The meaning of infinity

[steve_bank]

No. Numbers can have infinite decimals and still be computable. If there is a program that can compute the decimal to arbitrary precision, then the number is said to be computable.

And to correct an earlier post of yours: 0.999... is a real number. It denotes the limit of a particular geometric series, and that limit is 1.

How does 0.999... resolve to 1 as a limit?

 

[Kharakov]

All anyone needs to know about infinity is pi/2 and a natural log of #2 are at certain points of a certain infinite series....

x is the positive real root of x^n-x=1 as n-->infinity, x-->1 as n-->infinity x^n-->2 k= number of radicals

If you start with... well, as close to #1 as you can get with this set of equations, you get a natural log of #2.
\({log(2)}=\lim_{k\to\infty} \lim_{n\to\infty} \,\, \left(nx^{n-1}\right)^k \,\, \times \,\, \left(x-\sqrt[n]{1+\sqrt[n]{1+\sqrt[n]{1+\dots}}} \right)\)

If you start as close to #2s as possible (you can use #2)... you get (pi/#2)^#2 (pi over #2 to #2) at the other end (n = closest integer to 1)

n=#2, as does x (positive real root x^2-x=2) k=number of radicals
\({\frac{\pi}{2}}^2= \lim_{k\to\infty} \,\, (2^2)^{k} \,\, \times \,\, \left(2-\sqrt[2]{2+\sqrt[2]{2+\sqrt[2]{2+\dots}}} \right)\)


What is the etymology of this exceedingly simple joke about #1?

 

[Juma]

No. Numbers can have infinite decimals and still be computable. If there is a program that can compute the decimal to arbitrary precision, then the number is said to be computable.

Then you are not using same definition as speakpidgeon.

And to correct an earlier post of yours: 0.999... is a real number. It denotes the limit of a particular geometric series, and that limit is 1.

I have never said that they are different real numbers. They are different decimals though. (Sloppy languague)

 

[Juma]

There is nothing remarkable with ”uncomputable numbers”.

In a world that's becoming more digital by the picosecond I think it's a very, very remarkable notion.

And, just conceivably, if ever there's anything physical that would somehow feature a non-computable quantity, then it would imply that the universe itself is not computable. Some knowledgeable folks seem to hold that this would be a real puzzler. Parochially remarkable, if you like.

Its simply a result of the fact that some numbers have infinitely many decimals: you cannot print infinitely many decimals in finite time.

That's not the point. Such trivially computable numbers as 1/3 or 17/61 do have infinite decimal expansions that you could not all print. Computable means that you can compute it, not that you can do it within a finite time span.

This is similar to the notion of countable set, which really means that you can count the elements of the set but maybe not all within a finite time span. Some sets are not only countable, but also counted, like all integers between 1 and 10 for example.

And all numbers that have been computed are computable but the reverse isn't true.

Talk of computability proves faith in the meaningfulness of the notion of the infinite. Talk of non-computability proves faith in the reality of infinities.
EB

The ”proof” doesnt show that there more ”non-comoutable numbers ” than I told you so you would have to find some better example.

 

[steve_bank]

phil scot is correct. Some numbers and results of calculation end up with an infinite number of digits. In a calculation the number of digits is determined by the overall required precision of a calculation.

1/3 = 0.333... In a chain calculation 0.3 x 0.746 is not the same as 0.3333333 x 0.746.

 

[Bomb#20]

No. Numbers can have infinite decimals and still be computable. If there is a program that can compute the decimal to arbitrary precision, then the number is said to be computable.

And to correct an earlier post of yours: 0.999... is a real number. It denotes the limit of a particular geometric series, and that limit is 1.

How does 0.999... resolve to 1 as a limit?

"3.14" is simply a terse notation for a finite series -- it means "3 + 1/10 + 4/100". With me so far?

So "0.999..." is the same terse notation but this time for an infinite series -- it means "0 + 9/10 + 9/100 + 9/1000 + ...".

So let x = 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 90/10 + 90/100 + 90/1000 + ...

Therefore, 10x = 9/1 + 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 9/1 + x

Therefore, 9x = 9

Therefore, x = 1

So 0.999... = 0 + 1

 

[steve_bank]

No. Numbers can have infinite decimals and still be computable. If there is a program that can compute the decimal to arbitrary precision, then the number is said to be computable.

And to correct an earlier post of yours: 0.999... is a real number. It denotes the limit of a particular geometric series, and that limit is 1.

How does 0.999... resolve to 1 as a limit?

"3.14" is simply a terse notation for a finite series -- it means "3 + 1/10 + 4/100". With me so far?

So "0.999..." is the same terse notation but this time for an infinite series -- it means "0 + 9/10 + 9/100 + 9/1000 + ...".

So let x = 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 90/10 + 90/100 + 90/1000 + ...

