The meaning of infinity

[untermensche]

Phil Scott didn't say anything about any "series". He talked about all the rationals between 0 and 1. That's a set and more specifically that's an interval, not a series.

The rationals appear on a line. They are a series along that line.

The line approaches 1 but does not ever get there.

It is not bounded in the sense that 1 is a final destination.

The series is unbounded and endless.

Thinking about an infinite series as a set is just an abstraction.

It does not change the series in one way.

You have nothing but some magical thinking about the concept of a set.

A set can be imagined to contain an infinite series but in reality no such series can be contained.

Decimal numbers, real numbers etc are ALL abstract concepts.

But if translated into reality you could have approximately half an apple. You could definitely have some unknown fraction of an apple. In theory you could have every fraction possible. There is a fraction too small to have.

But you can't have infinite apples.

They would not fit in infinite universes.

Infinite apples is apples without end.

That is not something you could ever have. Not even something you could imagine.

 

[Juma]

Decimal numbers, real numbers etc are ALL abstract concepts.

But if translated into reality you could have approximately half an apple. You could definitely have some unknown fraction of an apple. In theory you could have every fraction possible. There is a fraction too small to have.

But you can't have infinite apples.

They would not fit in infinite universes.

Infinite apples is apples without end.

That is not something you could ever have. Not even something you could imagine.

So what? There are a lot of finite numbers of apples you cant have. As for example n=10^2000.

 

[untermensche]

Why can't you in theory have that amount?

You can't in theory have infinite apples.

It is not an amount.

 

[Phil Scott]

Why can't you in theory have that amount?

"In theory" means in your fictional la-la land. 10^2000 apples is a story for idiots who can't do maff. You can't have 10^2000 of anything outside of made up stuff. Do you need 10^2000 apples to do pharmacy? Thought not.

 

[untermensche]

Where do you suppose infinity is?

Hint: It's called fantasyland.

Are you saying it is impossible?

Infinite apples is infinitely more.

 

[Phil Scott]

Where do you suppose infinity is?

Hint: It's called fantasyland.

Are you saying it is impossible?

Infinite apples is infinitely more.

Where do you suppose 10^2000 apples is? That's fantasy woo-woo too.

 

[Speakpigeon]

Phil Scott didn't say anything about any "series". He talked about all the rationals between 0 and 1. That's a set and more specifically that's an interval, not a series.

The rationals appear on a line. They are a series along that line.

The line approaches 1 but does not ever get there.

It is not bounded in the sense that 1 is a final destination.

The series is unbounded and endless.

Thinking about an infinite series as a set is just an abstraction.

It does not change the series in one way.

You have nothing but some magical thinking about the concept of a set.

A set can be imagined to contain an infinite series but in reality no such series can be contained.

Sorry, we're talking about the set of rational numbers between 0 and 1. No series.

Bounded by 1 means that rationals between 0 and 1 are all inferior to 1. Nothing else. Please don't make up stuff. No "final destination".

So the set of rational numbers between 0 and 1 is bounded by 0 and 1.

We're not discussing any series. It's all about sets. If you're not prepared to argue about sex, sorry, about sets, then go elsewhere talk about whatever.

If you think there's a series of all the rationals between 0 and 1, please provide the formal expression of it or shut up. Short of a formal expression, there's nothing to discuss, let alone whether the series only approaches or indeed reaches 1.

So the set of rational numbers between 0 and 1 is infinite and it is bounded by 0 and 1. So, it is both infinite, in a way, and bounded, in another way.
EB

 

[Bomb#20]

...It's not a computable probability because there's no algorithm for looking at an arbitrary program and figuring out if it ever goes into an infinite loop. We can compute the first few digits of one of these constants pretty easily because a random procedure is much more likely to generate a short program than a long one, and with short programs it's usually pretty easy to prove whether they go into an infinite loop. But the digits get harder and harder to compute as you go deeper into the constant, and eventually you get to a point where you can't find a way to get any more digits.

Yes, I was aware of Chaitin's Constant but I have to admit my brain isn't properly trained to deal with the explanation of it provided by Wiki. It all seems to rest on the Turing machine and the halting problem. Well, I'll wait to get an upgrade to try that.

