Math Is Fun Forum

It reminds me of a bilogical experiment.

The existence of bacteria

At first people do not know bacteria.

People simply didn't see it - like you don't see or induce the last element

Then Paster evacuated a glass bottel containing soup.

His soup went bad much slower than usual.

He then concluded no matter anyone could see it or not, the logic is there must be something in the soup that lives with the air and make the soup bad.

And the same applies to my proof in Post 30. The last element exists by logic induction, no matter the mathematical induction can find it or not. If it cannot find it, it is the method's fault, not that the element does not exist.

Ultraforce, you need to understand the event first. Your event, if i get it correctly, is one but only one of A or C's letter reaches you. Since P{A or C}=P{A but not C}+P{C but not A} the calculation above is not right. when you add up probabilities you need to check if they have overlapping part first. if you add up two events like {A} and {C} you are counting {2 letters} twice. So an alternative would be P{A or C}=P{A}+P{C}-2P{2 letters} , and P{at least 1 letter}=P{A}+P{C}-P{2 letters}

orz, I don't even know what it is for in probability

Let me see,

f(z)=f1(z1)f2(z2) where f(z)=d[Pr(Z<z)]/dz

The event Z=y means Z1+Z2=y or Z2=y-Z1

The density of the event Z=y
would be integration of f1(z1)f2(y-z1) from z=-∞ to z=+∞

and what next?

Does it mean 6%±6%/8 instead?

Okay, here it is how it went wrong:

By traditional textbook pairing method, it seems possible to pair each and every element of Set B to corresponding one of Set C-

{1, 2, 3, 4, ...}
  |  |  |  | ...
{1, 2, 3, 4, ...}

But the question is- have the pairing exhausted Set B or Set C?

The answer is not.
When one tries to do the traditional pairing, they actually are doing inductive counting:
1 to 1 - 1 pairs
1 to 1 & 2 to 2 - 2 pairs
1 to 1 & 2 to 2 & 3 to 3 - 3 pairs
Good?

1 to 1 & 2 to 2 & 3 to 3 & ... & N to N - N pairs
So that
the same to above & N+1 to N+1 - N+1 pairs

Such inductive method seems powerful, but it underestimates the scale of the infinite sets.
Let us admit bluntly that
either Set B or Set C
has an amount of elements that is larger than any N you name, N as an integer

How can you finish pairing all elements of them with only N, O, P, Q, ... pairs?
I mean, no matter how far your counting and pairing go, they only appear to pair the elements over,
but actually they fail to cover the overwhelming amount of elements in either Set B or Set C

Take it in another way, when you draw the graph of 1/x^2, you draw it uprising against y axis, and you don't draw it intersecting y axis, because you know getting larger and larger doesn't mean reaching an infinite amount that exceeds any number, don't you? Same applies here.

The pairing is not complete. And this is why the proof is false. (this false proof fooled numerous folks to accept one set and its subset can have the same amount of elements just because they cannot prove it wrong)

Now let me show you some real proof. Just changing the method a little, we strictly define the pairing as complete.
By complete pairing, I define:
If one set A pairs to another set B completely, and it pairs to a third set C completely, which doesn't equal to B,
set A cannot pair to set B U set C.

Take it in simpler form:
A set cannot completely pair to its real subset.

This is not hard at all, anyone would think it makes sense unless they are fooled by Hilbert's Hotel. After all, it took a long time for people to accept Hilbert's Hotel after they failed to find an angle to prove it wrong.

But there is an angle, now.
Let us move on,
By simple, straightforward complete pairing definition,
we know/assume (whatever you call it) Set A is different than Set C
And here is something interesting about it
we cannot find which one element is different by traditional pairing,
1 to 1, 2 to 2, 3 to 3, ... all fine?

