Rational numbers == infinitely repeating sequences of digits

[lpetrich]

Something that goes on without end cannot have a final value.

Except that there is a theory of such final values:  Limit (mathematics). If an infinite sequence has a limit, then one can get arbitrarily close to it by going far enough in the sequence.

 

[bilby]

It is how infinite digits is defined.

It is digits without the possibility of a last digit.

Digits that go on without end.

A number has a value.

Something that goes on without end cannot have a final value.

0.9999... approaches 1 but it never has a final value so it does not equal 1.

You're really, really confident about that.

But still completely wrong.

Funny that. It's a common affliction of the faithful.

Why do you imagine that it takes a non-zero amount of time to evaluate the '...' in the expression '0.333...'?

Hint: It doesn't. The value is 1/3. That didn't take an infinite time to evaluate - I just did it in significantly less than a second.

Numbers have a value. 'final value' is an object of your faith, and is pure fiction. Nobody but you feels the need to invent the phrase 'final value'. There's just value. It's totally independent of time.

 

[Jokodo]

Again. Australia is a label of something real.

"Three" is not.

"Three" is not some other thing the label is pointing to.

It describes nothing else but itself.

It may have some other things it is mathematically similar to but "three" or "3" is only "three" or "3".

It is not something else. There is nothing else for it to be. There is nothing behind the label like "Australia".

It. Does. Not. Matter.

Numbers may or may not be real, under whatever definition of "real" you may have in mind. Things don't need to be real in order to be distinct from their labels. The character Gandalf and the name "Gandalf" are not the same thing despite the fact that both where created the self-same instant by J. R. R. Tolkien. The fact that Gandalf isn't real doesn't make it so that Gandalf and Tharkûn, as he's known to the Dwarves, are distinct entities.

Your argument that "3", "2.9..", "0b11" (i. e., binary 3) and the English word "three" are distinct entities - and must be because one has repeating digits while the others don't - is like saying that Tharkûn and Gandalf are different characters, and must be because one of them contains non-ASCII characters your phone display correctly.

There is something behind the word "Gandalf" even if it is a fictional character.

There is nothing behind the word "three", nothing behind the character "3".

Sure there is. A mathematical concept. Whether mathematical concepts are themselves fictional doesn't enter consideration - you've just admitted that fictional entities too are distinct from their labels with "Gandalf" != Gandalf.

 

[Jokodo]

As I'd posted earlier, untermensche seems to advocate  finitism. I don't find finitism very convincing, because I don't see any fundamental difference between a rule that generates all the elements of a finite set and one that generates all the elements of an infinite set.

A rule is not the same thing as the product of the rule.

In fact they are entirely separate.

1/3 is really a rule not a value.

"1/3" is a string containing two digits and a punctuation character. In the language of fractional notation, it refers to a number, the same number that's referred to by "0.4" in base 12, "0.333..." In Base 10, "0.010101..." In base 2. Semiology 101. Freshen up on your Saussure.

 

[untermensche]

There is something behind the word "Gandalf" even if it is a fictional character.

There is nothing behind the word "three", nothing behind the character "3".

Sure there is. A mathematical concept. Whether mathematical concepts are themselves fictional doesn't enter consideration - you've just admitted that fictional entities too are distinct from their labels with "Gandalf" != Gandalf.

It is defined.

That is how it exists. It does not exist before that.

I have said many times that all things in mathematics are defined. None of it exists without a human mind capable of having thoughts using conceptions.

But Gandalf is a person on the screen.

There is nothing but the definition of three.

You can't have "three" slap Harry Potter in a movie or even appear in a movie. All you could ever show is the symbol pointing to the definition. As if all you could show on screen of Gandalf were the letters that make up his name.

 

[untermensche]

As I'd posted earlier, untermensche seems to advocate  finitism. I don't find finitism very convincing, because I don't see any fundamental difference between a rule that generates all the elements of a finite set and one that generates all the elements of an infinite set.

A rule is not the same thing as the product of the rule.

In fact they are entirely separate.

1/3 is really a rule not a value.

"1/3" is a string containing two digits and a punctuation character. In the language of fractional notation, it refers to a number, the same number that's referred to by "0.4" in base 12, "0.333..." In Base 10, "0.010101..." In base 2. Semiology 101. Freshen up on your Saussure.

1/3 does not refer to anything but itself. There may be things that are equivalent to it but it does not refer to them. It refers only to itself.

It is also an operation that when carried out produces a different number.

 

[untermensche]

Why do you imagine that it takes a non-zero amount of time to evaluate the '...' in the expression '0.333...'?

Giving an equation that produces the infinite string of digits is not evaluating it.

It is merely imagining some way it can be produced.

