Paradox!

[Archimedes]

Imagine spending an hour drawing a 100m straight line.

In the first half hour you draw a 50m segment of the line.

In the next quarter hour you draw a 25m segment of the line.

In the next 7.5 minutes you draw a 12.m segment of the line.

Etc. etc.

For each subsequently half of the previous time you draw a correspondingly segment of the line half as long as the previous segment.

How long is the line after an hour has passed?

It doesn't make any sense to me to say that an hour will never pass. Of course an hour will pass. An hour will pass exactly one hour after you started drawing the line. So how long is the line then?

What step of ball manipulation is he performing when it becomes Noon?

Instead of answering my question with a question, how about answering my question.

In the scenario I described above, how long is the line after an hour has passed?

If you start an hour timer when you start drawing the line, how long is the line when the timer beeps to indicate an hour is up. Nothing is slowing down or stopping the timer. It merely counts down a simple single hour of time.

So what length is the line after 1 hour has passed?

 

[steve_bnk]

Is anyone really arguing 0.999999 is actually 1? It aint.
.

No but 0.9999... (Where ... indicates a never ending repetition) = 1

When dealing with variables in equations on the order of 10^-12 or 10^-15 taking 0.999999999999 to be one can have consequences. Likewise in Zeno's Paradox and beeros problem saying in the limit the distance is effectively crossed is an incorrect answer.

 

[Deepak]

Is anyone really arguing 0.999999 is actually 1? It aint.
.

No but 0.9999... (Where ... indicates a never ending repetition) = 1

When dealing with variables in equations on the order of 10^-12 or 10^-15 taking 0.999999999999 to be one can have consequences. Likewise in Zeno's Paradox and beeros problem saying in the limit the distance is effectively crossed is an incorrect answer.

First it was the real numbers and countably infinite sets - but now we're losing the rational numbers too. Poor little 1/3.

Dare I mention that between two finite points that we have a set of prime numbers strictly smaller than the natural numbers, but extended to infinity there are a countably infinite number of primes and therefore the primes are exactly as large as the natural numbers?

 

[steve_bnk]

Is anyone really arguing 0.999999 is actually 1? It aint.
.

No but 0.9999... (Where ... indicates a never ending repetition) = 1

When dealing with variables in equations on the order of 10^-12 or 10^-15 taking 0.999999999999 to be one can have consequences. Likewise in Zeno's Paradox and beeros problem saying in the limit the distance is effectively crossed is an incorrect answer.

First it was the real numbers and countably infinite sets - but now we're losing the rational numbers too. Poor little 1/3.

Dare I mention that between two finite points that we have a set of prime numbers strictly smaller than the natural numbers, but extended to infinity there are a countably infinite number of primes and therefore the primes are exactly as large as the natural numbers?

You can not assign a finite number toan infinity.


I believe you to be just repeatingsomething you read without understanding the application.


Give me a concrete example of a countable infinity and the theory, or how the theory resolves either of the paradoxes in the thread to a finite answer.


You can have set theories and algebrasthat manipulate infinite sets, but you you can not have an a variable taken to infinity with a finite value.

 

[Deepak]

Is anyone really arguing 0.999999 is actually 1? It aint.
.

No but 0.9999... (Where ... indicates a never ending repetition) = 1

When dealing with variables in equations on the order of 10^-12 or 10^-15 taking 0.999999999999 to be one can have consequences. Likewise in Zeno's Paradox and beeros problem saying in the limit the distance is effectively crossed is an incorrect answer.

First it was the real numbers and countably infinite sets - but now we're losing the rational numbers too. Poor little 1/3.

Dare I mention that between two finite points that we have a set of prime numbers strictly smaller than the natural numbers, but extended to infinity there are a countably infinite number of primes and therefore the primes are exactly as large as the natural numbers?

You can not assign a finite number toan infinity.


I believe you to be just repeatingsomething you read without understanding the application.


Give me a concrete example of a countable infinity and the theory, or how the theory resolves either of the paradoxes in the thread to a finite answer.


You can have set theories and algebrasthat manipulate infinite sets, but you you can not have an a variable taken to infinity with a finite value.

I don't understand what it means to 'assign a finite number to an infinity'.

If you mean I can't state that Sum[((-1)^x+1)/x, {x, 1, Infinity}] is equal to the natural log of 2 then well I think that you're absolutely positively wrong.

A countable set is a set which can be put into a one-to-one correspondence with a finite subset of the natural numbers. A countably infinite set is any set which can be put into one-to-one correspondence with the natural numbers. So the {rational numbers, the prime numbers, Galileo's paradox, ... \(\omega\)example of countably infinite sets.