But wait, we know there is difference already, shall we give up searching just because we cannot go to the end of the cave?
Come on, let's try a different approach:

We know there is at least one element that Set C has and Set A doesn't that makes Set C one element larger than Set A
And we pair them all up to find out who
For convenience, we name the missing element X, and it pairs to "-" to Set A U {-}
Suppose both A and C are ordered set from small to large.
We know X is somewhere in C, and we would like to put {-} in A U {-} to in a place corrsponding X
that all the elements before X in C and all the elements before - in A U {-} are the same

So all the elements before are {1, 2, 3, ...X-1}
Now, we have {1, 2, 3, ...X-1} U {-} in A U {-}
and {1,2,3,...X-1, X} in C
Here we go
Can there be an element X+1 in both A and C?
if so , we will have
{1, 2, 3, ...X-1} U {-} U {X+1} in A
remember, A does not include X
and it is absurd and contradictory
So there is no X+1

Hence we have proved the missing element is at the right end, the largest one in A is X-1 and the largest one in C is X
Also X and X-1 is the total amount of elements in Set C and Set A
X=∞
where
∞ > any N, N is any number one can name

And here we are not over yet
if X = ∞
X -1 = ∞
X -2 = ∞
X/2 = ∞
X/N=∞

from the other direction

1<∞
2<∞
100000000000000000000000000000000<∞
2^100000000000000000000000000000000<∞

There is no way for ∞ to transit to some finite integer
Yeah
C={1,2,3,...}U{...,X-3,X-2,X-1,X}

(My Paradox)

And do you still now believe it is a set of "all integers"

The only possible logic conclusion is there is no such "all integers" thing
The notion of {1,2,3,...} or {1}U{1,2}U{1,2,3}U...
is a self contradictory definition
it is like a definition of relationship
< and > and =
or a definition of A & B, where
A>B as well as B>A

Astonished?
You may be suprised to find there is actually no empirical evidence of "all integers"

When a poet looked upon the night sky and counts:
"one star, two stars,...countless stars!"
we call it romantic

But when a mathematician counts
"one, two, three, ... and theeeeeeeeeeeeeeeen, infinite of them!"

"We say that two sets have the same size if the elements of one set can be paired with the elements of the other set, with no repeats and no leftovers. "

Fine, I really like this definition, Ricky.

Let us pair the set
{1,2,3,...}  Set A
to
{2,3,4,...} Set B
by the rule of adding 1 to each of the element of Set A
perfect pairs, right? They have the same amount of elements, as definition.

Now let me show you something you won't find in a set theory book
Define
Set C = {1} U Set B
Without counting Set C over, we know Set C has more element than Set B.
Right?
Since Set B is included in Set C, but no the other way around.
It has one more, since the complimentary of {1} in Set C is Set B, and {1} has one element, do you agree?

Now we are facing a tough problem
Set A and Set C appear all the same
{1,2,3,...}
except
Set C has one more element than Set A
or
Set A has one less element than Set C

by transivity

I tried to explain it to you before in a simple way that how mathematicians like you misinterprete words, and simbols. And I show this simple logic disproof of your infinite set again:

Regardless of what particular infinite set you define
Try to shift all elements one position rightwards by some pratical algorithm (in this case +1)
Then add in the starting element lost in the shifting process (in this case 1)
And then it is undeniable logic truth that I have just created an infinite set with one element more than the old one
And now it is time to reveal what the old set lacks - the right end
And the right end is the infiniteth set revealed
sometimes infinity, sometimes infinitesimal
so the paradox is discovered.

"Let's say I have proven that proposition P holds for the value 1, and I know that if it holds for a number n, then it holds for n+1. "
This inductive model seems to stretch out the infinite set to you,
and you keep saying
Can you find any error in my set by this approach?
insead of asking
What other approach might prove my set wrong?

Ricky, have you encountered some classmate asking you the question:
"don't tell me your way and how you get it correct, tell me why my way is wrong try to work my way out to get the correct answer"
I heard these questions numerous times, and I have to admit, they are always much harder questions.

Nice as I am, I have to give you an answer, even though you might not be satisfied with it.
The misconception is
you mistake induction as infinite
Yes many had mistaken it,
But where is it mistaken?
You add in its infinite potential to imagine it as infinite
If we step firmly,
we should say inductive sets can never qualify infinite set
you have 1 element in it at first,
then by inductive process, you have 2,
you have 3,
you have 4,
just like day 1, day 2, day 3, day 4
but come on
when can you have the inductive set or sets to capture its ever changing nature
countless?
Every stage it is an Finite set.

This truth looks so boring
So most mathematicians like you say
Imagine(define) a set to include all element an inductive process can have, or the union of all inductive sets possible
like the
{1}U{1,2}U{1,2,3}U...
thing
In this way so can you create an infinite set without the dynamic inductive process and its each finite stage
You wanna chop off the process and leave the result
Fine,
just after you have done that, your set claiming to include "all positive integers" in it no longer adds in new elements like the inductive set, right?
Otherwise how can it be the union of all inductive set?
Then we are good to go
Go to four paragraphs above
And reread
"Regardless of what particular infinite set you define
Try to shift all elements one position rightwards by some pratical algorithm, in this case +1 ..."

