "the only limitation here is your imagination and possibly your skills in mathematics".
I find it hard to accept, but I am willing to be convinced, that a neat, technical, mathematical proof, which would shatter this "Zeno paradox type argument", exists.
This is what mitchell mckain seems to be suggesting.

No offense, Halliday, but I don't think it is relevant to this thread.

Infinity can do nothing but cause jeolousy in people, because our lives and the earth is finite. We will never see infinity so why worry about it?

Who is worrying about infinity?

Originally Posted by JaneBennet

Who is worrying about infinity?

Anyone who is thinking about infinity with anxiety, or concern. Maybe one person is experiencing the former, but certainly many are experiencing the latter.

The fact is infinity can only be used to propose abstractions, and not to solve real world problems like some suggest.

needless to say, this is still pretty cool though found it on wiki : The theorem concerns a thought experiment which cannot be fully carried out in practice, since it is predicted to require prohibitive amounts of time and resources. Nonetheless, it has inspired efforts in finite random text generation.

One computer program run by Dan Oliver of Scottsdale, Arizona, according to an article in The New Yorker, came up with a result on August 4, 2004: After the group had worked for 42,162,500,000 billion billion monkey-years, one of the "monkeys" typed, “VALENTINE. Cease toIdor:eFLP0FRjWK78aXzVOwm)-‘;8.t . . ." The first 19 letters of this sequence can be found in "The Two Gentlemen of Verona". Other teams have reproduced 18 characters from "Timon of Athens", 17 from "Troilus and Cressida", and 16 from "Richard II".[26]

A website entitled The Monkey Shakespeare Simulator, launched on July 1, 2003, contained a Java applet that simulates a large population of monkeys typing randomly, with the stated intention of seeing how long it takes the virtual monkeys to produce a complete Shakespearean play from beginning to end. For example, it produced this partial line from Henry IV, Part 2, reporting that it took "2,737,850 million billion billion billion monkey-years" to reach 24 matching characters:

RUMOUR. Open your ears; 9r"5j5&?OWTY Z0d…
Due to processing power limitations, the program uses a probabilistic model (by using a random number generator or RNG) instead of actually generating random text and comparing it to Shakespeare. When the simulator "detects a match" (that is, the RNG generates a certain value or a value within a certain range), the simulator simulates the match by generating matched text.[27]

Questions about the statistics describing how often an ideal monkey should type certain strings can motivate practical tests for random number generators as well; these range from the simple to the "quite sophisticated". Computer science professors George Marsaglia and Arif Zaman report that they used to call such tests "overlapping m-tuple tests" in lecture, since they concern overlapping m-tuples of successive elements in a random sequence. But they found that calling them "monkey tests" helped to motivate the idea with students. They published a report on the class of tests and their results for various RNGs in 1993.[28]

Originally Posted by AeDeAeMn0886

The fact is infinity can only be used to propose abstractions, and not to solve real world problems like some suggest.

I agree with your first statement, but not the second. Some real-world problems can indeed be solved by a proper application of the idea of infinity. For instance, Zeno’s paradoxes of motion (which are very much real-world “problems”) can be resolved by considering the idea of a convergent infinite series.

For instance, Zeno’s paradoxes of motion (which are very much real-world “problems”) can be resolved by considering the idea of a convergent infinite series.

I still do not quite understand convergent infinite series. Exactly how can an infinite series converge?

What's not to understand?

Consider the process of starting at 1 and progressively halving it; 1, 1/2, 1/4, 1/8,.... Do this for as long as you like.

Now add the first 4 terms, and note the answer. Add the first 10 terms and note the answer. Add the first 100 terms and note the answer.

See a pattern? You may postulate that, given world enough and time, that the sum will turn out to be 2. But since this side of eternity we are unable to iterate a process an infinite number of times, one says that as the number of terms obtained by our "halving algorithm" approaches infinity, the sum converges on 2.

This is (roughly) what what is meant by convergence of an infinite series.

