# Infinity -- Cyclical?

• B
In what may be an obvious observation in all accounts. I realised I did not have as good a grasp on infinity as I thought I had. For infinity to be infinite in must go on for ever(obviously, this I get), however, it must reach a point on where it begins to return to numbers it has already used. For example 123412341234 (1234 being infinity). This is where my problem lies, for infinity is infinite, therefore it should not run out of unused values in the first place?

What do you think, is it Cyclical or linear?

mfb
Mentor
however, it must reach a point on where it begins to return to numbers it has already used
No. You never "run out of numbers".

1234 is not infinity. It is an integer. 123412341234 is not 1234.

There are 10 digits in the decimal system, the smallest number of digits to express any number can be two (zero and one).

No. You never "run out of numbers".

1234 is not infinity. It is an integer. 123412341234 is not 1234.
Maybe I did not explain myself enough. I was substituting infinity for 1234, because I thought it would be easier to understand. By 'running out of number's' I mean that every conceivable number in every conceivable way(so, infinity), until the numbers begins to repeat. For example, should it be possible to print every digit of infinity,and you had the time to read it, you would begin to notice that the same series of digits begins to appear in the same order, 123412341235(where 1234 is said series of Digits(which in itself will be infinitely long)) being an example of this.

I apologise in my inability to put my thoughts on paper.

Maybe I did not explain myself enough. I was substituting infinity for 1234, because I thought it would be easier to understand. By 'running out of number's' I mean that every conceivable number in every conceivable way(so, infinity), until the numbers begins to repeat. For example, should it be possible to print every digit of infinity,and you had the time to read it, you would begin to notice that the same series of digits begins to appear in the same order, 123412341235(where 1234 is said series of Digits(which in itself will be infinitely long)) being an example of this.

I apologise in my inability to put my thoughts on paper.
Infinity has no digits. It's not a (natural) number.

Logical Dog
jbriggs444
Homework Helper
Maybe I did not explain myself enough. I was substituting infinity for 1234, because I thought it would be easier to understand. By 'running out of number's' I mean that every conceivable number in every conceivable way(so, infinity), until the numbers begins to repeat. For example, should it be possible to print every digit of infinity,and you had the time to read it, you would begin to notice that the same series of digits begins to appear in the same order, 123412341235(where 1234 is said series of Digits(which in itself will be infinitely long)) being an example of this.

I apologise in my inability to put my thoughts on paper.
One thought that you may have meant to express...

If you write down an unending string of numbers then eventually (within 10 digits) you will begin to re-use single digits that you have already written down. Similarly, you will eventually (within 100 digits or so) begin to re-use digit pairs that you have already written down. And so on. For any finite length n, you will eventually reach the point where the last n digits you have written [or will ever write] will have appeared previously in the string.

Another thought you may have meant to express...

If you write down an unending string of numbers then eventually you will have written down every unending digit string. That turns out to be false. There are more unending digit strings than there are digits in an unending digit string.

Cantor proved that a long time ago.

this can be extended to represent decimal numbers with components where b-1 and a-1 etc start

What I understand...You cannot have an infinite base, and therefore digits would have to be re used. Yuo cannot add subtract divide infinity unless you make some new system of numbers and define them clearly. Another interesting thing is if you use a base n greater than 10, then you have to come up with n - 10 new symbols for the an co efficients.

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I am afraid that I might portray myself an idiot, but here goes. I can understand the it don't exist thing, for that is an argument that I am aware of existing. But it has no digits? Surely it has some numeral representation? I was to keep pressing 1 ad infinitum, I just gave infinity a value. Please explain

Referring to micromass's comment

Just realised my mistake, if I was to do that, I would never get the whole number because by getting the whole value, I would therefore be able to add to it, increasing its value. Which I can again add to.

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One thought that you may have meant to express...

If you write down an unending string of numbers then eventually (within 10 digits) you will begin to re-use single digits that you have already written down. Similarly, you will eventually (within 100 digits or so) begin to re-use digit pairs that you have already written down. And so on. For any finite length n, you will eventually reach the point where the last n digits you have written [or will ever write] will have appeared previously in the string.

Another thought you may have meant to express...

If you write down an unending string of numbers then eventually you will have written down every unending digit string. That turns out to be false. There are more unending digit strings than there are digits in an unending digit string.

Cantor proved that a long time ago.
I was thinking the bottom one, I was not aware it had been disproven.Thank you.

Mark44
Mentor
There are 10 digits in the decimal system, the smallest number of digits to express any number can be two (zero and one).
???
Whatever you're trying to say here isn't clear. The smallest positive integer is 1, which is expressed as one digit. Are you trying to get across some more weighty idea than this?

mfb
Mentor
I think Bipolar Demon means you can express numbers in binary, with just 0 and 1.

Mark44
Mentor
I think Bipolar Demon means you can express numbers in binary, with just 0 and 1.
That thought occurred to me, but there was no indication that he/she was talking about any system other than decimal.

I think Bipolar Demon means you can express numbers in binary, with just 0 and 1.
That thought occurred to me, but there was no indication that he/she was talking about any system other than decimal.
Yes, it is unclear. I meant to say that the distinct symbols used to represent numbers in the hindu arabic notational system cannot be less than two or infinite if we restrict the base and the coefficients to be Natural numbers.. So in base two: 0 1, are the only symbols used, we cannot have base of 1. All numbers will be expressed as strings of zeros and ones, or of a higher base but the base itself will be a natural and not infinity, and not less than the natural number 2 if we restrict the base and the coefficients to be naturals.

in this image, if a and b are in integers then b cannot be less than two and function as a representation of an integer other than zero.

