- This topic has 48 replies, 6 voices, and was last updated 12 years, 7 months ago by
.
-
Posts
-
>>>Some interesting maths i saw on tv the other day
>>>about inifinity [some snipped]
>>>Generally the proof that there are different sizes of infinity
>>>is that an infinity number starting with .11 is a different
>>>subset of numbers than an infinity subset starting with .12
>>>and then the superset containing both infinity numbers
>>>can be an infinity number starting with .1 ,
>>>it would contain both .11 and .12 …..
>>>So they are all infinity but there are
>>>different sizes of infinity.I don’t know what they discussed in the TV program,
but this explanation is not correct.While the set of all real numbers with decimal expansion
beginning with .1 contains both the set of numbers beginning
with .11 as well as the set of numbers beginning with .12,
each subset being infinite in size . . . this does NOT make
the set of real numbers beginning with .1 “a larger infinity”.In fact, the set of all real numbers with decimal
expansion beginning with .1 actually has the same
infinite cardinality as the sets of numbers beginning
with either .11 or .12 EVEN THOUGH the set of all real
numbers beginning with .1 properly contains both of them.All three sets have the same infinite cardinality, the
cardinality of the continuum.There ARE infinities of different sizes, but this is not it.
Here all three are the same size.Steven
I can’t say i understood the whole concept 100% but it made sense to me because if you have an apple (.11) and it has infinate components, atoms, subatomic particles, energy, and even smaller parts we dont’ even know of… then you have another apple (0.12) that has infinate components too. Then you have 10 apples in a bag the bag would be the 0.1 ? The bag has infinate components but it must be a bigger infinity than the infinate components of one apple.
Can you shed some light on where i’m going wrong? Is there a basic version you can give me that illustrates how there can be different sized infinities?
>>>I can’t say i understood the whole concept 100%
>>>but it made sense to me because if you have an apple (.11)
>>>and it has infinate components, atoms, subatomic particles,
>>>energy, and even smaller parts we dont’ even know of…
>>>then you have another apple (0.12) that has infinate components too.
>>>Then you have 10 apples in a bag the bag would be the 0.1 ?
>>>The bag has infinate components but it must be a bigger infinity
>>>than the infinate components of one apple.Nope. Sorry. They are the same size infinity.
Even though it appears as though the bag contains 10 times as much
infinite stuff, the number of components of the bag
is in fact no different the number of components of any one apple.
This might be a little bit of a mind-f*k but it is in fact the truth,
and I can prove it mathematically.But don’t take this personally. To the newcomer, infinity is
difficult for the mind to comprehend. The first mathematical
pioneer to unlock the secrets of infinity, in fact spent
so much time trying to figure out the mystery, that he actually
went crazy and had to be institutionalized!>>>Can you shed some light on where i’m going wrong?
>>>Is there a basic version you can give me that
>>>illustrates how there can be different sized infinities?Yes, but you have to first understand when two infinities
are the same. This understanding is essential to be
able to explain when two infinities are *not* the same.
So it would require several little mini-lessons, and I’m
not sure it is something of interest to people who read this board.But nonetheless, let me start off with what I would call “lesson #1”.
If there is interest, I’ll continue. Otherwise, consider it
a step to getting your feet wet.Lesson #1:
Forget the infinite world for a moment and step back to
the everyday world where things are finite and we can count
things and come up with amounts that are not infinite.Let me pose to you a possible “Zen” riddle:
Suppose you had two actual jars of marbles sitting on a desk.
The first jar is filled with blue marbles.
The second jar is filled with red marbles.
How could you SHOW ME that the first jar has the same
number of marbles as the second jar?
The only stipulation I give, is that you are NOT allowed
to count all the marbles in either jar.Believe it or not, figuring out the answer to this Zen riddle
is the KEY in the first step to understanding infinity.I am dead serious about this, and not joking even a little.
Steven
Thanks for your reply… I’d say just take one marble out of each jar at the same time, then another out of each jar, etc, not counting though, and if when you take the last marble out of the red jar, if you are taking the last one out of the blue jar then there must be an equal amount….
>>>Thanks for your reply… I’d say just take one marble
>>>out of each jar at the same time, then another out
>>>of each jar, etc, not counting though, and if when
>>>you take the last marble out of the red jar,
>>>if you are taking the last one out of the blue jar
>>>then there must be an equal amount….Exactly.
