I What are the implications of infinity?

[david2]

hi,

say you have an infinite distance. is half of that distance also infinite? or 1/10000000000000 of that distance. and so on and so on.

i suspect it must be because if you say that half of infinity is for example 10000 km then infinty must be 20000 km, which it is not.

so, if this proposition (that every part of infinity is infinite) is true then i come to the following conclusion:

the universe cannot be infinite because we can move around. if every bit of infinity is infinite then it would not be possible to move let's say 10 cm. it would take forever.

am i missing something?

 

[jfizzix]

For starters, there are an infinite number of non-overlapping length intervals in one meter, but humans have no trouble crossing such distances.

 

[david2]

so these intervals in real life (not a moot point) must have a finite lenght. not?

 

[phinds]

the universe cannot be infinite because we can move around. if every bit of infinity is infinite then it would not be possible to move let's say 10 cm. it would take forever.

This canard has been around for a couple of thousand years. Google Zeno's Paradox.

 

[david2]

thx phinds , will check it out.

 

[mfb]

if every bit of infinity is infinite

It is not.
Infinity is not a real number. There are extensions of the real numbers where you can work with infinity as numbers. In these extensions, infinity divided by a finite number is (edit: ±) infinity. You cannot cross 10% of an infinite universe. But you don't have to do that to move 10 cm. Infinity divided by infinity is undefined.

 

[david2]

interesting

 

[Janosh89]

Riemann (misspelt?) used numbers and there correspondence ,gridwise , to quantify and even add infinities together.
thus the set of fractions _ 1/2 1/3 etc were not an unbounded infinity but the set of decimal equivalents were...

 

[jbriggs444]

Riemann (misspelt?) used numbers and there correspondence ,gridwise , to quantify and even add infinities together.
thus the set of fractions _ 1/2 1/3 etc were not an unbounded infinity but the set of decimal equivalents were...

The set of fractions 1/2, 1/3, etc is bounded above by 1/2 and below by zero. The set is clearly bounded. It is equally clear that the number of fractions in the set exceeds any finite bound. The same applies for the set of equivalent decimal numerals. (0.5, 0.333..., 0.25, 0.2, ... )

It is not clear what you are trying to say. Perhaps that the set of all rational fractions is countably infinite but that the set of all infinite decimal strings is uncountably infinite? [That's Cantor, not Riemann]

 

[Drakkith]

so these intervals in real life (not a moot point) must have a finite lenght. not?

That's right. When I walk to the store, that distance is finite and it takes me a finite amount of time to cross it. It's important to understand that infinity describes, what I like to call, a process (I'm sure mathematicians have a proper term for it though). If I walk to the store at 1 meter per second, then every second I go a finite distance of 1 meter. That's obvious of course. But, what happens if I start dividing this length into equal finite segments? Well, if I divide it into 2 equal segments, then every second I cross 2 segments of 1/2 meter each. The total distance is still the same, and it still takes the same amount of time to go 1 meter. Also, each segment has a different start and end point. If the start of segment 1 is at $x=10$ then the end of segment 1 is at $x=10.5$. Segment 2 would then go from 10.5 to 11.

Now, what happens if we keep increasing the number of segments? The length of each segment decreases and the number of segments per meter increases. So I'd have to cross more and more segments as I divide the total distance up into more and more pieces. Note that each segment still has a start and an end point, each is still finite in size, and the number of segments is still a regular old number.

But let's keep going. Let's keep dividing it up, further and further. Am I forced to stop at some point? Is there some number of segments that I suddenly cannot go above? No, there is not. I can divide my 1 meter segment into more and more pieces and the number of segments increases without end. Infinity describes this process. It is the concept that something can continue happening without end. In this case, I can continue to increase the number of segments in my 1 meter length to any arbitrary amount. I don't suddenly have to stop at 10⁵⁰ segments. I can continue to 10⁵⁰ + 1 and even beyond.

In this case, we say that there are an infinite number of segments and the behavior of these segments is that they get smaller and smaller as their length goes to zero. They become point-like. Another description of these segments is that they become infinitesimal in size. Note that infinitesimal is not zero any more than infinity is a number. It merely describes the behavior of something as it gets smaller and smaller. If there is no minimum size, then the length of any segment can anything. We can make their lengths as small as we want as long as it's non-zero.

Be aware that just because we've started with a finite number of line segments and made them smaller and smaller, this doesn't mean that reality is "discrete". As far as we know, things like length and distance are continuums, meaning that there is no inherent minimum length that something can be. In our math, in order to properly go from a discrete number of line segments, where each line segment is non-zero in size, to a continuum, we need concepts like infinity and infinitesimals.

That's my understanding at least.

 

[jbriggs444]

That's right. When I walk to the store, that distance is finite and it takes me a finite amount of time to cross it. It's important to understand that infinity describes, what I like to call, a process (I'm sure mathematicians have a proper term for it though).

