God And Infinity, Worlds Without Number Are Numbered To God?

[asbestosman]

Is it possible that it shouldn't be taken so literally?

Certainly. Even so, there is a sensible way to discuss infinity. If Rob doesn't want to discuss the mathematical senses of infinity, that's fine but he keeps making false assertions about the mathematics of it.

[mfbukowski]

Certainly. Even so, there is a sensible way to discuss infinity. If Rob doesn't want to discuss the mathematical senses of infinity, that's fine but he keeps making false assertions about the mathematics of it.

You mean there's someone wrong on the internet? Not possible.

[asbestosman]

You mean there's someone wrong on the internet?   Not possible.

Which of all the internet users is right? None of them, for they are all wrong.Let the mods be true and every internet user a liar.

[Rob Osborn]

No. He has a rule that changes them all at once. Add 1 mod 10.

He is not racing against anything.

Really. Each row would be infinite and many would have no repeating pattern. How are you, a person who doesn't believe in the infinite imagine so "easily" and infinite list of column where each column is infinitely long?

There would be repeating patterns in certain rows- that's not hard to imagine. Remeber- every possibility for every column and sum of all columns, is already down. The proof I would ask Cantor, is to show me that number. Now of course this is ludicrous- we both know that. But we also both know that any number, no matter how long it is can only be of finite length anyway so the whol infinite number is ludicrous in and of itself. But we both agree to play along with these "infinite" numbers to show a point. My point is that a truly infinite list- if it were possible (of the which it isn't) would already contain every possibility for every colomun and sum of all those columns. It is mathematically incorrect to believe that a number can be generated that would not already be in the list. It's like saying that the list doesn't really account for all possibilities and thus it is not infinite at all. Remember that even in Cantor's argument even he can't do it all at once- there is no end to numbers and as such there can be no proof as he claims.

In fact, there is no such list of all possibilies no matter what you think you can imagine.

Why not? Its not hard to compute that in the first column of my list there are 10 possibilities. In the next column there would be 100 and the next a 1000 and so on and so forth after that manner. That is just plain simple elementary math.

But real numbers are given by infinite lists of digits.

I care about what exists in the abstract, not what can be known in your "can-I-count-it-out-loud-before-I-die-sense".

In one sense maybe- but it is all abstracts. For example in my work we measure things all the time in the 0-1" range. For a 1/4 inch we measure it as .25 with no zeroes after it we even write it down as just that. We don't have to put an infinite amount of zeroes after the 5. Remember that everything is in realation with some "limit". We can speak in hundreths, thousands, ten thousandths, etc. but each has a defined limit where there are no more zeroes as it becomes a matter of redundancy and carries with it no new information. In the real world- the physical tangible world, those real numbers mean nothing other than a means of communication between us so we both know what area we are defining.

What?? This is totally confused. After five beats the number written down by the drummer would be 5. But the diagonal isn't what is in the fifth row. There can't be two things that are fifth in the list. That would ruin what it means to have listed everything.

Besides, the diagonal is infnite in length and so the generated number would be also. The drum would never get to write anything down for THAT real number.

Now you are starting to get it. You see- the numbering system we have is one that defines powers and relations of "10". Numbers have a value that we have ascribed to each one. For instance we know what a 1,000 apples would be. In all of math we use numbers as a type of value. The drum beats I was using is to show the principle of the natural numbers in a real world scenerio- not some meaningless abstract on paper and in our imaginations. My point there is to show that the natural numbers are best defined as a mere simple mark- each one being the exact same and then they only get meaning after we add them to a sum or total of what we want to define. That is where cardinality comes into play- the size of a set. If I beat on the drum a 100 times and quit- that cardinality would be a 100. But, when looking at something having no end, the cardinality is not known and can't be either.

In the case of countable sets like the natural numbers or the rational numbers, we can do it by a fixed method so that any specific number will eventually be counted.

In the case of the real numbers, there is no method that has that property (that any specific real number would eventually be reached by a single counting method)..