Lesson #2:
And this exactly the manner in which you have to use
to count / making comparisons when dealing with infinity,
because if say the jars contained infinitely many marbles,
there is no way you could sit down and count all the
marbles in any one jar.Thus if both of the jars had infinitely many marbles,
and we could pair each blue marble to a red marble,
so that each blue or red marble has a corresponding mate,
then we would say that the infinities are the same size.So now the first basic example:
Suppose I now have “magic marble jars” that each contain
infinitely many marbles. The first jar is filled with
blue marbles, the second is filled with red marbles.For ease in identification, all the blue marbles have
black numbers painted on them, as do the red marbles.The blue marbles are painted with the numbers:
1, 2, 3, 4, 5, 6, 7, 8, etc. on to infinityThe red marbles are painted with the numbers:
2, 4, 6, 8, 10, 12, etc. on to infinity“Zen riddle #2”:
Can you pair each blue marble to a red marble
and vice-versa, so that no marble from either jar
is missing a mate? If so, then the two infinities
are the same size.If you understand the example in the previous post,
and understand the counting method, then you should
be able to answer this.Warning: If you do it correctly, your answer should
“surprise” you, as it perhaps goes against intuition,
which fails here (since intuition is based on our
experience with the finite world).Steven
They would have the same infinity.
Because it is infinite another partner
marble will always appear?
??????????????????????????
AdelHello Adel,
Nice to see you jump into the discussion. 😉
I’ll make this post slightly longer to give you
guys something to chew on for awhile, as I’m
leaving on a road trip today, and will be away
from the computer for a day or so.Lesson #3:
You are correct.
They have the same size infinity.Recall blue marbles: {1, 2, 3, 4, 5, etc.}
Recall red marbles: {2, 4, 6, 8, 10, etc.}We pair them as follows:
Blue 1 with Red 2
Blue 2 Red 4
Blue 3 Red 6
Blue 4 Red 8
Blue 5 Red 10
Etc.Every blue marble has a red marble mate
and vice-versa.The way to check this is to randomly pick
either a blue marble or red marble and ask
if you know the name of its mate.Ex: Who do you mate with Blue Marble #71?
Ans: Red Marble #142Ex: Who is paired with Red Marble #200?
Ans: Blue Marble #100Since you can always answer this question . . .
Everybody is paired up, A for B, so therefore
we say that these infinities have the same size!So look at this again:
Blue: {1, 2, 3, 4, 5, 6, …}
Red: {2, 4, 6, 8, 10, 12, …}We have just shown that while the number of both
blue and red marbles are infinite, there are
exactly the SAME number of them.Of course, this is where the mind starts to rebel.
It says, BUT WAIT, there are marbles in the “blue jar”
that are not in the “red jar”! There must be more
blue marbles!Nope. That’s not true.
Just because there are blue marbles that are not
in the red marble jar does NOT mean that there are
MORE of them! This idea is only true when dealing
with finite numbers of things. It is NOT true
when dealing with infinite numbers of things.As we saw in this example, you can indeed have
“extra” items in one infinite collection as compared
to another infinite collection, but in fact
both infinite collections are the same size!OK, so let’s briefly check/review and see if you
fully understand so far. Try the following 3 examples.Example #1:
Blue: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .}
Red: {5, 6, 7, 8, 9, . . .}In other words, with two identical jars of marbles,
one blue and one red, I reached into the red jar and
took the marbles labeled 1, 2, 3, 4 AND THREW THEM IN THE TRASH.Surely there must now be less red marbles than blue marbles, right?
Wrong.
In fact, the infinities are the same size.Show me how to pair them up, so everybody has a mate.
Example #2:
Blue: {1, 2, 3, 4, 5, . . .}
Red: {0, 1, 2, 3, 4, . . .}In this example, I took two identical jars of marbles,
and I bought an extra red one. I labeled the extra one
with the label “0”, and threw it into the red jar.Surely there must be more red marbles now, right?
Wrong.
In fact, the infinities are the same size.Show me how to pair them up, so everybody has a mate.