More to the point, I don't think you understand the implications of your statements - .999... not being one. This is an artifact of decimal notation which makes it impossible to represent specific numbers without requiring an infinite number of digits. See my example of 1/3. So tell me, does 1/3 not exist since '.333' extended an arbitrary number of digits (say '.' followed by 10^15 '3's) is not actually equal to 1/3 (for the people who accept the system of math used the world over .333...)?

 

[Tom Sawyer]

More to the point, I don't think you understand the implications of your statements - .999... not being one. This is an artifact of decimal notation which makes it impossible to represent specific numbers without requiring an infinite number of digits. See my example of 1/3. So tell me, does 1/3 not exist since '.333' extended an arbitrary number of digits (say '.' followed by 10^15 '3's) is not actually equal to 1/3 (for the people who accept the system of math used the world over .333...)?

It's a shorthand notation for it which is good enough in 99.999..N% of the situations where it's used and the infinte set can be used to represent the finite amount of digits without any problems.

Sometimes, however, you get situations like the "paradox" from this thread where there's an inconsistency between using an infinite set to represent a finite operation. It's not a paradox, it's a category error where you've run into one of the situations where using an infinite set to handle a finite equation doesn't work out. It's the same as how if you keep writing out 0.3333333333..., you never quite get to 1/3. Generally that's not a worry but if you run into one of the rare instances where it is, you haven't discovered a paradox, you've discovered one of the times where the shorthand isn't as useful as it usually is.

 

[Jimmy Higgins]

What step of ball manipulation is he performing when it becomes Noon?

What step of addition (0.9 + 0.09 + 0.009 + 0.009 + ....) is being performed when 0.999999... Becomes 1.0000000?

Or do you want to start arguing that 0.9999999999999... Isn't the same thing as 1.0000000?

I'm arguing, much like Sawyer is, that 0.9999... doesn't reach 1.00 in a physical process. The OP is mixing real world with mathematical world. In the real world, the number of balls in the vase is ever increasing. It only allegedly decreases once you start using fake infinite numbers.

 

[Deepak]

More to the point, I don't think you understand the implications of your statements - .999... not being one. This is an artifact of decimal notation which makes it impossible to represent specific numbers without requiring an infinite number of digits. See my example of 1/3. So tell me, does 1/3 not exist since '.333' extended an arbitrary number of digits (say '.' followed by 10^15 '3's) is not actually equal to 1/3 (for the people who accept the system of math used the world over .333...)?

It's a shorthand notation for it which is good enough in 99.999..N% of the situations where it's used and the infinte set can be used to represent the finite amount of digits without any problems.

Sometimes, however, you get situations like the "paradox" from this thread where there's an inconsistency between using an infinite set to represent a finite operation. It's not a paradox, it's a category error where you've run into one of the situations where using an infinite set to handle a finite equation doesn't work out. It's the same as how if you keep writing out 0.3333333333..., you never quite get to 1/3. Generally that's not a worry but if you run into one of the rare instances where it is, you haven't discovered a paradox, you've discovered one of the times where the shorthand isn't as useful as it usually is.

I disagree. .333, .333333333333333333333, and .333333333333333333333333333333333333333333333333333333333333333333333 are all shorthands for 1/3. The number .333... has an exact value and is the only way to accurately represent that number in base 10.

I could just as well point out that 1/2 or .5 represented in base 5 is actually .222...

This is merely an artifact of the representation of the number in a specific base, and in no way changes whether the value is definite and the properties of the number are preserved (that is to say that it is still a rational number capable of being represented by a fraction of two base 5 integers).

 

[Deepak]

What step of ball manipulation is he performing when it becomes Noon?

What step of addition (0.9 + 0.09 + 0.009 + 0.009 + ....) is being performed when 0.999999... Becomes 1.0000000?

Or do you want to start arguing that 0.9999999999999... Isn't the same thing as 1.0000000?

I'm arguing, much like Sawyer is, that 0.9999... doesn't reach 1.00 in a physical process. The OP is mixing real world with mathematical world. In the real world, the number of balls in the vase is ever increasing. It only allegedly decreases once you start using fake infinite numbers.

See my above post - base 10 decimal representations of numbers no more map to the 'real world' than any other.

x=.999...
10x = 9.999...
10x - x = 9
x = 1

 

[steve_bnk]

Is anyone really arguing 0.999999 is actually 1? It aint.
.