Don't let numbers play tricks on you, it is easy to want to see this. In a set of real numbers like those between 0-1 we know that starting at the decimal place we can count from that point down in each column forever and ever just as you can count the natural numbers the same way. There is another way to look at this to add some contrast. Suppose I asked you to count all the natural numbers but my only rule was that you must first start with the biggest number and count backwards. That we know is impossible. Well then- you get the drift. In counting anything one must start at one end or known finite place and count from that point. No wof course we both know that whether it be a real numbering system or natural numbering system, you could keep counting and adding more columns and rows forever and ever and never reach any limit because the numbering system is just that- limitless.

[asbestosman]

Rob, what you fail to realize is that the extra zeros are there whether you write them down or not. Also, we are exactly saying that the list doesn't account for all possibilities despite being infinite. In fact you yourself admit that the list only contains the numbers between 0 and 1. If the list were really infinite, wouldn't it contain the number 42 as well? Or are you telling me that you understand that we can have an infinite list that does not contain all possibilities? Now, I expect you'll tell me it contains all possibilities between 0 and 1, but you are merely asserting that without proving it. Your assertion that if it's missing a number between 0 and 1 means that it's not really an infinite list is fallacious because even you accept that a list can be infinite and leave out some numbers. I'll even give you an example of a list that is infinite but leaves out some numbers within the range: the list of all even integers (no odds are included. Another: The list of all prime integers. Another: The list of all powers of 1/2 (1/2, 1/4, 1/8, 1/16, etc) which leaves out numbers like 5/16. Therefore we know that a list can be infinite and yet leave out some numbers. We've tried many times to point out which numbers your list omits, but you refuse to listen--you just keep asserting that since your list is infinite it must contain those numbers.

Edited January 26, 2012 by asbestosman

[mfbukowski]

Which of all the internet users is right? None of them, for they are all wrong.Let the mods be true and every internet user a liar.

Hey man, speak for yourself.

I am always right. Wanna fight about it?

[Tarski]

But we also both know that any number, no matter how long it is can only be of finite length anyway so the whol infinite number is ludicrous in and of itself.

So the exact ratio of the circumference of a circle to its diamter (what we call pi) is not a number? How do you write that down as a decimal expansion of finite length.

Its not hard to compute that in the first column of my list there are 10 possibilities. In the next column there would be 100 and the next a 1000 and so on and so forth after that manner. That is just plain simple elementary math.

Well you will never hit 1/3 then will you? That needs to be an infinite length decimal expansion of isn't 1/3 but rather some other prefectly fine number.

If there is to be any hope of listing all real numbers, then you have better come to grips with the fact that we need to represent them all with an infinite list of digits even if some of use only zeros after a certain point:

.2 = .2000000000000000000000000000000000000000000000000000000.... FOREVER

Now the number of possibilites for just the first number (row) is infinite since it could have been pi. Or it could have been .01010101010101010101010101010101010101.. Repeat PATTERN forever.

Every row is infinite in length.

If you think that every number has a finite decimal expansion (meaning eventually all zeros) then you exclude pi, the golden ratio, the square root of 2 and a whole lot more real numbers--even 1/3.

You exclude way more numbers that way than you are including.

Elementary math test:

What is the tenth digit in the decimal expansion of the rational number 23/100 ??

If you answer this I can make my point.

Oh again, if you can really explain what your list is then you should be able to easily explain what the diagonal looks like. Tell me that and I will hand you the missing number.

Give it a try. Show me or just describe to me a clear rule for the diagonal of your list. Write out the first 20 digits.

Edited January 26, 2012 by Tarski

[Rob Osborn]

So the exact ratio of the circumference of a circle to its diamter (what we call pi) is not a number? How do you write that down as a decimal expansion of finite length.

Don't forget that the numbering system we use in not precise for all things. It is an abstract and the ratio we use sometimes becomes imprecise- were not perfect nor have we a perfect mathematical system. Sure, Pi is a number that we can use to "some degree" of precision when using our abstract numbers to relate. I could write pi as 3.14 or even as .7854% in which i often do in my work But, whatever I do work with I create a limit or "end" to that number I work with.

Well you will never hit 1/3 then will you? That needs to be an infinite length decimal expansion of isn't 1/3 but rather some other prefectly fine number.