Example #3:
Blue: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, . . .}
Red: {10, 20, 30, 40, 50, . . .}Here I took two identical jars of marbles, and
from the red jar: I threw away 90% of them into the trash!
In fact, I threw away infinitely many!
All red marbles with labels 1-9, 11-19, 21-29, etc. to infinity
from the red jar are all thrown into the trash!
Here infinitely many have been removed from the red jar 😉Surely there must be more blue marbles HERE, right?
Wrong.
In fact, the infinities are the same size!Show me how to pair them up, so everybody has a mate.
Have fun,
StevenP.S. Assuming your mind can wrap itself around the
above concept–which is no easy task for most
people I think–and you can start to see that
all the above infinities are the same size . . .
Then you may start to believe that
they always ARE the same size, and that you
can’t have a larger infinity. But interestingly
enough, that is not true either! (devilish grin)
But to have any hope to understand that, you need
to fully understand the above examples. You have
to completely understand when they are the same,
so that you can understand and appreciate when they are not.Example #1
Blue 1 with Red 5
2 6
3 7
etc..
20 24
21 25Example #2
Blue 1 with Red 0
2 1
3 2
etc..
20 19
21 20Example #3
Blue 1 with Red 10
2 20
3 30
etc..
9 90
10 100Is it because the pairs have a “relationship”?
Adel>>>Is it because the pairs have a “relationship”?
>>>AdelExcellent. Gold Star!
All of your examples are correct, and you went
one step farther and noticed something on your
own that I even wasn’t going to mention (because
it is perhaps a little too mathematical).So before I go on, let me just address what you
mentioned for your curiosity.So EXTRA INFO FOR ADEL:
Mathematically, we would say that two infinities
have the same size if you can find what is
called “a bijective function” between the two.
This is just a fancy way of saying that you
can come up with a pairing relationship between
the two, so that each item in the first list
has exactly one mate in the second and vice versa.
So an aside, the relationships you discovered on
your own are as follows:Example 1: f(x) = x + 4
Example 2: f(x) = x – 1
Example 3: f(x) = 10 * x—
But to avoid scaring people off using math,
the key idea is that if each item in the first
group has exactly one mate in the second group,
and vice-versa, so that nobody is left out,
then we say that the two infinities are the same size.Next post: Introduce the idea of a “larger infinity”
Steven
Lesson #4:
Now we are going to consider the possibility
of an infinity that is larger than the one
we just looked at. I have to warn you in advance
though . . . this idea and the argument are a
real mind trip and it may take a few readings of
this before it sinks it.And don’t worry if you don’t fully understand it,
the person who first observed this was driven mad
and had to be institutionalized. I’m not kidding
in the slightest bit.If, on the other hand, you can understand this . . .
then you’ve really deserve congratulations at
being able to understand a VERY complex idea.First review:
With infinite jars of marbles, we looked at some examples.
Blue: {1, 2, 3, 4, 5, . . .}
Red1: {2, 4, 6, 8, 10, . . .}
Red2: {5, 6, 7, 8, 9, . . .}
Red3: {0, 1, 2, 3, 4, . . .}
Red4: {10, 20, 30, 40, 50, . . .}We showed in the previous posts, that all of these jars
which contain infinitely many marbles, all contain the
same amount. They are the same size infinity.This infinity is called the “COUNTABLE INFINITY”.
It is the “smallest infinity” that we can have.
Of course, at this point, you should start to hopefully
feel that all infinities are the same size. After all,
if you take some marbles away or if you add some marbles–
even if you add or subtract away an infinite amount,
then the infinity stays the same size.Remember how we showed this.
And this is crucial.We can show that two infinities are the same size
IF we can find a way to pair the items up, so that
nobody is left over. It’s like the night at the
roller rink when you put the girls on one side of the
rink and the boys on the other, and you pair everybody
up, so that hopefully no boy or girl is left by themselves.Why it is crucial that you understand this, is
that the way in which you show that something is
a larger infinity is you show that NO MATTER WHICH CLEVER
WAY you come up with (to pair everybody), there will always
be people left over with no mates.So to give an example of a larger infinity and
show you that it is indeed a larger infinity,
we have to:1. First, describe our infinite collection.