The modern mathematical notion of infinity does not include the notion of a process. Roughly speaking, the idea of a process is the idea of a "potential infinity". As you suggest, this is the notion that no matter how far you go, you can go farther or no matter how finely you divide, you can divide more finely.

Gregor Cantor put a foundation under the notion of a completed infinity. This is the idea of a collection of infinitely many things. For instance the set of all the natural numbers has infinitely many members. We talk about this set as a fixed thing, not as a continuously incrementing process.

It can be a difficult thing to wrap one's head around. It took me about a week in my first formal course on real analysis before the idea gelled and I could stop thinking of the Peano axioms as describing an unending process and start thinking of them as defining the properties of a completed whole.

 

[david2]

hi again,

thank you all for your answers.It got me thinking.

I also saw some other threads about the infinity subject. very interesting albeit a bit difficult for me to grasp instantly.

btw this is a great site. very informative.

 

[phinds]

hi again,

thank you all for your answers.It got me thinking.

I also saw some other threads about the infinity subject. very interesting albeit a bit difficult for me to grasp instantly.

btw this is a great site. very informative.

For more fun with infinity, Google "Hilbert's Hotel"

 

[david2]

i will, thanks again

 

[mustang19]

The implications of infinity include:

1. 0.999... Equals 1
2. Infinity equals -1/12
3. The explosion principle applies to mathematics

 

[phinds]

The implications of infinity include:

1. 0.999... Equals 1
2. Infinity equals -1/12
3. The explosion principle applies to mathematics

Why is #3 a result of infinity? Do you contend that it does not exist WITHOUT infinity?

 

[WWGD]

...There is a bijection between an infinite set S and a proper/strict subset of S, e.g.., between the natural numbers and the even numbers.

 

[Drakkith]

2. Infinity equals -1/12

Infinity does not equal $-\frac{1}{12}$. What does equal $-\frac{1}{12}$ is the Riemann Zeta Function evaluated at $x=-1$: https://plus.maths.org/content/infinity-or-just-112

 

[sumar]

one question 10cm must be a percent of infinity right?
and if so does that mean that every time you move you are moving a percent of infinity?
at lest that's my understanding

 

[WWGD]

Not in the standard sense of the word; a is x% of y if (a/y)*100= x, but , in the standard Reals, infinity is not a number, so an expression (a/$\infty$) has no meaning, unless you "massage it" with limits.

 

[Mark44]

one question 10cm must be a percent of infinity right?

No.

and if so does that mean that every time you move you are moving a percent of infinity?

No. A percentage is a ratio. What you are saying essentially is that $\frac {10} \infty$ is some positive number, but that isn't a meaningful fraction.

As a limit, $\lim_{n \to \infty} \frac {10} n = 0$
No matter what specific number you put in the numerator, the limit is still zero.

at lest that's my understanding

 

[sumar]

Not in the standard sense of the word; a is x% of y if (a/y)*100= x, but , in the standard Reals, infinity is not a number, so an expression (a/$\infty$) has no meaning, unless you "massage it" with limits.

No.
No. A percentage is a ratio. What you are saying essentially is that $\frac {10} \infty$ is some positive number, but that isn't a meaningful fraction.

As a limit, $\lim_{n \to \infty} \frac {10} n = 0$
No matter what specific number you put in the numerator, the limit is still zero.

thanks

 

[sumar]

there are two types of infinity countable and uncountable right?

 

[WWGD]

there are two types of infinity countable and uncountable right?

No; remember Cantor's theorem $|A| < |2^{A}|$

EDIT: For the sake of completeness, likely overkill for the question of whether there are levels of infinity between countable and uncountable, the issue is undecidable, though by above theorem there are more than two levels of infinity..

 

[Mark44]

there are two types of infinity countable and uncountable right?

Your original question about whether 10 was some percentage of infinity has nothing to do with sets being either countably infinite or uncountably infinite.

 

[bahamagreen]

From Wiki - Thompson's Lamp
"Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of this infinite series of time intervals is exactly two minutes.
The following question is then considered: Is the lamp on or off at two minutes?"

 

[PeroK]

From Wiki - Thompson's Lamp
"Consider a lamp with a toggle switch. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of this infinite series of time intervals is exactly two minutes.
The following question is then considered: Is the lamp on or off at two minutes?"

This seems to me the sort of confusion between mathematics and physics/reality that leads nowhere. A lamp is not a mathematical object and cannot, even theoretically, be switched on/off an infinite number of times in any finite time interval.

Thompson's lamp simply confuses a pure mathematical object (an infinite sequence) with a real-world physical process (switching a lamp on and off) and is, therefore, nonsensical. Although, I imagine a good number of clever people have vexed themselves over it.

 

[bahamagreen]

This seems to me the sort of confusion between mathematics and physics/reality that leads nowhere. A lamp is not a mathematical object and cannot, even theoretically, be switched on/off an infinite number of times in any finite time interval.