Still, the way you explain it, it seems you've just explained how to compute it and we don't actually compute just because it would "get harder and harder", which seems to be a good allegory for computing infinite series of digits.
EB

The "harder and harder" isn't the operative constraint; the "you can't find a way to get any more digits" is. I haven't actually explained how to compute Chaitin's Constant -- the procedure I described relies on a machine being able to figure out whether another machine halts. There's no way to do that in the general case, even though we know how to write a program to check if another program halts that will work 99.99% of the time. Whether a given machine halts may depend on the answer to an unsolved math problem. For example, it's a piece of cake to write a program that will run forever only if the Goldbach Conjecture is true, and we don't know if it's true. The first twenty digits of Chaitin's Constant won't depend on whether that program halts, because the probability of randomly generating it is so low; but somewhere in the first thousand digits there's surely a digit that depends on whether that program halts. Now, no doubt there's an unknown program somewhere out there in possible-program-space that's a better mathematician than any human mathematician who ever lived, and maybe that program is able to solve the Goldbach Conjecture and compute Chaitin's Constant to more decimal places than we can. But that's just delaying the inevitable -- even that program will eventually hit a brick wall. There are other even harder math problems out there, that it too is unable to solve. There is no limit to the subtlety of mathematics. Some math problems are logically undecideable from any finitely-explainable set of reasoning procedures*. This is what Goedel proved.

(* Even plain vanilla algebra only on whole numbers actually relies on infinitely many axioms. But the reasoning system still counts as finite, because there's a finite procedure for testing whether a given formula is an axiom or not. So any alleged proof can be checked for correctness in a finite amount of time. That's what "finitely-explainable set of reasoning procedures" means.)

 

[untermensche]

Phil Scott didn't say anything about any "series". He talked about all the rationals between 0 and 1. That's a set and more specifically that's an interval, not a series.

The rationals appear on a line. They are a series along that line.

The line approaches 1 but does not ever get there.

It is not bounded in the sense that 1 is a final destination.

The series is unbounded and endless.

Thinking about an infinite series as a set is just an abstraction.

It does not change the series in one way.

You have nothing but some magical thinking about the concept of a set.

A set can be imagined to contain an infinite series but in reality no such series can be contained.

Sorry, we're talking about the set of rational numbers between 0 and 1. No series.

Bounded by 1 means that rationals between 0 and 1 are all inferior to 1. Nothing else. Please don't make up stuff. No "final destination".

So the set of rational numbers between 0 and 1 is bounded by 0 and 1.

We're not discussing any series. It's all about sets. If you're not prepared to argue about sex, sorry, about sets, then go elsewhere talk about whatever.

If you think there's a series of all the rationals between 0 and 1, please provide the formal expression of it or shut up. Short of a formal expression, there's nothing to discuss, let alone whether the series only approaches or indeed reaches 1.

So the set of rational numbers between 0 and 1 is infinite and it is bounded by 0 and 1. So, it is both infinite, in a way, and bounded, in another way.
EB

I know.

You think by some magic that calling an infinite series a set you have changed one thing about the series.

You still have the same problem of translating an infinite series into something real. You still have endless elements to contend with.

Your magic solution is so much rubbish!

It is an insult to the intelligence.

 

[Speakpigeon]

Note: just to be clear: the non-computability of such numbers is a property of the problem stated (that states which value we want to compute) not a property of numbers (as in the set of reals).

As far as I understand it, Chaitin is a way of identifying numbers that are not computable. So, non-computability is a property of those numbers unless you can explain why Chaitin doesn't work.
EB

I suspected that you should believe something like that. Thats why i wrote the post.
There is nothing special with the numbers themselves, its just a problem of identifying which number.

Problem?! You're not making sense here. Where would the problem be exactly?

Me, I think Chaitin is a solution, not a problem. If there's a problem, it is entirely yours: Whether you could prove Chaitin's method for identifying non-computable numbers doesn't work. If you can't do that, then where's the "problem" exactly?

It has nothing incommon with incommensrable numbers, transcendet numbers etc.