2. Try to show that it is the same size infinity
by trying to pair the items up A for B with our
jar of infinitely many blue marbles as we did in
the previous post.3. Demonstrate that no matter which way you
would try to pair them up, you will always have
“extras” that can’t be paired.This requires a little bit more brainpower, and
requires that you understand the examples in
the previous post #3 very well.I expect that the few people that may have
followed me to this point may get lost here,
but I’ll try my best.OK, so let’s do it.
Step 1: Give the example
Take a ruler that is one meter long.
Draw a line from the point that says 0 meters to
the point that says 1 meter.The line that you drew has a *ton* of points on it.
0 is a point (the left end)
1 is a point (the right end)
.5 is a point (in the middle)
.17832 is a point (sort of over to the left)You should hopefully be able to see that there are
infinitely many points on the line segment you drew!
To see this, simply pick any decimal string between
0 and 1, and that corresponds to a unique point.
Since there are infinitely many different such
decimal strings you can come up with, you have
infinitely many points.Now the question is:
Is this the countable infinity?
In other words, we already saw that the different
jars of marbles all had infinity many and they
were the same size infinity . . . so is it true
that there are same number of points on that line
segment as there are in the jar of blue marbles
that we had labeled {1, 2, 3, 4, 5, . . .} ???Step 2: Try to come up with a pairing scheme with
the blue marblesThe logical argument is as follows:
IF the number of points is the countable infinity,
then there is *some* pairing scheme that works.
I don’t know what the pairing scheme would be,
but if it is the countable infinity, then there
must be one.For the sake of argument
you tell me that you’ve found a pairing scheme:Blue 1 .63785621 . . . (some infinite string)
Blue 2 .54155483 . . .
Blue 3 .43778006 . . .
Blue 4 .28054332 . . .etc. (this list of assignments is infinitely long)
Now you tell me that you’ve been a good matchmaker
and your list has everybody paired up.Now here is the crucial idea.
Given your list, I will demonstrate
that you left somebody off of the list!Specifically:
I will create an infinite decimal string that you missed.
It will look like a number .xxxxx… that you don’t have
in your list.Here’s how I do it, and construct my infinite decimal:
Look at your list again:
Blue 1 .63785621 . . . (some infinite string)
Blue 2 .54155483 . . .
Blue 3 .43778006 . . .
Blue 4 .28054332 . . .etc. (this list of assignments is infinitely long)
What is the first digit for Blue 1 above?: 6
I will pick arbitrarily the number 2.What is the second digit for Blue 2 above?: 4
I will pick arbitrarily the number 5What is the third digit for Blue 3 above?: 7
I will pick arbitrarily the number 1What is the fourth digit for Blue 4 above?: 5
I will pick arbitrarily the number 8Doing so, I construct the decimal string .2518 . . .
on to infinity.This number is NOT IN THE LIST.
BUT WAIT, you argue, the matchmaking list I gave you
is infinitely long! How do you KNOW that this number
is not in the list?STEP 3: Argue it is not in the list
Well, take a look at my number.
It is not paired with Blue marble #1, because
the first digit is different.
It is not paired with Blue marble #2, because
the second digit is different than the second
digit of Blue marble #2’s mate
It is not paired with Blue marble #3, because
the third digit is different than the third
digit of Blue marble #3’s mate.
ETC.You can not say that my number is paired with
somebody in your list, say blue marble #78531,
because my number has a different digit in
the 78531st place.So therefore it is not in your list.
Notice that it DOESN’T MATTER WHICH LIST YOU GIVE ME,
I can always do this same thing and identify
somebody not in the list.Thus you can not pair up a blue marble with each
point on that line segment, because there will always
be points that don’t have “blue marble mates”.There are always leftovers, so this infinity is larger
than the countable infinity.Thus the number of points on the 1 meter long line segment
is a larger infinity than the countable infinity.This larger infinity is called the “continuum infinity”.
Thus any finite number is smaller than
the countable infinity which is smaller than
the continuum infinity.The countable infinity and the continuum infinity
are both infinities but the continuum infinity is
strictly larger than the countable infinity.[Pipe in a young Keanu Reeves with a surfer accent]:
“Woah dude!”If you are lucky enough to understand this post
and the preceding posts, you now have an understanding
that very few do.I’ll pause here, and let it digest a bit, and see
if anyone has any comments. 😉Cheers,
StevenYeahhhhhhh…
My first gold star in math (I barely squeaked
by high school chemistry) I remember my
teacher saying that blondes were too stupid
to understand math anyway (my hair naturally
darkened after graduation, I wonder if he
scared my blonde away)haha.Yeahhhhhhh Adel
Ok finally got a chance to get back on the PC and catch up on my missing lessons…
I follow your ideas but have some questions to check my understanding…
Countable infinity, or just countable means that the each number in the set is an integer correct? No decimals?