"Evaluating it" has no rational meaning here.

The digits repeat infinitely.

By definition there can be no final digit.

A thing with no possibility of a final digit cannot have a final value.

Period.

 

[Jokodo]

There is something behind the word "Gandalf" even if it is a fictional character.

There is nothing behind the word "three", nothing behind the character "3".

Sure there is. A mathematical concept. Whether mathematical concepts are themselves fictional doesn't enter consideration - you've just admitted that fictional entities too are distinct from their labels with "Gandalf" != Gandalf.

It is defined.

That is how it exists. It does not exist before that.

Quite unlike Gandalf, who existed long before Tolkien was even born, right?

I have said many times that all things in mathematics are defined. None of it exists without a human mind capable of having thoughts using conceptions.

Unlike Gandalf...

But Gandalf is a person on the screen.

Erm, no. The actor playing Gandalf might be though, but, to slightly paraphrase you, "there's nothing but the actor".

There is nothing but the definition of three.

You can't have "three" slap Harry Potter in a movie or even appear in a movie.

(Wrong universe, but never mind) Sure you can. You can have a sheep run across the screen and say it's the number 3.

All you could ever show is the symbol pointing to the definition. As if all you could show on screen of Gandalf were the letters that make up his name.

So things that can't be shown don't exist (like gravity, air, etc.), while fictional characters that can be imagined to have a physical form are real???

My 8 year old has a better sense of reality than that - at least since 2013 at that!

 

[Jokodo]

Why do you imagine that it takes a non-zero amount of time to evaluate the '...' in the expression '0.333...'?

Giving an equation that produces the infinite string of digits is not evaluating it.

It is merely imagining some way it can be produced.

"Evaluating it" has no rational meaning here.

The digits repeat infinitely.

By definition there can be no final digit.

A thing with no possibility of a final digit cannot have a final value.

Period.

"digits" are not a property of numbers. They are a property of (some) representations of numbers in (some) languages used to represent numbers, such as the language of decimal notation.

It's quite easy to prove, though, that the number referred to by "0.9..." must be 1, not despite but because of the fact there is "no final digit": In decimal notation, all numbers of of the format "0" + "." + n * "9" have values of 1 - <the number represented by "0" + "." + (n-1) * "0" + "1", that is their difference from 1 is a number with a "1" in the position of the final "9" and only "0"s up to there. Since 0.9..." has no final "9", it follows that the difference between that number and 1 has nothing but "0"s after the decimal point. Thus, it is 1-0 = 1. QED.

 

[untermensche]

Why do you imagine that it takes a non-zero amount of time to evaluate the '...' in the expression '0.333...'?

Giving an equation that produces the infinite string of digits is not evaluating it.

It is merely imagining some way it can be produced.

"Evaluating it" has no rational meaning here.

The digits repeat infinitely.

By definition there can be no final digit.

A thing with no possibility of a final digit cannot have a final value.

Period.

"digits" are not a property of numbers.

A digit is a number.

3 is a defined number.

You can't say a digit repeats infinitely AND has a final digit.

That is simple contradiction.

 

[Jokodo]

"digits" are not a property of numbers.

A digit is a number charcter used in identifying numbers.

Fify. And it's language specific: in base 16, "a" is a digit.

3 is a defined number.

3 is, but "3" isn't. In base 2, it's not even a valid digit.

You can't say a digit repeats infinitely AND has a final digit.

Category error. A digit doesn't have digits. It has hooks and lines and if we're very generous code points (more precise would be the code points are external to the digits and mapped to them).

That is simple contradiction.

Argument based on equivocation.

 

[untermensche]

It is defined.

That is how it exists. It does not exist before that.

Quite unlike Gandalf, who existed long before Tolkien was even born, right?

No you are right here but this was your point:

Things don't need to be real in order to be distinct from their labels.

You are now saying Gandalf is not distinct from his definition or label pointing to the definition.

Just like "three".

 

[untermensche]

Fify

All you have done is given another definition not shown mine is wrong.

You can't say a digit repeats infinitely AND has a final digit.

Category error. A digit has hooks and lines and if we're very generous code points (more precise would be the code points are external to the digits and mapped to them).

Again, all you have done is provided your own idiosyncratic capricious definition, not shown my definition is in error.

That is simple contradiction.

Argument based on equivocation.

Conclusion based on burying your head in the sand and only looking at other topics and ignoring everything said.

 

[Jokodo]

No you are right here but this was your point:

Things don't need to be real in order to be distinct from their labels.

You are now saying Gandalf is not distinct from his definition or label pointing to the definition.

Just like "three".

I'm not saying that. I'm pointing out it's a logical consequence of your ramblings. To a normally debeloping 5-Year-old, this shows their absurdity.

