Math Is Fun Forum

Hehe, you simply don't accept {1, 2, 3, 4, 5, 6,...}# has one more element than {1,2,3,4,5...} because you suppose since they both have infinite amount, they are the same. Don't you?

It seems that religion is like the complex number, it has a good chance not to be true, but it can function well in a sense. And to function well, it requires people to believe it true, which is a dilema.

But one more?  How can you conclude which set has more?

You are not reading, Ricky.

The conclusion comes from the fact that its subset just has the same. And I did not define it, for I don't like to pre-define or pre-assume things the way mathematicians do.

Again, you use naive matching from left one by one again.

" smallest* such set"
Good, quite interesting, then you agree there are several such sets? Then what distinguishes the smallest one to any larger one? The largest integer? The largest integer+1 ? Come on, I wanna know what is the smallest set missing so to locate the end. Don't wanna talk about it? Ban us from talking about it?  That's all what mathematicians can do?

Still, thank you for your advice.

terms such as "one more" are nonsensical

So you mean infinite amount is just the amount of how many integers? That's what you call mapping?

Well, that's rediculous too. Because  the infinite amount of integers is also preassumed. You count 1, then you count to 2, you count to 3, and then you assume you can count to any number. But in order to do this, you have to get a pool of infinite integers so to ensure you can have an integer as large as you want out of this pool. Otherwise how can you?

The infinite integers assumption bear the same problem. I can easily duplicate this argument by assignning one set that contains all the natural numbers plus 1, and add in the number 1 in it so it now has infinite+1 numbers due to matching.

{1, 2 , 3, 4, 5, ...}
matching by +1=
{2, 3,  4, 5, 6, ...}
above plus the set {1}=
{1, 2, 3, 4, 5, 6,...}#
Now that the subset of {1,2,3,4,5...}#, {2,3,4,5...},  is a 1-1 match to the original {1,2,3,4,5...}, the new set clearly has got 1 more element in it.
What?
You wanna argue that since you have found the pattern between the two by 1-1,2-2, 3-3, and so on, so that you are convinced that they have got all the same elements?

Is this naive inference what matching technique all about? I just see you have got 1 head, 1 chunk,  two arms, and 2 legs and therefore I conclude you and me are the same? The matching technique just naively infers equivalence by incomplete information or one sided story. It can only handle a finite amount of matching from left and never suceeds in handling the whole, even allowing that this finite amount can grow.

And my technique, on the contrary, is not only much more grounded than the matching technique but it also can spot the largest integer, discovering the largest integer problem.

What I understand is that in dealing with infinity, mathematicians always apply two meanings. They apply the real infinity subtly to demonstrate they have eliminated the residual error in case such as 1=.999..., and they prevent themselves from being spotted wrong at the last digit by forcing challengers only to think in an counting way, applying the growing infinity, which is a fake one.

I think unless you had some knowledge on the distinguishment between a real infinity and a potential one (growing one) raised by Aristotle thousands of years ago, you would never realize the mistake and the jumps from one to the other the mathematicians have made in set theory.

1     D     
2     1
4     2
8     4
16   8

Taking difference of 2^k is fairly easy, but it needs specifying steps, which seems hard. In the end, they abandon 2^k and resort to fit a line with the function e^(bx), which makes less sense. However, you can fit a line with 2^(ax) and the relationship between a and b is

b/a=ln2

Or b/a=log(2/e) , whatever log,
or b log(e)=a log(2)

whichever you find easy to memory.

Well, traditional defense against inquiry on a "last 9" is always like this:

You cannot find a last 9, can you? Since infinity is forever, you cannot get to that (count to that).

That's a false proof though, as I refuted before, because the concept of infinity is already created unable to be reached by count.

The disproof is unsolid, but what about the proof?

My simple technique, in solid logic, but only to strengthen the intuition of 10*.999 marking the last 9:

How many 9's does .999... have?