You would discover why it is an illusion to you
that your infinite set doesn't contain the infiniteth element or the right end
Your only defense would be going back to the inductive process, back from the beginning
{1} "I haven't seen an infinite number!"
{1,2} "I haven't seen an infinite number, not yet!"
{1,2,3} "not yet at all"
n, n+1 "they are defined as integers, not infinite numbers"

Ricky, your way and my way get different answers, and I know one part of your reasoning is flawed
That if you use n and n+1 only
You are burried in the "infinite" inductive loop
If you have a computer program or anything like that to run it
You are in the "dead loop"
{1}
{1,2}
{1,2,3}
...
{1,2,3,...,n}
not infinite yet?
...
{1,2,3,...,n,...,m}
still not infinite?
...
{1,2,3,...,n,...,m,...,q}
not infinite, no way!

And there is nowhen you can achieve your ideal infinite set
which contains all inductive sets possible (really?)

To get away from stucked by dead loop
You make a jump-
You just claim you can get or define such an infinite set,
which in fact is never(literally) a product of inductive growing
And it has now invited the infiniteth element demon  into it, (as I proved)
which innocent n and n+1 algorithm had never ever and done and will never ever do. (indeed you won't find "infinity" in n 'n' n+1)

And here is where your way (or the way chosen by mathematics community) is flawed, Ricky.

"Either an integer is a finite distance away from zero, or it is an infinite distance away.  I am taking all the integers that are a finite distance away, and I call this my set that I posted in #9.  What is wrong with this?"

The problem is, Ricky, the conception of integers.

When people think of positive integers, they think of 1, 2, 3, and so on

And the "so on" is the problem

Is it finite or infinite?

It has a finite side that every integer is finite, just as you said.

Yet it appears infinite that the growing process might go on and on, without a boundary. (and this is what you want me to admit on infinte set)

This is the most wide-spreading misconception of infinity, I have to say.

Aristotle name it as potential infinity, he referred to time as evidence

Let's examine more carefully the two sides.

The former is experimental, the latter is imagination.

When you observe time or days, you do see the pattern

1, 2, 3, 4, 5, and might go on.

But actually, the cardinals 1, 2, 3, 4, 5 and perhaps more don't represent absolute quantity.

The fact is, you cannot put 5 days together, shuffle them, seperate them, group them as you do with 5 apples.

Are 5 days of 1, 2, 3, 4, 5 the same as the 5 days of 1, 3, 5, 7, 9? Just as you arbitarily choose 5 apples with almost the same quality?

We all know time changes, it cannot be the same.

The major difference is, 5 apples are 5 apples, they are independent, true 5 individuals, free of consequential or consecutive relations.
whereas 5 days, are actually 1 day mutated 5 times, each following day is the previous day evolved.

When you try to group {Day1, Day2, Day3, Day4, Day5, ...} and think you are forming a good example of integer set,
you are mistaken.

You are like forming {you in yesterday, you at today, you at tommorrow, ...} but not {1 of you, duplication of you, triple of you...} or even {1 days, 2 days, 3 days, 4 days, ...}

The latter is a list of quantities, whereas the former is a list of cardinals. Cardinals doesn't necessarily represent quantities.

It is really an illusion to form a list of integers by forming a list of cardinals like these, since they represent only one thing changing.

Since cardinals are not good example or proof of existence, we can think of quantities

when you believe you can form a set as

{1 apples, 2 apples, 3 apples, ...} , which includes all the possibilities of amount of apples

You have included infinite apples at the right end, which I have proved.

Please don't tell me you haven't finished listing the amount of apples imaginarily

Only cardinals like date have this right not to be shown at the same time, since they are actually only one, since changing the appearance needs time. But true quantities can be gathered at the same time, all at now.

If you just need a palse at x=0, there is a quasi-function called delta function which might be a good candidate but you need to multiply it by 30. If you think it doesn't look like a function, erf (error function) might be of help. You just need to set the sigma really small and it behaves like a palse.

Ricky, if you trust induction algorithm is all that entails infinity, my inductive algorithm can produce decimals any long thus is infinity as well.