Anyway, here's another version of the famous "Hilbert's Hotel"

Two identical twins arrive together at the pearly gates, having recently died. St. Peter apologizes, and says there has been a clerical error - while both are entitled to enter Heaven, only one may. St. P. selects one, and tells the other he must go to Hell; but not to worry, as he may exchange places with his twin each Feb 29.

"But that's not fair" says the excluded twin, "I only get to spend 1 day in 4 years in Heaven, while my brother stays there mostly all the time".

"Not to worry" says St. P, "in the long run you will both spend exactly the same amount of time in Heaven"

Originally Posted by Liongold

The concept of infinity is a conundrum. Normal logic lends itself to contradictions when dealing with infinity. An example describes this:

Take two lines AB and CD. Obviously, the number of points on them is equal to infinity.

If we were to add the number of points together, we get the seemingly ridiculous answer of infinity + infinity = infinity, even though the number of points on new line AD must be greater than those on AB and CD.

Wouldn't this mean that we must apply new kinds of rules to understand infinity better? That is to say, a new system of mathematics or axcioms that applies only to infinity.

Assuming this were so, what kind of axioms do you think could exist?

If you really want to study "infinity" you need to learn about cardinal and ordinal numbers. For a reasonable introduction see Naive Set Theory by Paul Halmos.

That should clear up the multitude of misconeptions that are evident in your later posts on the subject.

The way this was explained to me was a mathematician and an engineer are placed in a room across from a stunning example of the opposite gender. Each is told to half the remaining distance to the person every 5 seconds.

The mathematician explains that he will never reach the person.

Then the engineer is asked if they can reach the person.

"Sure", says the engineer with a smile, "I can get there for all practical purposes."

Originally Posted by hokie

The way this was explained to me was a mathematician and an engineer are placed in a room across from a stunning example of the opposite gender. Each is told to half the remaining distance to the person every 5 seconds.

The mathematician explains that he will never reach the person.

Then the engineer is asked if they can reach the person.

"Sure", says the engineer with a smile, "I can get there for all practical purposes."

That is all that matters. ; )

Originally Posted by JaneBennet

Some real-world problems can indeed be solved by a proper application of the idea of infinity. For instance, Zeno’s paradoxes of motion (which are very much real-world “problems”) can be resolved by considering the idea of a convergent infinite series.

At the risk of wandering off thread again I would like to ask whether the present could ever be reached if time, and, of course, the universe in some form, had always existed? In other words can such a question be answered "by a proper application of the idea of infinity?"

Originally Posted by Halliday

Originally Posted by JaneBennet

Some real-world problems can indeed be solved by a proper application of the idea of infinity. For instance, Zeno’s paradoxes of motion (which are very much real-world “problems”) can be resolved by considering the idea of a convergent infinite series.

At the risk of wandering off thread again I would like to ask whether the present could ever be reached if time, and, of course, the universe in some form, had always existed? In other words can such a question be answered "by a proper application of the idea of infinity?"

The answer is yes. It has nothing to do with the notion of "infinity" in any of the forms studied by mathematicians.

[

The answer is yes. It has nothing to do with the notion of "infinity" in any of the forms studied by mathematicians.

Then how can mathematicians answer the question?

Originally Posted by Liongold

For instance, Zeno’s paradoxes of motion (which are very much real-world “problems”) can be resolved by considering the idea of a convergent infinite series.

I still do not quite understand convergent infinite series. Exactly how can an infinite series converge?

It can converge point wise, it can oonverge uniformly, it can converge in measure. There are lots of ways that it can converge. You can look at the sequence of partial sums or you can look at more advanced summability methods.

To understand the notion of convergence properly you need the machinery of elementary point set topology. It is helpful to have encountered the notion of a net in order to extend the notion beyond just sequences.

Infinity is quite an interesting subject. I remember one problem that i stumbled upon. It was about a square that with every mutation got an equilateral triangle in the middle of each of the sides of the square. The side of the triangle was one third of the side of the square. Quite hard to explain.
In the next mutation the same thing was made with all the sides, old and newly formed with the new length of sides.
The results considering the area and the circumference you got after an infinite amount of mutations was quite surprising.

I had another niceish infinity thing but it is and remains forgotten...