Sure, I can hit a number that that comes as close as one wants to be represented for the expansion of 1/3. It all depends on where you want your limit. So yes- I can hit that number- which ever one you want to make 1/3. Just realize that according to this system of abstract numbers- we will never have a perfect representation of this number using the count of "10". Like was said by someone else- if we use a base of "9" to number then of course it would be easy- .3

If there is to be any hope of listing all real numbers, then you have better come to grips with the fact that we need to represent them all with an infinite list of digits even if some of use only zeros after a certain point:

Nobody i know can write or compute with an endless line of zeros. Those endless 0's mean nothing in this abstract system. We only define actual numbers by what size or relation we want to use such as 25 thousandths- .025. I could write it .0250 if I was using it in relation to ten thousandths, etc etc. Its just a matter of how we use it at this point but in reality.

Now the number of possibilites for just the first number (row) is infinite since it could have been pi. Or it could have been .01010101010101010101010101010101010101.. Repeat PATTERN forever.

And my list would contain that pattern

Every row is infinite in length.

Yes it is.

If you think that every number has a finite decimal expansion (meaning eventually all zeros) then you exclude pi, the golden ratio, the square root of 2 and a whole lot more real numbers--even 1/3.

You exclude way more numbers that way than you are including.

I am not saying this, I recognize that some numbers can go on forever. Their end just can't be "knoiwn" and we will nbever use it like that in an infinite form to calculate math.

Elementary math test:

What is the tenth digit in the decimal expansion of the rational number 23/100 ??

If you answer this I can make my point.

Oh again, if you can really explain what your list is then you should be able to easily explain what the diagonal looks like. Tell me that and I will hand you the missing number.

Give it a try. Show me or just describe to me a clear rule for the diagonal of your list. Write out the first 20 digits.

Not every number can be expressed perfectly in this numbering system so what is your point?

[Zeta-Flux]

Rob,

(1) Just because we cannot write a number "precisely" (which I take to mean, with all of its decimal digits) does not mean that we cannot precisely name the number. The number 1/3 is precisely defined, even if it doesn't have a finite decimal expansion. Just as 1/2 is precisely defined. As is 1/7.

(2) Just because we cannot write out all of the decimal digits, all at once, does not mean that we cannot write out any finite number of them. So we CAN go as far as we like.

(3) That means we can compare two numbers. If we ever get to a place where their digits are different, the numbers are different [ignoring one minor technical point].

(4) I will again repeat my claim (with a minor correction): Pi/10 does not occur on your list. If you don't believe me, then tell me where it does occur.

Further, even if you change your rule so Pi/10 does now occur, I can describe a number which you can check will not appear on your list. (As can Tarski, asbestosman, and others.)

[asbestosman]

Rob, one thing you need to realize is that a list with an infinite number of rows has no "infinitieth" row (in other words, no last row). If any number you claim is on your list occurs on the infinitieth row, it actually isn't on your list at all. That's why we claim that 1/3 isn't on your list even though there's a perfectly sensible way to represent 1/3 in decimal notation. Yes, it has an infinite decimal expansion of 3's. See Zeta-Flux's points 1-3.

Edited January 27, 2012 by asbestosman

[asbestosman]

Hey man, speak for yourself.

I am always right.   Wanna fight about it?

Are you a Mod?

No?

Then . . . die. (with a nod to Dan Aykroyd)

[Tarski]

Don't forget that the numbering system we use in not precise for all things. It is an abstract and the ratio we use sometimes becomes imprecise- were not perfect nor have we a perfect mathematical system. Sure, Pi is a number that we can use to "some degree" of precision when using our abstract numbers to relate. I could write pi as 3.14 or even as .7854% in which i often do in my work But, whatever I do work with I create a limit or "end" to that number I work with.

Sure, I can hit a number that that comes as close as one wants to be represented for the expansion of 1/3. It all depends on where you want your limit. So yes- I can hit that number- which ever one you want to make 1/3. Just realize that according to this system of abstract numbers- we will never have a perfect representation of this number using the count of "10". Like was said by someone else- if we use a base of "9" to number then of course it would be easy- .3

Nobody i know can write or compute with an endless line of zeros. Those endless 0's mean nothing in this abstract system. We only define actual numbers by what size or relation we want to use such as 25 thousandths- .025. I could write it .0250 if I was using it in relation to ten thousandths, etc etc. Its just a matter of how we use it at this point but in reality.