 

[Jokodo]

All you have done is given another definition not shown mine is wrong.

Category error. A digit has hooks and lines and if we're very generous code points (more precise would be the code points are external to the digits and mapped to them).

Again, all you have done is provided your own idiosyncratic capricious definition, not shown my definition is in error.

That is simple contradiction.

Argument based on equivocation.

Conclusion based on burying your head in the sand and only looking at other topics and ignoring everything said.

Sure, you can invent your own private language in which the words "digit" and "number" have your own private meaning; most three-year-olds go through a phase like that. But if you think you can use that to prove anything about the concepts others are talking about when using those words, I'll just redefine "the world's biggest idiot" to refer to you and you only, and have thereby proven that you're the world's biggest idiot in the conventional sense.

 

[bilby]

Why do you imagine that it takes a non-zero amount of time to evaluate the '...' in the expression '0.333...'?

Giving an equation that produces the infinite string of digits is not evaluating it.

It is merely imagining some way it can be produced.

"Evaluating it" has no rational meaning here.

The digits repeat infinitely.

By definition there can be no final digit.

A thing with no possibility of a final digit cannot have a final value.

Period.

What The Fuck is a "final value"?

Numbers don't have "final values". They have values.

 

[untermensche]

Why do you imagine that it takes a non-zero amount of time to evaluate the '...' in the expression '0.333...'?

Giving an equation that produces the infinite string of digits is not evaluating it.

It is merely imagining some way it can be produced.

"Evaluating it" has no rational meaning here.

The digits repeat infinitely.

By definition there can be no final digit.

A thing with no possibility of a final digit cannot have a final value.

Period.

What The Fuck is a "final value"?

Numbers don't have "final values". They have values.

A final value is distinct from an ever changing value.

If 3's go on infinitely then the value is ever changing.

It is never reaching a final value.

There can never be a final 3.

- - - Updated - - -

All you have done is given another definition not shown mine is wrong.





Again, all you have done is provided your own idiosyncratic capricious definition, not shown my definition is in error.

Argument based on equivocation.

Conclusion based on burying your head in the sand and only looking at other topics and ignoring everything said.

Sure, you can invent your own private language in which the words "digit" and "number" have your own private meaning; most three-year-olds go through a phase like that. But if you think you can use that to prove anything about the concepts others are talking about when using those words, I'll just redefine "the world's biggest idiot" to refer to you and you only, and have thereby proven that you're the world's biggest idiot in the conventional sense.

I assure you I invented none of the definitions I use.

I am not the first to say that 3 is a number and a digit. And define those words as merely a symbolic representation of value. Which is of course how they have been defined to me many times.

 

[bilby]

What The Fuck is a "final value"?

Numbers don't have "final values". They have values.

A final value is distinct from an ever changing value.

If 3's go on infinitely then the value is ever changing.

It is never reaching a final value.

There can never be a final 3.

This is abject nonsense. A number doesn't have a changing value.

You can't say 'It's OK, officer, I was only driving at 10mph, because I stopped evaluating the "100" on my speedo after the first two digits'. That's fucking insane.

The number represented by "100" has a value of 100. It never has any other value.

The number represented by "0.999..." has a value of 1. It never has any other value. "Final value" is total horseshit, made up by you to try to rationalise your gross errors. I challenge you to find any instance of any mathematician anywhere who has used this phrase in the way you are using it here - to refer to a number.

 

[untermensche]

What The Fuck is a "final value"?

Numbers don't have "final values". They have values.

A final value is distinct from an ever changing value.

If 3's go on infinitely then the value is ever changing.

It is never reaching a final value.

There can never be a final 3.

This is abject nonsense. A number doesn't have a changing value.

If you try to express the number you will get an ever changing number.

You will add 3's forever.

It will never finish.

If you don't express the number but merely imagine it has expressed you have violated the definition which says it can never be fully expressed. Infinite digits means there can never possibly be a last digit.

0.333... is not a value. It has no final value.

It is something else. It is approaching a value. It can by definition never get there.

 

[bilby]

This is abject nonsense. A number doesn't have a changing value.

If you try to express the number you will get an ever changing number.

You will add 3's forever.

It will never finish.

If you don't express the number but merely imagine it has expressed you have violated the definition which says it can never be fully expressed. Infinite digits means there can never possibly be a last digit.

0.333... is not a value. It has no final value.

It is something else. It is approaching a value. It can by definition never get there.

I didn ask for more rambling nonsense from you. I challenged you to find any instance of any mathematician anywhere who has used the phrase "final value" in the way you are using it here - to refer to a number.

Can you meet that challenge, or is this entirely your own stupidity?