Thompson's lamp simply confuses a pure mathematical object (an infinite sequence) with a real-world physical process (switching a lamp on and off) and is, therefore, nonsensical. Although, I imagine a good number of clever people have vexed themselves over it.

I was thinking of taking the sequence S = 1 − 1 + 1 − 1 + 1 − 1 + · · · and making a function so that each number indicates a y-axis value of either +/-1 and the x-axis position of each subsequent number is half as translated along the x-axis as the former, but it looks like there is a complication with restriction on the domain of x in the positive direction? Usually sequences as functions take the positive integers indexed to the numbers in the sequence as their domain, but I want to make the domain include the /2 rationals... can I do that without reference to the approached limit? I'm not seeing how being on the business side of a limit necessarily restricts the domain on the other side...

I'm not assigning time for the x-axis to prevent this from being a physical problem, but rather just a mathematical problem - x is just something, but what of x (and y) past the limit?

 

[PeroK]

I was thinking of taking the sequence S = 1 − 1 + 1 − 1 + 1 − 1 + · · · and making a function so that each number indicates a y-axis value of either +/-1 and the x-axis position of each subsequent number is half as translated along the x-axis as the former, but it looks like there is a complication with restriction on the domain of x in the positive direction? Usually sequences as functions take the positive integers indexed to the numbers in the sequence as their domain, but I want to make the domain include the /2 rationals... can I do that without reference to the approached limit? I'm not seeing how being on the business side of a limit necessarily restricts the domain on the other side...

I'm not assigning time for the x-axis to prevent this from being a physical problem, but rather just a mathematical problem - x is just something, but what of x (and y) past the limit?

A sequence is a mathematical object. Let's take two examples:

$1, 1/2, 1/4, 1/8 \dots$

This sequence has the property that it converges to a limit - in fact, it converges to 0. This is a mathematically well-defined property.

Note that the limit is not part of the sequence; it is a property of the entire sequence. In particular, it is not true that "the sequence eventually reaches 0 at infinity". This may be intuitively how non-mathematicians think of a limit. But, it is mathematically not sound.

However, when modelling a physical process by such a sequence, this allows you to say that eventually the physical process reduces to 0. To be more precise, you could say that eventually the process reduces to the point where it is no longer measurable and is in all practical terms 0.

Your sequence:

$1, 0, 1, 0 \dots$

Has the property that it does not converge. It has no limit.

If you model a physical process by this sequence, then there is no natural end state. There is no mathematical limit that could be assigned to the process once it has terminated.

This means that is you want to include the end state, then you must change the model somehow.

In any case, this says nothing paradoxical about either the mathematics or the physics. Simply that you don't have a fully viable model of the physical situation that includes an end state.

 

[Khashishi]

so, if this proposition (that every part of infinity is infinite) is true

This proposition is not true in number systems containing infinity. In the hyperreal number system, every infinite number has a multiplicative inverse. Following, any real number can be represented as a fraction of two infinite numbers. Therefore, your conclusions don't follow.

 

[bahamagreen]

A sequence is a mathematical object. Let's take two examples:

$1, 1/2, 1/4, 1/8 \dots$

This sequence has the property that it converges to a limit - in fact, it converges to 0. This is a mathematically well-defined property.

Note that the limit is not part of the sequence; it is a property of the entire sequence. In particular, it is not true that "the sequence eventually reaches 0 at infinity". This may be intuitively how non-mathematicians think of a limit. But, it is mathematically not sound.

However, when modelling a physical process by such a sequence, this allows you to say that eventually the physical process reduces to 0. To be more precise, you could say that eventually the process reduces to the point where it is no longer measurable and is in all practical terms 0.

Your sequence:

$1, 0, 1, 0 \dots$

Has the property that it does not converge. It has no limit.

If you model a physical process by this sequence, then there is no natural end state. There is no mathematical limit that could be assigned to the process once it has terminated.

This means that is you want to include the end state, then you must change the model somehow.

In any case, this says nothing paradoxical about either the mathematics or the physics. Simply that you don't have a fully viable model of the physical situation that includes an end state.

My sequence was 1, -1, 1, -1...
I'm putting it as a function where the count number of the individual numbers in the sequence are not counted as 1, 2, 3... but something like 1/1, 1/2, 1/3... so that although the numbers in the sequence do not have a limit, the count numbers do. In this kind of function where the numbers of the sequence are y and the numbers of the count (indexes) are x, this makes x approach a limit. The numbers of the sequence do not have a limit, but the counts do...

I think this is analogous to a version for Zeno's paradox where instead of counting the decreasing distance measures we count the runner's steps and have him run with increasingly smaller running strides of increasing frequency, and asking which foot hits the ground after the limit. In both the original and in this version of the paradox, Zeno does run through the limit, so asking about what happens past the limit is legitimate... still in the domain of x isn't it?

 

[jbriggs444]

run with increasingly smaller running strides of increasing frequency, and asking which foot hits the ground after the limit.

We should probably take the discussion of supertasks, convergence and infinite cardinalities to another thread.