Well, a tough question?
But first, no matter how many, the amount is neither finite nor changing.
(The precise version is that
it is not finite and stable.
it is not finite but growing.
it can be infinite but growing, but too weird to be accepted
it can then only be infinite and stable)

Now let's see the amount of 9's here
.999...
0.1*.999...
=.0999...
Notice the amount of 9's still remains the same here
But how about this?
.0999...+.9
=.999...#
Look! Look! Look!
We have just added in one 9 to the so-called stable infinite amount of 9's, now no matter how many 9's the original .999 has, the new .999..., marked as .999...# has Just 1 More 9 than that!

Not accepting it? Refuting it? Anyone claiming just being divided by 10 get  .999... lose one 9? If no one. Continue:

Let's just compare
the original
0.999...
and the new
0.999...#
with just one 9 more

Discompose the two from the left,
we discover that they follow the same structure
9*10[sup]-1[/sup]+9*10[sup]-2[/sup]+...
And if sorted by meaning,
the added 9 for the latter is plays just the same role as the first 9 in the former, so in this way both first 9's are the same. They are so identical that we don't say the latter one has one more 9 on the first digit or the former one has one less 9 on the first digit. (Or you get 18*10[sup]-1[/sup] or 0*10[sup]-1[/sup]

Now where is the one more 9 on .999...#?
Well, taking another angle helps:
Where is the one 9 missing in .999 compared to .999...#?

It cannot be missing on a digit before a 9 both have:
...  9...   (.999...)
...99...   (.999...#)

Note if it was lacked, it could only mean 0 at that decimal and would contradict no 0 after dot in .999. (making something like .0999 or .999...09...)

And typing a space doesn't help because the 9 before the typed space stands for the same thing as the former 9 in .999...# thus .999 doesn't lack this 9. So we just wipe out the space for simplicity.

Let me just go back to the 9 in the words "before a 9 both have".  Can we just simplify this to "before a digit which is filled by 9 in both case"? And the missing 9 in .999... is just a zero at some digit where a 9 is present in .999...#, allowing both have the same amount of digits.

Let's find the 0 and 9 combination, by the clue that there is no 9 and 9 combination after this.

The 9 in 0 and 9 combination is the last 9 in .999...#. The only 9 the .999... lacks from .999...# is the last 9 in .999...#!

The last 9 in .999...# is found! So does the last 9 in .999...! (The 9 the same to the 9 before the last 9 in .999...#)

Wait a minute! How can you say a last 9?
From beginning, I have proven the dissertation
there is no 9 and 9 combination after 0 and 9 combination.
And what does no 9 and 9 combination mean? It means either the digit is both filled with 0's or there isn't such a digit at all. Either way, the 9 in 0 and 9 combination stands for a 9 in .999...# which has no 9 following it whatsoever! That's literally what the last 9 means.

But no 9 and 9 combination, can be 9 and 0 combination ( other than 0 and 0 or nothing and nothing) -OK, then what's next? nothing ? or 9 and 9 combination again. The former, who lacks one 9? the latter, the same contradictory to full 9's.

Err, why assuming 0 and 9 combination? Because 0.999... doesn't have at least one 9 0.999...# has ( for it has one 9 less). And every 9 in 0.999...#  stands for both an existed decimal and the coefficient 9 for that decimal (9*10[sup]-i[/sup]), and it's wise to assume 0.999 has no more decimals than 0.999...#, therefore a one 9 lack means a 0 coefficient at the decimal for which 0.999...# has 9 coefficient, a 0 and 9 combination.

Hence, Logically, the last 9 is located, whose existence, is of course proven.

OK, the last 9 virus, is back! What  does it stands fpr ? What does the 9's before it stands for? Delima comes!

But this time, no one can shrink off this delima by simply denying the existence of the last 9.
Because it is not only "imagined", it is proved.

Yeah you are right my friend. The score is only a preliminary proof that you have studied hard enough. One should demonstrate the real ability through talking to the potential employer.