Originally Posted by Halliday

[

The answer is yes. It has nothing to do with the notion of "infinity" in any of the forms studied by mathematicians.

Then how can mathematicians answer the question?

Without breaking a sweat.

Originally Posted by DrRocket

Originally Posted by Halliday

[

The answer is yes. It has nothing to do with the notion of "infinity" in any of the forms studied by mathematicians.

Then how can mathematicians answer the question?

Without breaking a sweat.

That's cleared that up then!
This argument about the present in a universe where time stretches back to infinity has come up before and has been dismissed without clear (mathematical or otherwise) explanation. Mitchell McKain is one poster who appeared to dismiss this idea as nonsense, in a recent post, in the Religion sub forum.
I was simply asking, within a thread about infinity, if the idea had any merit.

Originally Posted by Halliday

Originally Posted by DrRocket

Originally Posted by Halliday

[

The answer is yes. It has nothing to do with the notion of "infinity" in any of the forms studied by mathematicians.

Then how can mathematicians answer the question?

Without breaking a sweat.

That's cleared that up then!
This argument about the present in a universe where time stretches back to infinity has come up before and has been dismissed without clear (mathematical or otherwise) explanation. Mitchell McKain is one poster who appeared to dismiss this idea as nonsense, in a recent post, in the Religion sub forum.
I was simply asking, within a thread about infinity, if the idea had any merit.

No, you were not. You posed a rather poorly formulated question that had no merit. It is not even close to being an idea.

Your question has no more depth than merely asking how there can be a 0 on the number line despite the fact that there is neither a smallest nor a largest real number.

No, you were not. You posed a rather poorly fomulated question that had no merit. It is not even close to being an idea.

Your question has no more depth than merely asking how there can be a 0 on the number line despite the fact that there is neither a smallest nor a largest real number.

I noticed there was quite a bit of information, on the Internet, about this question with no depth!
Here is one example from William L. Craig, of the University of Munich, writing in a philosophy journal. He begins by saying that this old argument has been expounded by thinkers such as Kant and that it may be roughly schematized thus:
"If the series of past events is infinite, then an infinite number of events must elapse before the present moment could arrive.
But it is impossible for an infinite number of events to elapse.
Therefore, if the series of past events is infinite, the present moment could not arrive.
But the present moment has arrived.
Therefore,the series of past events cannot be infinite."
I have said, in other posts, that I am simply a layperson with an interest in science.
Maybe I should not have raised this topic in the maths sub forum but the thread was about infinity and threads often wander around a topic and do not always stay true to the original question.
I first asked the question, about the merits of the above argument, ages ago, in an astronomy sub forum thread about the BBT versus the SST. Much later I asked the same question in a Religion sub forum thread. I did not feel the question was answered, in either thread, so I asked the same question here.
I apologise for having put forward a poorly fomulated (sorry formulated) question that had no merit or depth. I did not realise that such a question had to be expressed as if I was presenting a Ph.D thesis on the work of Cantor.
A kid can ask whether space and time are infinite-that does not mean the question is stupid or meaningless!

Originally Posted by Halliday

No, you were not. You posed a rather poorly fomulated question that had no merit. It is not even close to being an idea.

Your question has no more depth than merely asking how there can be a 0 on the number line despite the fact that there is neither a smallest nor a largest real number.

I noticed there was quite a bit of information, on the Internet, about this question with no depth!
Here is one example from William L. Craig, of the University of Munich, writing in a philosophy journal. He begins by saying that this old argument has been expounded by thinkers such as Kant and that it may be roughly schematized thus:
"If the series of past events is infinite, then an infinite number of events must elapse before the present moment could arrive.
But it is impossible for an infinite number of events to elapse.
Therefore, if the series of past events is infinite, the present moment could not arrive.
But the present moment has arrived.
Therefore,the series of past events cannot be infinite."
I have said, in other posts, that I am simply a layperson with an interest in science.
Maybe I should not have raised this topic in the maths sub forum but the thread was about infinity and threads often wander around a topic and do not always stay true to the original question.
I first asked the question, about the merits of the above argument, ages ago, in an astronomy sub forum thread about the BBT versus the SST. Much later I asked the same question in a Religion sub forum thread. I did not feel the question was answered, in either thread, so I asked the same question here.
I apologise for having put forward a poorly fomulated (sorry formulated) question that had no merit or depth. I did not realise that such a question had to be expressed as if I was presenting a Ph.d thesis on the work of Cantor.
A kid can ask whether space and time are infinite-that does not mean the question is stupid or meaningless!