You're not helping yourself here. Where is it I asserted anything like what you suggest? We're talking about non-computable numbers. Who suggested commonality with transcendent numbers or whatever?

If that can help, I understand that the probability is overwhelmingly for a non-computable number to be transcendental. But it's not clear to me that the "definition" we have allows to infer that non-computable number are all necessarily transcendental. However, the definition I think leaves some room for interpretation between numbers actually non-computable versus numbers we don't know how to compute. In the latter case, we could have non-transcendental numbers that would be non-computable based on some identification. In the former case, all non-computable numbers would be all transcental, I think.
EB

 

[Speakpigeon]

Sorry, we're talking about the set of rational numbers between 0 and 1. No series.

Bounded by 1 means that rationals between 0 and 1 are all inferior to 1. Nothing else. Please don't make up stuff. No "final destination".

So the set of rational numbers between 0 and 1 is bounded by 0 and 1.

We're not discussing any series. It's all about sets. If you're not prepared to argue about sex, sorry, about sets, then go elsewhere talk about whatever.

If you think there's a series of all the rationals between 0 and 1, please provide the formal expression of it or shut up. Short of a formal expression, there's nothing to discuss, let alone whether the series only approaches or indeed reaches 1.

So the set of rational numbers between 0 and 1 is infinite and it is bounded by 0 and 1. So, it is both infinite, in a way, and bounded, in another way.
EB

I know.

You think by some magic that calling an infinite series a set you have changed one thing about the series.

You still have the same problem of translating an infinite series into something real. You still have endless elements to contend with.

Your magic solution is so much rubbish!

It is an insult to the intelligence.

Here we go again. You haven't lasted beyond ONE post! It's just coitus interruptus. Not very polite, that.

So, we've been there before...

You had all the time to explain yourself and you haven't. Countless people have died trying to get you to articulate something cogent. You're parroting yourself ad libitum. Nauseating. You never produce anything like an argument, a piece of knowledge, or even a piece of your own expertise. You're just parroting yourself without a fail. You have one absurd belief that infinity doesn't exist in the physical word and you have no proof whatsoever. You're only argument is to claim that the word "infinite" means without end, which is no even true, and if it was, it would still be an idiotic argument. It's a bit short to argue that infinity doesn't exist. You're attention span also is very short. No way anyone can have a rational debate with you. It always ends up in the same litany of your deadbeat mantra that an infinite past is impossible because "it would never complete".
Parrot.
EB

 

[untermensche]

Where do you suppose infinity is?

Hint: It's called fantasyland.

Are you saying it is impossible?

Infinite apples is infinitely more.

Where do you suppose 10^2000 apples is? That's fantasy woo-woo too.

I don't imagine them anywhere.

But it is an amount of apples. So while an astronomical amount still an amount. So in theory they may possibly exist somewhere.

Unlike infinite apples.

Not an amount.

Not something that could exist even in theory.

 

[untermensche]

Sorry, we're talking about the set of rational numbers between 0 and 1. No series.

Bounded by 1 means that rationals between 0 and 1 are all inferior to 1. Nothing else. Please don't make up stuff. No "final destination".

So the set of rational numbers between 0 and 1 is bounded by 0 and 1.

We're not discussing any series. It's all about sets. If you're not prepared to argue about sex, sorry, about sets, then go elsewhere talk about whatever.

If you think there's a series of all the rationals between 0 and 1, please provide the formal expression of it or shut up. Short of a formal expression, there's nothing to discuss, let alone whether the series only approaches or indeed reaches 1.

So the set of rational numbers between 0 and 1 is infinite and it is bounded by 0 and 1. So, it is both infinite, in a way, and bounded, in another way.
EB

I know.

You think by some magic that calling an infinite series a set you have changed one thing about the series.

You still have the same problem of translating an infinite series into something real. You still have endless elements to contend with.

Your magic solution is so much rubbish!

It is an insult to the intelligence.

Here we go again. You haven't lasted beyond ONE post! It's just coitus interruptus. Not very polite, that.

So, we've been there before...