And my list would contain that pattern

Yes it is.

I am not saying this, I recognize that some numbers can go on forever. Their end just can't be "knoiwn" and we will nbever use it like that in an infinite form to calculate math.

Not every number can be expressed perfectly in this numbering system so what is your point?

answer the question. What is the diagonal of your list?

[mfbukowski]

Are you a Mod?

No?

Then . . . die. (with a nod to Dan Aykroyd)

<shriveling away like a slug with salt on it....>

[Rob Osborn]

answer the question. What is the diagonal of your list?

Well, with my modified rules there would be no number. What is the objective- to find a number sequence not in the list. But my list would contain every possible sequence. The rules should be that one couldn't go to the next column until all the sum of previous columns were listed with all possible outcomes. It is only through this way that we can ever be certain that the list really did contain all the numbers possible.

[Rob Osborn]

Rob,

(1) Just because we cannot write a number "precisely" (which I take to mean, with all of its decimal digits) does not mean that we cannot precisely name the number. The number 1/3 is precisely defined, even if it doesn't have a finite decimal expansion. Just as 1/2 is precisely defined. As is 1/7.

I agree

(2) Just because we cannot write out all of the decimal digits, all at once, does not mean that we cannot write out any finite number of them. So we CAN go as far as we like.

I agree- that is why we have a limits with math numbers- whatever we choose them to be.

(3) That means we can compare two numbers. If we ever get to a place where their digits are different, the numbers are different [ignoring one minor technical point].

I agree.

(4) I will again repeat my claim (with a minor correction): Pi/10 does not occur on your list. If you don't believe me, then tell me where it does occur.

My list would contain every possible number between 0-1. So if you have a number to squeeze in there it should be there already

Further, even if you change your rule so Pi/10 does now occur, I can describe a number which you can check will not appear on your list. (As can Tarski, asbestosman, and others.)

I am not sure what number you could come up with that wouldn't already be in there.

[Zeta-Flux]

Thanks for your comments. Since we seem to agree on all of the points except the last one, let me ask you one simple question.

Give me the rule you have that lists the numbers. If you could include the first 20 numbers, that would be helpful. Finally, tell me where 1/3 occurs on your list.

Thanks,

Zeta-Flux

[Cold Steel]

This discussion of numbers proves nothing, does it?

God numbers all of his creations, which are numberless to men. When prophets in the Old Testament spoke of "all nations" and "all the nations of the earth," it wasn't literal. They didn't mean North and South America or even Asia. And John said of the Beast, "and power was given him over all kindreds, and tongues, and nations." Yet in context we don't expect him to have power in the Western Hemisphere. The Bible is centered in Jerusalem and the Book of Mormon in wherever it took place. Thus the three days of light and darkness did not apply to the Old World.

So continue debating numbers if you wish. You're just concepts that are child's play to God.

.

[Rob Osborn]

Thanks for your comments. Since we seem to agree on all of the points except the last one, let me ask you one simple question.

Give me the rule you have that lists the numbers. If you could include the first 20 numbers, that would be helpful. Finally, tell me where 1/3 occurs on your list.

Thanks,

Zeta-Flux

Ok. Rules must be laid out to properly understand number sequence and probability. In a way we must reverse engineer what Cantor is asking or stating. His proposal is to find a missing number not already in the list. For example- on a smaller level lets say with the numbers 1-10 that we were to make a partial list thinking we had listed "all" the numbers. So we construct a method similar to the diagonal argument to do just that. Here is the list-

1

2

3

4

6

7

8

9

10

Going through this list we start at the first and go through all until we get to 10 and find that "5" has been left out. This is simple 1st grader math here So we would find that the number 5 has been left out of the list. Using this same principle and rule we would start at .0 and work from there every possibility in this first column until we got to .9. If all possibilities are there we can't write any new number and must then press forward to the next column- those numbers between .00-.99 if there isnt any missing, we must press from there to the next set of .000-.999

The principle agreeing with the rule here is that if we are trying to find a "number not already in the list" then we must use logic and this process to go through every possibility from a known satrting point. For instance- If I had a list of 4 digit numbers and the missing number was 6,421 the diagonal argument isn't going to tell me that. The only way to find it is to go through every sequence until we find it. That means going through every number from a known starting location until all probabilities have been exhausted and then, if there was a sequence left out we shall know what number it was.