Kids ask a lot of stupid and meaningless questions. But philosophers ask even more, and on a regular basis. Kant's musings have no mathematical merit.

The question has nothing to do with the nature of the infinite. It is quite simply no more deep than asking how zero can exist on the number line, which is unbounded in both directions.

Kids ask a lot of stupid and meaningless questions. But philosophers ask even more, and on a regular basis. Kant's musings have no mathematical merit.

The question has nothing to do with the nature of the infinite. It is quite simply no more deep than asking how zero can exist on the number line, which is unbounded in both directions.

I stand corrected and can only offer a grovelling apology for having dared to compare the views of an intellectual giant like yourself with those of Kant and other assorted second-raters.

Originally Posted by Halliday

"If the series of past events is infinite, then an infinite number of events must elapse before the present moment could arrive.
But it is impossible for an infinite number of events to elapse.
Therefore, if the series of past events is infinite, the present moment could not arrive.
But the present moment has arrived.
Therefore,the series of past events cannot be infinite."

Consider this.

Suppose a series of events happened an hour ago. The first event E1 happened an hour ago and lasted half an hour. The second event E2 happened hour an hour ago and lasted a quarter of an hour. The third event E3 happened a quarter of an hour and lasted one-eighth of an hour. So we have a series of events {E1,E2.E3.…} which is clearly infinite.

If the abovequoted balderdash were true, the infinite series of events would never lead to the present moment. This is clearly absurd. It is therefore possible for some infinite series of events – such as this one – to lead to the present.

I suggest that Halliday stop polluting this Mathematics subforum with poorly thought-out ideas and pretending they have any intellectual merit whatsoever. I have PMed the moderator to have this thread locked. It is clearly not going anywhere as long as stupid ideas keep being spouted.

Originally Posted by JaneBennet

Originally Posted by Halliday

"If the series of past events is infinite, then an infinite number of events must elapse before the present moment could arrive.
But it is impossible for an infinite number of events to elapse.
Therefore, if the series of past events is infinite, the present moment could not arrive.
But the present moment has arrived.
Therefore,the series of past events cannot be infinite."

Consider this.

Suppose a series of events happened an hour ago. The first event E1 happened an hour ago and lasted half an hour. The second event E2 happened hour an hour ago and lasted a quarter of an hour. The third event E3 happened a quarter of an hour and lasted one-eighth of an hour. So we have a series of events {E1,E2.E3.…} which is clearly infinite.

If the abovequoted balderdash were true, the infinite series of events would never lead to the present moment. This is clearly absurd. It is therefore possible for some infinite series of events – such as this one – to lead to the present.

I suggest that Halliday stop polluting this Mathematics subforum with poorly thought-out ideas and pretending they have any intellectual merit whatsoever. I have PMed the moderator to have this thread locked. It is clearly not going anywhere as long as stupid ideas keep being spouted.

I do see what he is saying about infinity. In other words, an infinite amount of events cannot have passed. Because we will never know what they were.

That would be the iron clad definition of infinity. In theory the fact that we do not know back to infinity, could leave open the possibility that our world is only say 157 years old. All this was just created by God for a test.

We could say some unknowable amount of events have occurred back yonder. But that is all we could say about the infinity of past.

Yet we are here, and an infinite number of events has occurred, if only in the last 157 years, and these events will continue to occur in the future forever.

Sincerely,


William McCormick

Mod note I remind you, yet again, this is a Mathematics subforum. If you want to talk about philosophy, we have a dedicated subforum. I will, within reason, allow discussions on the philosophy of math, but this discussion falls outside reason.

Thread locked