I have not computed with bases other than integers, so someone once mentioned base of root two, how would that work? wouldn't n in the image then need to be a real number and not integer?

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is it possible to have say, 100 represented as base of root two following the same pattern? can all real numbers be used instead of 10? like 0.5 or root 2?

what about for rational and irrational bases? if we use root two, how will we come up with a system? will we restrict the coefficients (a in the picture to be naturals?)

sorry if this is not constructive

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smokingwheels
Maybe I did not explain myself enough. I was substituting infinity for 1234, because I thought it would be easier to understand. By 'running out of number's' I mean that every conceivable number in every conceivable way(so, infinity), until the numbers begins to repeat.
Not really. Here is an infinite string of digits with no repeating pattern of the type you think about:
11010010001000010000010000001....

Not really. Here is an infinite string of digits with no repeating pattern of the type you think about:
11010010001000010000010000001....
My point is it must happen eventually. As an example,At some point 1101001001 will appear again further along in the string. But what I am proposing Is that the entire infinite string will keep repeating again, within the infinite string.

Are you familiar with the Infinite Hotel Paradox, presented by David Hilbert. Within this hotel is obviously an infinite amount of rooms. Imagine you are the receptionist, and you have the list of number's of rooms in front of you. We already know that you can fit an infinite number if customers in who are in the queue even though all the rooms are taken.

So, let's just imagine for a moment that you write the room numbers as a string (123456789...) and you leave that piece of paper alone for a while. You get rooms for the infinite number of customers, even though all the rooms are full (infinity + infinity= infinity). And then you get the piece of paper with the infinity string and then looks at lists of the numbers of rooms. Even though you copied every figure correctly, the list has change(the is another infinite amount of rooms) but you notice that what you wrote on the paper is visible, one after the other, an infinite amount of times in the new list of rooms.

Hence infinity is Cyclical.

However I am aware that this is likely complete rubbish, as for infinity to be Cyclical it must have an end of sorts, becuase even though what is in the paper is an infinite list of rooms, must have an End to repeat. But as I said earlier infinity cannot end because as soon as you know it, it can be added to and anything x or+ infinity is infinity. Which of course would make infinity linear.

This is what I was struggling with. However badly I portrayed my thoughts, you guys must have an idea:

Is it linear or Cyclical?

Staff Emeritus
My point is it must happen eventually.
And you're wrong. You were presented with a counterexample.

I'm not sure what the next step is, if a counterexample won't change your mind. Whatever we are doing now, it's not mathematics.

nikkkom and smokingwheels
And you're wrong. You were presented with a counterexample.

I'm not sure what the next step is, if a counterexample won't change your mind. Whatever we are doing now, it's not mathematics.
Oh yeah I know about what jbriggs said that cantor disproved. I understand that I am wrong, I've said since the beginning I probably am. I just dont understand how it is wrong, I read the material jbriggs provided on the wiki.

I just can't wrap my head around it for some reason.

I mean come on, it fits perfectly with the eternal recurrence concept I just found( https://en.m.wikipedia.org/wiki/Eternal_return ) just in maths form.

smokingwheels
jbriggs444
Homework Helper
Oh yeah I know about what jbriggs said that cantor disproved. I understand that I am wrong, I've said since the beginning I probably am. I just dont understand how it is wrong, I read the material jbriggs provided on the wiki.
The clarified claim that you made and the counter-example are far simpler than that. No need to invoke Cantor.

You indicated that an infinite string that includes "1101001001" [sic] once must include it again infinitely many times. That's simply false. It would be true (with probability one) for an infinite string that is created "at random". But for an infinite string constructed according to a particular rule, it is simply false.

Example: 110100100100000000000...

Rule: The first 10 digits are "1101001001" and the rest are all zeroes.

The clarified claim that you made and the counter-example are far simpler than that. No need to invoke Cantor.

You indicated that an infinite string that includes "1101001001" [sic] once must include it again infinitely many times. That's simply false. It would be true (with probability one) for an infinite string that is created "at random". But for an infinite string constructed according to a particular rule, it is simply false.

Example: 110100100100000000000...

Rule: The first 10 digits are "1101001001" and the rest are all zeroes.
I think there has been a misunderstanding. I was using that as an example because it had already been provided. And when the guy said 'the counter example' I assumed he meant your cantor one,which helped(thanks).

When I say infinity I near infinity in its natural,random form, with no structure or rule. The binary stuff kept being used for some reason, I probably should have clarified. Maybe this will change things.

I thank you for all of your guys politeness which me, I must appear to be a village idiot. I am 19 and I have nothing more than Gcse's. Which will eventually be rectified. I plan on getting a degree in physics.

Staff Emeritus
When I say infinity I near infinity in its natural,random form, with no structure or rule
Now you're redefining things.

Now you're redefining things.
Where did I define it differently? In my original post I said infinity. I did not give it structure or rule. I assumed wrongly that infinity would be taken as it random form because of that. In fact I said "every conceivable number in every conceivable way" in a later post. I said 123412341234 because it was easier than infinityinfinityinfinity. And I said I had substituted it for this. And because of this I assumed that when I said infinite string, I assumed that would be taken as random too.

I apologise for this.

jbriggs444