Therefore, 10x = 9/1 + 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 9/1 + x

Therefore, 9x = 9

Therefore, x = 1

So 0.999... = 0 + 1

An out of use but applicable term, gobbledygook. x = 0.999... No matter how many terms you use 1/x will never = 1. As the number of terms get large it will approach 1 asymptoticaly but it will be > 1. Take your calculator and calculate 1/0.9, 1/0.99,1/0.999... to the max digits on your calculator and see what happens. All arithmetic is finite. I nelieve entropy also applies to math. In every calculaion there can be loss of information. Rounding and truncation. Entropy as defined by Shannon in Information Theory.


https://www.merriam-webster.com/dictionary/gobbledygook.

Definition of gobbledygook
: wordy and generally unintelligible jargon

 

[Underseer]

Does anything in math "mean" anything if we don't actually apply it to something in the real world (e.g. physics)? I thought it was all abstractions. Isn't that the fun of it?

Anyway, the funniest thing about infinity and meaning are William Lane Craig's various arguments against "actual infinity," which get pretty hilarious:



I'm beginning to suspect mister Craig never took a freshman calculus class, because his understanding of infinity seems to be around jr. high level math.

Although I must confess that his attempts to disprove Relativity were probably funnier.

 

[Underseer]

"3.14" is simply a terse notation for a finite series -- it means "3 + 1/10 + 4/100". With me so far?

So "0.999..." is the same terse notation but this time for an infinite series -- it means "0 + 9/10 + 9/100 + 9/1000 + ...".

So let x = 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 90/10 + 90/100 + 90/1000 + ...

Therefore, 10x = 9/1 + 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 9/1 + x

Therefore, 9x = 9

Therefore, x = 1

So 0.999... = 0 + 1

An out of use but applicable term, gobbledygook. x = 0.999... No matter how many terms you use 1/x will never = 1. As the number of terms get large it will approach 1 asymptoticaly but it will be > 1. Take your calculator and calculate 1/0.9, 1/0.99,1/0.999... to the max digits on your calculator and see what happens. All arithmetic is finite. I nelieve entropy also applies to math. In every calculaion there can be loss of information. Rounding and truncation. Entropy as defined by Shannon in Information Theory.


https://www.merriam-webster.com/dictionary/gobbledygook.

Definition of gobbledygook
: wordy and generally unintelligible jargon

Holy fuck, are you for real? If you think it's less than one, then you don't understand what infinity is at all.

It is one.

The decimal number system just has certain numbers that can be represented in more than one way. It doesn't matter if you can't wrap your head around that. Do we really have to go through this idiotic conversation yet again? And I thought Craig's arguments were bad.

 

[DBT]

Infinity: something without boundary in time or space.....as a definition of ''infinite'' - good enough for me.

 

[steve_bank]

"3.14" is simply a terse notation for a finite series -- it means "3 + 1/10 + 4/100". With me so far?

So "0.999..." is the same terse notation but this time for an infinite series -- it means "0 + 9/10 + 9/100 + 9/1000 + ...".

So let x = 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 90/10 + 90/100 + 90/1000 + ...

Therefore, 10x = 9/1 + 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 9/1 + x

Therefore, 9x = 9

Therefore, x = 1

So 0.999... = 0 + 1

An out of use but applicable term, gobbledygook. x = 0.999... No matter how many terms you use 1/x will never = 1. As the number of terms get large it will approach 1 asymptoticaly but it will be > 1. Take your calculator and calculate 1/0.9, 1/0.99,1/0.999... to the max digits on your calculator and see what happens. All arithmetic is finite. I nelieve entropy also applies to math. In every calculaion there can be loss of information. Rounding and truncation. Entropy as defined by Shannon in Information Theory.


https://www.merriam-webster.com/dictionary/gobbledygook.

Definition of gobbledygook
: wordy and generally unintelligible jargon

Holy fuck, are you for real? If you think it's less than one, then you don't understand what infinity is at all.

It is one.

The decimal number system just has certain numbers that can be represented in more than one way. It doesn't matter if you can't wrap your head around that. Do we really have to go through this idiotic conversation yet again? And I thought Craig's arguments were bad.

Infinity is not a number. limit x-> inf 1/[2 +1/x] is 0.5 but never gets there.For practical purposes I would use 0.5. Infinities are introduced in calculus.

In electronics models taking infinite limits is common to reduce complexity in some cases. Parts of equations may evaluate to zero in some cases. Well familiar with infinies.


Again show me where any number of digits ofn 0.999.. ever result in 1. I gave you aa real example. It is a function of numbers and finite arithmetic.



For 1/x for any number of digits of 0.999.. the result will never ever be 1. In the limit as the number of terms go to infinity it will always be > 1 however small the difference may be. To make your case that 0.999... = 1 you must refute my example.

At some point for all practical reasons 0.9999.. becomes 1 as the difference is too small to affect anything.

Is there a number of terms for 0.999... for which 1 - the series ever equals zero? Another way to look at it.

 

[Juma]

Holy fuck, are you for real? If you think it's less than one, then you don't understand what infinity is at all.

It is one.

The decimal number system just has certain numbers that can be represented in more than one way. It doesn't matter if you can't wrap your head around that. Do we really have to go through this idiotic conversation yet again? And I thought Craig's arguments were bad.