You had all the time to explain yourself and you haven't. Countless people have died trying to get you to articulate something cogent. You're parroting yourself ad libitum. Nauseating. You never produce anything like an argument, a piece of knowledge, or even a piece of your own expertise. You're just parroting yourself without a fail. You have one absurd belief that infinity doesn't exist in the physical word and you have no proof whatsoever. You're only argument is to claim that the word "infinite" means without end, which is no even true, and if it was, it would still be an idiotic argument. It's a bit short to argue that infinity doesn't exist. You're attention span also is very short. No way anyone can have a rational debate with you. It always ends up in the same litany of your deadbeat mantra that an infinite past is impossible because "it would never complete".
Parrot.
EB

Your position is laughable.

You think if you call infinite elements a set you suddenly have less than infinite elements.

You somehow forget we are talking about translating something into the real world. You still have infinite elements in your set. You still have elements without end to contend with.

About as stupid a position as is possible.

Totally worthless shit!

And you are the only person trying to pass off this bad position.

 

[Speakpigeon]

...It's not a computable probability because there's no algorithm for looking at an arbitrary program and figuring out if it ever goes into an infinite loop. We can compute the first few digits of one of these constants pretty easily because a random procedure is much more likely to generate a short program than a long one, and with short programs it's usually pretty easy to prove whether they go into an infinite loop. But the digits get harder and harder to compute as you go deeper into the constant, and eventually you get to a point where you can't find a way to get any more digits.

Yes, I was aware of Chaitin's Constant but I have to admit my brain isn't properly trained to deal with the explanation of it provided by Wiki. It all seems to rest on the Turing machine and the halting problem. Well, I'll wait to get an upgrade to try that.

Still, the way you explain it, it seems you've just explained how to compute it and we don't actually compute just because it would "get harder and harder", which seems to be a good allegory for computing infinite series of digits.
EB

The "harder and harder" isn't the operative constraint; the "you can't find a way to get any more digits" is. I haven't actually explained how to compute Chaitin's Constant -- the procedure I described relies on a machine being able to figure out whether another machine halts. There's no way to do that in the general case, even though we know how to write a program to check if another program halts that will work 99.99% of the time. Whether a given machine halts may depend on the answer to an unsolved math problem. For example, it's a piece of cake to write a program that will run forever only if the Goldbach Conjecture is true, and we don't know if it's true. The first twenty digits of Chaitin's Constant won't depend on whether that program halts, because the probability of randomly generating it is so low; but somewhere in the first thousand digits there's surely a digit that depends on whether that program halts. Now, no doubt there's an unknown program somewhere out there in possible-program-space that's a better mathematician than any human mathematician who ever lived, and maybe that program is able to solve the Goldbach Conjecture and compute Chaitin's Constant to more decimal places than we can. But that's just delaying the inevitable -- even that program will eventually hit a brick wall. There are other even harder math problems out there, that it too is unable to solve. There is no limit to the subtlety of mathematics. Some math problems are logically undecideable from any finitely-explainable set of reasoning procedures*. This is what Goedel proved.

(* Even plain vanilla algebra only on whole numbers actually relies on infinitely many axioms. But the reasoning system still counts as finite, because there's a finite procedure for testing whether a given formula is an axiom or not. So any alleged proof can be checked for correctness in a finite amount of time. That's what "finitely-explainable set of reasoning procedures" means.)

So, it's not really non-computable. Rather, it's non-computable in all probability. So, logically computable but probably not in practice, somewhat like the non-zero probability for a particle to be anywhere in the universe. Non-measurable in any place, sort of. And yet it has to be somewhere when we do measure it.
EB

 

[Speakpigeon]

Here we go again. You haven't lasted beyond ONE post! It's just coitus interruptus. Not very polite, that.

So, we've been there before...

You had all the time to explain yourself and you haven't. Countless people have died trying to get you to articulate something cogent. You're parroting yourself ad libitum. Nauseating. You never produce anything like an argument, a piece of knowledge, or even a piece of your own expertise. You're just parroting yourself without a fail. You have one absurd belief that infinity doesn't exist in the physical word and you have no proof whatsoever. You're only argument is to claim that the word "infinite" means without end, which is no even true, and if it was, it would still be an idiotic argument. It's a bit short to argue that infinity doesn't exist. You're attention span also is very short. No way anyone can have a rational debate with you. It always ends up in the same litany of your deadbeat mantra that an infinite past is impossible because "it would never complete".
Parrot.
EB

Your position is laughable.