No but 0.9999... (Where ... indicates a never ending repetition) = 1

When dealing with variables in equations on the order of 10^-12 or 10^-15 taking 0.999999999999 to be one can have consequences. Likewise in Zeno's Paradox and beeros problem saying in the limit the distance is effectively crossed is an incorrect answer.

First it was the real numbers and countably infinite sets - but now we're losing the rational numbers too. Poor little 1/3.

Dare I mention that between two finite points that we have a set of prime numbers strictly smaller than the natural numbers, but extended to infinity there are a countably infinite number of primes and therefore the primes are exactly as large as the natural numbers?

You can not assign a finite number toan infinity.


I believe you to be just repeatingsomething you read without understanding the application.


Give me a concrete example of a countable infinity and the theory, or how the theory resolves either of the paradoxes in the thread to a finite answer.


You can have set theories and algebrasthat manipulate infinite sets, but you you can not have an a variable taken to infinity with a finite value.

I don't understand what it means to 'assign a finite number to an infinity'.

If you mean I can't state that Sum[((-1)^x+1)/x, {x, 1, Infinity}] is equal to the natural log of 2 then well I think that you're absolutely positively wrong.

A countable set is a set which can be put into a one-to-one correspondence with a finite subset of the natural numbers. A countably infinite set is any set which can be put into one-to-one correspondence with the natural numbers. So the {rational numbers, the prime numbers, Galileo's paradox, ... \(\omega\)example of countably infinite sets.

More to the point, I don't think you understand the implications of your statements - .999... not being one. This is an artifact of decimal notation which makes it impossible to represent specific numbers without requiring an infinite number of digits. See my example of 1/3. So tell me, does 1/3 not exist since '.333' extended an arbitrary number of digits (say '.' followed by 10^15 '3's) is not actually equal to 1/3 (for the people who accept the system of math used the world over .333...)?

I am well aware by experience of the issues of numerical analysis. I have used fractional arithmetic coded into microprocessor software. there is no such thing in general as an exact number.

1/3 does exist within measurement error. 1/3 of cake. But you illustrate the problem with 1/3.

Zeno's paradox without any physical limitations leads to an infinite sequence and diminishingly small but non zero remaining distances at each step.


Show me how the theory yields a finite answer to either of the paradoxes in the thread. Beero claims 42 balls for his problem. show me how the infinite steps of Zeno's paradox can be reduced to a finite number.

 

[Tom Sawyer]

More to the point, I don't think you understand the implications of your statements - .999... not being one. This is an artifact of decimal notation which makes it impossible to represent specific numbers without requiring an infinite number of digits. See my example of 1/3. So tell me, does 1/3 not exist since '.333' extended an arbitrary number of digits (say '.' followed by 10^15 '3's) is not actually equal to 1/3 (for the people who accept the system of math used the world over .333...)?

It's a shorthand notation for it which is good enough in 99.999..N% of the situations where it's used and the infinte set can be used to represent the finite amount of digits without any problems.

Sometimes, however, you get situations like the "paradox" from this thread where there's an inconsistency between using an infinite set to represent a finite operation. It's not a paradox, it's a category error where you've run into one of the situations where using an infinite set to handle a finite equation doesn't work out. It's the same as how if you keep writing out 0.3333333333..., you never quite get to 1/3. Generally that's not a worry but if you run into one of the rare instances where it is, you haven't discovered a paradox, you've discovered one of the times where the shorthand isn't as useful as it usually is.

I disagree. .333, .333333333333333333333, and .333333333333333333333333333333333333333333333333333333333333333333333 are all shorthands for 1/3. The number .333... has an exact value and is the only way to accurately represent that number in base 10.

I could just as well point out that 1/2 or .5 represented in base 5 is actually .222...

This is merely an artifact of the representation of the number in a specific base, and in no way changes whether the value is definite and the properties of the number are preserved (that is to say that it is still a rational number capable of being represented by a fraction of two base 5 integers).

Right, that's my point. It's an artifact representing it in a certain way. The number never "doesn't exist" or whatever that was about, it's just that some ways don't express it as accurately as other ways. You get situations such as "Sometimes decimal notation isn't as accurate a way of describing irrational numbers as using fractions is. One third is an example of this under base 10". If you keep adding a 3 to the right of the decimal point, you never actually get to 1/3rd, since there is no finite number which reaches it and the inaccuracy of using a decimal always leaves you with a small discrepency when comparing to 1/3. That difference is too trivial to worry about, however, so it's fine to simply express it as 0.333... and pretend that you wrote an infinite number of threes in order to account for this limitation of the system you're using. However, if you find yourself with a case where that difference does matter, you haven't found yourself a paradox, you've found yourself a case that falls within this small margin of error. The fact that 99.9999...N% of the cases where you'd use it don't run into that (but not 9/9 of the cases, since fractional notation isn't as accurate as using decimals to represent the small-but-non-zero chance of a discrepency) doesn't mean that something odd is happening during the few times that you do.