On my list, the numbers between 0-1 we should not find "1/3" on the list because that is not something in that set. However, if we were to expand it into it's closest decimal equi=valent then we should find every number as close as we could possibly get to it which would of course be .33333333333333333333333... but then you get the drift.

[Tarski]

On my list, the numbers between 0-1 we should not find "1/3" on the list because that is not something in that set. However, if we were to expand it into it's closest decimal equi=valent then we should find every number as close as we could possibly get to it which would of course be .33333333333333333333333... but then you get the drift.

There is no closest. But of course we can get close to any real number. The rational numbers (which do for a countably infinite set) are dense in the set of real numbers (which is uncountably infinite).

Quit changing the goals posts. 1/3 isn't in your list. Close isn't good enough.

But you are being silly anyway because we can include 1/3 and all other rationals if we just do it more cleverly--something you haven't seen yet.

The only thing you are clumsily showing is something we already know. Namely that one can list the set of all rational numbers whose decimal expansion is finite. None of those are left out. So what? That isn't all the real numbers. (it misses pi and whole lot more). (if you were smarter you could list all rationals but we already knew that too).

[Rob Osborn]

There is no closest. But of course we can get close to any real number. The rational numbers (which do for a countably infinite set) are dense in the set of real numbers (which is uncountably infinite).

Quit changing the goals posts. 1/3 isn't in your list. Close isn't good enough.

But you are being silly anyway because we can include 1/3 and all other rationals if we just do it more cleverly--something you haven't seen yet.

The only thing you are clumsily showing is something we already know. Namely that one can list the set of all rational numbers whose decimal expansion is finite. None of those are left out. So what? That isn't all the real numbers. (it misses pi and whole lot more). (if you were smarter you could list all rationals but we already knew that too).

Ok then brains, show me what you are getting at.

[Rob Osborn]

I will bet that no one can come up with a number not in my list.

[Tarski]

I will bet that no one can come up with a number not in my list.

1/3

[Rob Osborn]

1/3

You keep asserting that but never show why. This is getting ridiculous.

[JDave]

This seems vaguely like a rerun of a bad soap opera on the "Mathematics" cable station (or sub-forum ). I wonder why....

No one is going to convince Rob that 1/3 isn't on his list if he has the fundamental disagreement that arbitrarily close is, indeed, close enough. I will make one small effort, but really have no interest in continuing

Rob, as you have stated, if you were representing your list in base 3, then 1/3 would have a finite representation of 0.1. It would appear quite early in your list, and would not require any appeal to "well, it's arbitrarily close enough". It really is there, look, it's the first number in the list!! So if you can create your list in base 3 and see the number, but create your list in base 10 and not see the number, that means in base 10 the number is NOT in your list. And obviously switching everything to base 3 doesn't solve anything because then the question would be, where is 1/2?

[Tarski]

You keep asserting that but never show why.

Because it is obvious and everyone here sees it but you, including 3 mathematicians.

1/3 has an infnitely decimal representation while none of your rows is infinite. All of the numers on your list differ from 1/3.

To be 1/3 is not the same as being almost 1/3. Equal is equal.

This is getting ridiculous.

It sure is! But not for the reasons you think.

It is also ridiculous for the reason that you are trying to make something pedantic out of the poetic writtings of some ancient bronze age priests.

It is almost as if I said to my wife that I asked her countless times to not burn the toast and then she starts talkning about cardinality to prove me wrong.

It is all silly no matter how you look at it.

It is silly to bring up mathematical notions in connection with those scriptures but then when you do it anyway, you do it crazy wrong.

Your fear of perfectly finite and exact notions of infinity seems to be driving you into some kind of batty denialism.

I can one up you in nuttiness. Watch this.:

Hey Rob, ya know what? Not only is there no such thing as infinity, there is no such thing as three. All I see are some apples, some dollars and some shapes on a piece of paper. No one has every actually seen or proved "three" itself so it doesn't exist.

Edited January 27, 2012 by Tarski