Continuum infinity, is related to 2 different points and the space between them?
So it’s kind of like listening to a drum LIVE, where by the sound could be analyzed mathmatically as being a continuum infinity because each beat of the drum could be divided into a start and a beginning (before the silence either side of the beat) which could then be divided into even more smaller parts infinatly, kind of like a sine wave and you could zoom in to any point along that sine wave, infinate points similar to as you could on you’re ruler example.
This would be different to listening to the same drum beat on a CD or mp3, or anything digital, where by the sampling rate kind of limits the different number of points along the sine wave of the beat on the cd. The sample rate is chosen so that it is so high that the human ear can’t detect the difference between the live and the digitally recorded version. You could however increase the sample rate so that it were 10x what was required to be undetectably different for the human ear to notice between the live and digital drum beat. You could even increase the sample rate infinately times for the digital recording but it would always be a smaller infinity than the number of points along the LIVE drum beat sine wave.
Am i correct this far?
So…. The sampling rate, being a countable infinity, is always playing catchup to the continuum infinity and can never match it no matter how fast is grows.
But then going back to my apple in a bag idea… If you zoomed in at the single apple with an imaginary impossibly strong microscope you would see n atoms in the apple and 10n atoms in the bag… and then you zoom in a little further and find n +m particles in the apple and in the bag there are 10n + 10m particles (m would be some formula but since they are all apples the formula for a single apple and the apples in the bag would be nearly identical – the apples were from the same tree 😉 So then if you zoomed in at the same rate on the one apple or the bag the one apple’s infinity (considering we are counting the visible particles) would always be playing catchup with the bag’s infinity so therefore could it not be said the infinities were different size? The infinities were growing at different rates.
Hmmm, but then if the single apple was zoomed in on at a faster rate well then the infinities could be the same size. where the continuum infinity would always be larger.
Is there any merit in what i am saying or am i making the mistake that the particles in the bag and the apple are approaching infinity and not actually infinity?
Does the secret of the continuum infinity’s greatness lay in the fact that there is no counting, no disceting or dividing? Does the “one” experience, experienced in the now similar to the continuum’s infinity where the dualistic view is a countable infinity?
Can we actually get a snapshot of infinity in time? Is it real or is it imaginary?
WHAT IS INFINITY!??!?!? haha 🙂
Also…
Are there any other kinds of infinities apart from countable and continuum?
Now when somebody says they are infinatly wise i will ask them is that countable infinity or continuum infinity. hahah we will see just how wise they are. Maybe they will say both 🙂
Oh, that’s wasn’t very nice of him to say!
I think that everyone has the ability to
be successful in math. I think the major
problem is just the way that it is taught.
There is too much ridicule indoctrinated
at the lower levels and not enough
encouragement, promotion of curiosity,
and giving students the joy of self-discovery.Instead of a dry subject with a bunch of rules
that have “no use” in everyday life, if it
were presented as observing abstract patterns
and then trying to discover a logical process
to verify or disprove the observed pattern,
I think more people would find it both
interesting and natural (not difficult).Steven
P.S. Having seen you before, I can’t
really imagine you with blonde hair. That’s
quite a transformation. But maybe with
such a cruel comment, you did indeed shift
your inner structure to create a change in
your appearance, lol. You were already
becoming an alchemist, and you didn’t know it.The only thing I can grasp is that
because the blue marbles (1,2,3,4)
are whole and the points are decimals
that there is no fixed relationship
(formula)that can be used to pair
them together? Make them countable?If you break up the marble into
degrees (break up the sphere)
would you be able to pair those
with the points on the ruler?Ahhhhhhh, I don’t get it but am having
lots of fun!!!!!!!!By the way, I like myself better as a
brunette!! Lol Steven – you’re the bestAdel
- You must be logged in to reply to this topic.