It's the same thing with the "paradox" of this thread. Generally, an ever-increasing series of finite numbers can be expressed as an infinite set without any worry because the difference between the two is too trivial to care about and it doesn't affect your calculations. If you find yourslef in oen of the rare cases which fall within the margin of error caused by the limitations of using this method, you haven't found a paradox, you've found a category error where representing the finite series with an infinite set wasn't correct.

 

[steve_bnk]

What step of ball manipulation is he performing when it becomes Noon?

What step of addition (0.9 + 0.09 + 0.009 + 0.009 + ....) is being performed when 0.999999... Becomes 1.0000000?

Or do you want to start arguing that 0.9999999999999... Isn't the same thing as 1.0000000?

I'm arguing, much like Sawyer is, that 0.9999... doesn't reach 1.00 in a physical process. The OP is mixing real world with mathematical world. In the real world, the number of balls in the vase is ever increasing. It only allegedly decreases once you start using fake infinite numbers.

See my above post - base 10 decimal representations of numbers no more map to the 'real world' than any other.

x=.999...
10x = 9.999...
10x - x = 9
x = 1

10 X 0.999... is not defined in algebra. You can not algebraically add, subtract, multiply, and divide infinites.

if a variable results in 0.9999... you must pick a finite truncation point. In numerical methods the truncation point can affect error propagation.

x = .999
10X = 9.99
10X - x = 9.991
x(10 - 1) = 9.991
x = .991

In a real world calculation the number of digits for .999.. is selected to maintain the required number of significant digits in the results. in this case if all that is needed is auacury to xx.x it works.

 

[Jimmy Higgins]

What step of ball manipulation is he performing when it becomes Noon?

What step of addition (0.9 + 0.09 + 0.009 + 0.009 + ....) is being performed when 0.999999... Becomes 1.0000000?

Or do you want to start arguing that 0.9999999999999... Isn't the same thing as 1.0000000?

I'm arguing, much like Sawyer is, that 0.9999... doesn't reach 1.00 in a physical process. The OP is mixing real world with mathematical world. In the real world, the number of balls in the vase is ever increasing. It only allegedly decreases once you start using fake infinite numbers.

See my above post - base 10 decimal representations of numbers no more map to the 'real world' than any other.

x=.999...
10x = 9.999...
10x - x = 9
x = 1

But at what point does the person with his balls in the vase get to Noon?

 

[Archimedes]

I'm arguing, much like Sawyer is, that 0.9999... doesn't reach 1.00 in a physical process. The OP is mixing real world with mathematical world. In the real world, the number of balls in the vase is ever increasing. It only allegedly decreases once you start using fake infinite numbers.

But the numbers 0.999...
or 0.33333.... or 0.142857142857142857... don't refer to physical processes. The dots indicating recurring decimals do not mean "keep adding these digits forever after the dots". They mean "there are an infinite amount of these digits after the dots".

The dots means there IS an infinite amount of 3s or 9s after that point, not that there WILL BE when you get round to typing them all out.

- - - Updated - - -

But at what point does the person with his balls in the vase get to Noon?

When the clock strikes 12.

 

[Deepak]

What step of ball manipulation is he performing when it becomes Noon?

What step of addition (0.9 + 0.09 + 0.009 + 0.009 + ....) is being performed when 0.999999... Becomes 1.0000000?

Or do you want to start arguing that 0.9999999999999... Isn't the same thing as 1.0000000?

I'm arguing, much like Sawyer is, that 0.9999... doesn't reach 1.00 in a physical process. The OP is mixing real world with mathematical world. In the real world, the number of balls in the vase is ever increasing. It only allegedly decreases once you start using fake infinite numbers.

See my above post - base 10 decimal representations of numbers no more map to the 'real world' than any other.

x=.999...
10x = 9.999...
10x - x = 9
x = 1

10 X 0.999... is not defined in algebra. You can not algebraically add, subtract, multiply, and divide infinites.

if a variable results in 0.9999... you must pick a finite truncation point. In numerical methods the truncation point can affect error propagation.

x = .999
10X = 9.99
10X - x = 9.991
x(10 - 1) = 9.991
x = .991

In a real world calculation the number of digits for .999.. is selected to maintain the required number of significant digits in the results. in this case if all that is needed is auacury to xx.x it works.