Infinity is not a number. limit x-> inf 1/[2 +1/x] is 0.5 but never gets there.For practical purposes I would use 0.5. Infinities are introduced in calculus.

In electronics models taking infinite limits is common to reduce complexity in some cases. Parts of equations may evaluate to zero in some cases. Well familiar with infinies.


Again show me where any number of digits ofn 0.999.. ever result in 1. I gave you aa real example. It is a function of numbers and finite arithmetic.



For 1/x for any number of digits of 0.999.. the result will never ever be 1. In the limit as the number of terms go to infinity it will always be > 1 however small the difference may be. To make your case that 0.999... = 1 you must refute my example.

At some point for all practical reasons 0.9999.. becomes 1 as the difference is too small to affect anything.

Is there a number of terms for 0.999... for which 1 - the series ever equals zero? Another way to look at it.

No. There isnt a finite numbrer of terms. Since the 0.999... has an infinite number of terms.

 

[Speakpigeon]

In a world that's becoming more digital by the picosecond I think it's a very, very remarkable notion.

And, just conceivably, if ever there's anything physical that would somehow feature a non-computable quantity, then it would imply that the universe itself is not computable. Some knowledgeable folks seem to hold that this would be a real puzzler. Parochially remarkable, if you like.



That's not the point. Such trivially computable numbers as 1/3 or 17/61 do have infinite decimal expansions that you could not all print. Computable means that you can compute it, not that you can do it within a finite time span.

This is similar to the notion of countable set, which really means that you can count the elements of the set but maybe not all within a finite time span. Some sets are not only countable, but also counted, like all integers between 1 and 10 for example.

And all numbers that have been computed are computable but the reverse isn't true.

Talk of computability proves faith in the meaningfulness of the notion of the infinite. Talk of non-computability proves faith in the reality of infinities.
EB

The ”proof” doesnt show that there more ”non-comoutable numbers ” than I told you so you would have to find some better example.

Sorry, it comes out as all garbled on my computer screen I can't make out what you say here.

Still,

Show

I can't compute what you say.

Some hacker's idea of a good joke, I guess. Or something else.

What kind of keyboard do you use?
EB

 

[Speakpigeon]

Infinity: something without boundary in time or space.....as a definition of ''infinite'' - good enough for me.

Infinity is not necessarily something "without boundary".

Never was.
EB

 

[DBT]

Infinity: something without boundary in time or space.....as a definition of ''infinite'' - good enough for me.

Infinity is not necessarily something "without boundary".

Never was.
EB

Not necessarily?

 

[Speakpigeon]

No. Numbers can have infinite decimals and still be computable. If there is a program that can compute the decimal to arbitrary precision, then the number is said to be computable.

And to correct an earlier post of yours: 0.999... is a real number. It denotes the limit of a particular geometric series, and that limit is 1.

How does 0.999... resolve to 1 as a limit?

"3.14" is simply a terse notation for a finite series -- it means "3 + 1/10 + 4/100". With me so far?

So "0.999..." is the same terse notation but this time for an infinite series -- it means "0 + 9/10 + 9/100 + 9/1000 + ...".

So let x = 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 90/10 + 90/100 + 90/1000 + ...

Therefore, 10x = 9/1 + 9/10 + 9/100 + 9/1000 + ...

Therefore, 10x = 9/1 + x

Therefore, 9x = 9

Therefore, x = 1

So 0.999... = 0 + 1

A kind of Hilbert's Hotel where all the guests would have to be dressed to the nines.

Obviously,

Show

some people won't get the joke.

EB

 

[Speakpigeon]

Infinity: something without boundary in time or space.....as a definition of ''infinite'' - good enough for me.

Infinity is not necessarily something "without boundary".

Never was.
EB

Not necessarily?

There might be things infinite that would be "without boundary". There might be other things, however, and just as obviously, that would be infinite in some way and yet finite, or not "without a boundary", in some other way. So the notion of "boundary" is a red herring here.
EB

 

[DBT]

Not necessarily?

There might be things infinite that would be "without boundary". There might be other things, however, and just as obviously, that would be infinite in some way and yet finite, or not "without a boundary", in some other way. So the notion of "boundary" is a red herring here.
EB

I'm not sure how something can be both ''infinite in some way and yet finite''

 

[Speakpigeon]

Not necessarily?

There might be things infinite that would be "without boundary". There might be other things, however, and just as obviously, that would be infinite in some way and yet finite, or not "without a boundary", in some other way. So the notion of "boundary" is a red herring here.
EB

I'm not sure how something can be both ''infinite in some way and yet finite''

Very easy. Most things we think of as examples of infinity are bounded in some way.

It's much more difficult to come up with examples of things that would be infinite without any boundary at all. I only know of two examples.
EB

 

[Phil Scott]

Then you are not using same definition as speakpidgeon.

Well, I'm not sure he's got a definition! But he's now looking at the correct wikipedia page, which leads to the correct definition, and by that, uncomputability isn't about the mere fact that some decimals are infinite.

I have never said that they are different real numbers. They are different decimals though. (Sloppy languague)

Ack. Sorry. Total misattribution on my part there.