Thank you.

No one could answer?

That's a tough question?

" r has no solution here."
_it should be many any solutions instead of no solution.
0r=5 has no solution, whereas 0r=0 has, but no single determined solution.

I guess this can make your irrelevant case possible in a way.

Let's see why this works.
Suppose you are telling jokes to a friend, you have told one joke, and s/he doesn't laugh. Then you continue to tell jokes, but no matter how many jokes you have told, s/he still behaves like a dump wood.
Are we sure that your friend's good mood is irrelevant to how many jokes you tell?
Sure, irrelevant. No correlation

Now we can remodel this.
Everytime you tell a joke, your friend laughs.
If we divide the time interval to be the time required for your to tell a joke
Then we can come up with the data that how many jokes you tell during such a period and how many laughs your friend has in the mean period.
Then
1 1 1 1 ...
1 1 1 1 ...
Are you sure that when x=1  y is guaranteed to be 1? And if x=0 y==0? If you come up with such a conclusion and you are guaranteed that they share the same exact probability, they syncronize. They are totally related. They can determine each other. So they are totally related.

Note, if you use the cumulative jokes and laughs, you get out of the 0/0 problem and get it 1.

Now you are stuck with the fact that your friend's laugh is in the middle. How can you judge this? Well, they have come up with the correlation formula, where
correlation=covariance/(stdv(X)stdv(Y)]
covariance=Sum(Xi-Xmean)(Yi-Ymean)/n
stdv(X)=Sum(Xi-Xmean)

They have made a good reason on two extreme cases. If X stays the same irrespective of Y's change, we know that they are irrelavent, and the formula gives the 0 result; If Y is a linear combination of X, which means Y=a+bX, the formula returns 1. And further they have used triangular property to prove that r^2 is no large than 1

But that's the best the correlation defination can give. It isn't a truth or fact, it is just an arbitary and artificial index made by staticians which happens to enjoy these three properties, which gives it some sense. But it has its pitfalls as well.

How about Y=X^2 or Y=2^X, they are totally relevant but without a correlation 1. The correlation index cannot capture a non-linear relationship.

Another pitfall is that it cannot have a denominator as 0. This comes to the pitfall of linear language. Just recall the words I have written:

If X stays the same irrespective of Y's change, we know that they are irrelavent, and the formula gives the 0 result; If Y is a linear combination of X, which means Y=a+bX, the formula returns 1.

Swap X and Y, it makes sense as well. But why don't I use them equally in the sentence? Because I am handicapped by the language to. The language flows linearly, from the subject to the object, hindering me to express the mutual relationship. The only way out is to state the inverse as well. But that only means this direction+ that direction, not so perfect to reflect the nondirectional truth.

That's how people go with slope or ratio. people can only say y will change 2 if x change 1 or x will change .5 if y change 1 -they have to simplify the dual process to a single directional one to think. That results in slope's inability to capture a horizontal line. You have to jump out of dY/dX and use another direction dX/dY instead or switch to the more complicated undirectional vector thinking (dX,dY)=k(1,2).

However, this is not a big issue. A function, at the beginning, bares the role of discovering a causal relationship. How do we know whether two things, one is the cause of another or they are both the result of some other thing? Bacon puts it:
If one increases and the other increases or decreases correspondingly, we can infer they have a causal relationship.
Or at least we are sure they have a conditional relationship. (which is just weaker than a causal relationship and avoids the debate which is which's cause or whether there exists a third factor as their shared cause)

But what happens to two constant things? The earth has a constant amount of carbon element as it has a constant travelling speed, almost. But can you say that one is the other's cause or they share a common cause? Not necessarily, right?

So here is the key. You have to show some variation to prove such an at least conditional relationship. Conditions means more than one possibilities. So back to the correlation case, either an 0 and 0 sample  or an 2 and 2 sample has to be added to the 1 and 1 samples in order to make sense to predict an if-then relationship.

e.g.