A baker is contracted to make 9 cakes for a party. His trainee accidentally drops 6 cakes loading them into the delivery van. What is the ratio of delivered cakes to ordered cakes?

.333...

A contractor brings 300 bricks to a worksite. 150 bricks are used to build a barbecue in his client's back yard. Expressed in base 5 what is the ratio of remaining bricks to the starting amount of bricks?

.222...

I can't disabuse you of any neopythagorean hangups you may have about math, but the limitations of your chosen applied math field do not carry to all math.

If I define x to equal .999... or .333... there's no ambiguity about what the value is, which is distinct from identifying a finite number of digits as a repeating decimal. If the value is defined or calculated then I have no qualms about using the operations I have. Just like the conditional convergence example using the harmonic series.

But at what point does the person with his balls in the vase get to Noon?

Like Archimedes said - at noon. That is to say, at step \(\omega\)

 

[steve_bnk]

1/3 is an infinite process by repeated long division.

3 goes into 1.0 .3 times. ...

 

[Deepak]

1/3 is an infinite process by repeated long division.

3 goes into 1.0 .3 times. ...

What is the ratio of delivered cakes to ordered cakes?

 

[Artemus]

If two real numbers are inequal there must be a third real number that lies between the two in value. What number lies between 0.999... and 1?

0.999... = 1

 

[Deepak]

If two real numbers are inequal there must be a third real number that lies between the two in value. What number lies between 0.999... and 1?

0.999... = 1

An infinite number of zeros followed by a 1. See they're not equal!

 

[steve_bnk]

What step of ball manipulation is he performing when it becomes Noon?

What step of addition (0.9 + 0.09 + 0.009 + 0.009 + ....) is being performed when 0.999999... Becomes 1.0000000?

Or do you want to start arguing that 0.9999999999999... Isn't the same thing as 1.0000000?

I'm arguing, much like Sawyer is, that 0.9999... doesn't reach 1.00 in a physical process. The OP is mixing real world with mathematical world. In the real world, the number of balls in the vase is ever increasing. It only allegedly decreases once you start using fake infinite numbers.

See my above post - base 10 decimal representations of numbers no more map to the 'real world' than any other.

x=.999...
10x = 9.999...
10x - x = 9
x = 1

10 X 0.999... is not defined in algebra. You can not algebraically add, subtract, multiply, and divide infinites.

if a variable results in 0.9999... you must pick a finite truncation point. In numerical methods the truncation point can affect error propagation.

x = .999
10X = 9.99
10X - x = 9.991
x(10 - 1) = 9.991
x = .991

In a real world calculation the number of digits for .999.. is selected to maintain the required number of significant digits in the results. in this case if all that is needed is auacury to xx.x it works.

A baker is contracted to make 9 cakes for a party. His trainee accidentally drops 6 cakes loading them into the delivery van. What is the ratio of delivered cakes to ordered cakes?

.333...

A contractor brings 300 bricks to a worksite. 150 bricks are used to build a barbecue in his client's back yard. Expressed in base 5 what is the ratio of remaining bricks to the starting amount of bricks?

.222...

I can't disabuse you of any neopythagorean hangups you may have about math, but the limitations of your chosen applied math field do not carry to all math.

If I define x to equal .999... or .333... there's no ambiguity about what the value is, which is distinct from identifying a finite number of digits as a repeating decimal. If the value is defined or calculated then I have no qualms about using the operations I have. Just like the conditional convergence example using the harmonic series.

But at what point does the person with his balls in the vase get to Noon?

Like Archimedes said - at noon. That is to say, at step \(\omega\)

10 x 0.999... is not defined in algebra as you attempted to do. You can make up any definitions you like, but your example resulting in 0.999.. = 1 is not algebraically correct. Quantitatively 0.999... has no meaning.

algebraically correct-

i = .999, .9999, .99999..

j = 10 x.
k = 10x -x
l = k/10

Defines l = .991, .9991. .99991....


Bases? hex and binary are second nature.

You still have not have provided a quantitative answer to beero's problem with your infinites.. either that or say there is no finite answer.

1/3 = .333... ok, but so what? How does that apply to the op?

(1/3) * 5 = ?

.333... x 5 has no value. .3 x 5 = 1.5, .33 x 5 = 1.65, .333 x 5 = 1.665