You think if you call infinite elements a set you suddenly have less than infinite elements.

You somehow forget we are talking about translating something into the real world. You still have infinite elements in your set. You still have elements without end to contend with.

About as stupid a position as is possible.

Totally worthless shit!

And you are the only person trying to pass off this bad position.

We've JUST been there before...

I spent the necessary time to ask you specific questions and you always failed to provide any appropriate answer. You never explained yourself. You failed to keep track of the debate. It's effectively impossible to have a rational conversation with you. Everybody ends up frustrated and flustered, and more importantly none the wiser. I even went so far as to explain to you what it was you could do, and in fact what you had to do to explain yourself. I'm still waiting for you even to acknowledge this post, let alone provide any sensible answer. I guess the main point is that you are intellectually devious. Never a straight answer. Never. Most of what you say comes out as irrelevant to what you pretend to be responding to. You just ignore the bits you don't like. It's not a debate. It's a sick joke. It's literally a sick joke.
EB

 

[untermensche]

Your position is a joke.

You have some delusion that calling an infinite series a set or calling anything with infinite elements a set you have reduced the number of elements or somehow allowed infinite elements to be expressed.

It is so stupid you should be ashamed to even say it.

That you don't know it is such a worthless position is a little strange. I often wonder what is at the other end of my comments.

 

[steve_bank]

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.

You can approach infinity but you can not get there. You are evading my question. 0.999... can only be evaluated at a finite number of terms. Hoe can an infinie, uncountable, number of terms magically = 1?

If what you say is true then 1 - [0.999...] = 0? Infinity is approached as a technique but we can not get there. Infinite and reaching infinity are mutually exclusive.

We say the limit as x ->(approaches) inf 1/x is zero. Infinity is approached but 1/x is not evaluated at infinity. The function never reaches exactly zero. Infinite and finite are mutually exclusive. You can never reach infinity. 0.999... has an infinite number of terms, meaning an uncountable number of terms.

What is 0.555... ? I might round to .6, .56 or .556 depending on calculations.
How about 0. 123123123...? what does that equal when you take it to infinity?

You can not resolve and infinite series like 0.999.. to a finite number, in this case 1.

 

[Speakpigeon]

Your position is a joke.

You have some delusion that calling an infinite series a set or calling anything with infinite elements a set you have reduced the number of elements or somehow allowed infinite elements to be expressed.

It is so stupid you should be ashamed to even say it.

That you don't know it is such a worthless position is a little strange. I often wonder what is at the other end of my comments.

We're not discussing series...
EB

 

[Speakpigeon]

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

You can approch infinity but you can not get there.

If space is infinitely divisible, which it may be for all we know, then I can only stand at a place which is an infinite number of points of space from your own place. So, I'm literally at infinity and there's nothing I can do not to just like I'm always at one centimetre of some place...

limit as x ->(approaches) inf 1/x is zero. Infinity is approached but 1/x is not evaluated at infinity. The function never reaches exactly zero. Infinite and finite are mutually exclusive. You can never reach infinity. 0.999... has an infinite number of terms, meaning an uncountable number of terms.

0.999... definitely has a countable set of digits.

That's something you shouldn't ignore!
EB

 

[steve_bank]

bomb

Make 0.999... 0.777...


So let x =77/10 + 7/100 + 7/1000 + ...

Therefore, 10x = 70/10 + 70/100 + 70/1000 + ...

Therefore, 10x = 7/1 + 7/10 + 7/100 + 7/1000 + ...

Therefore, 10x = 7/1 + x

Therefore, 7x = 7

Therefore, x = 1

So 0.777... = 0 + 1 ?

What you did is show the distributive property.

x = a + b
k*x = k*a + k*b
k*x = k*[a + b]
k/k = [a + b]/x
1 = [a + b]/x

You took a stab at it and it did not work, happens to everybody. Improving math skills in large part is trial and error. Failure leads to insight and new doors to walk through.