∀x in R, if x > 0 and x < 1 then x = 0.5

[wtf]

Now, I know what you may be thinking

"Did he fire six shots, or only five?"

, what about "numbers" that are irrational numbers like pi that go on infinitely? I am not listing the infinities!

How about familiar rational numbers like 1/3 = .3333... or 1/7 = .142857142857...? Are they omitted from your list as well? In that case, you admit that all you're doing is listing the terminating decimals, which do in fact form a countably infinite set.

[Magnolia5275]

Now, I know what you may be thinking

"Did he fire six shots, or only five?"

, what about "numbers" that are irrational numbers like pi that go on infinitely? I am not listing the infinities!

How about familiar rational numbers like 1/3 = .3333... or 1/7 = .142857142857...? Are they omitted from your list as well? In that case, you admit that all you're doing is listing the terminating decimals, which do in fact form a countably infinite set.

If it's not finite then it's not a number. If something keeps going on forever without terminating, then it is simply the output of a non-ending function. My list will go through every number and every output of a number function such as pi.

That said, there really is no limit to what can be listed. You can turn any number into binary machine code that can then represent anything you can imagine. So that makes it possible to list every bit of information stored on planet earth, which would also include all the number functions we know of as well as the number functions we have yet to discover. Is that not a solution?

[wtf]

If it's not finite then it's not a number. If something keeps going on forever without terminating, then it is simply the output of a non-ending function.

1) So familiar rationals like 1/3 and 1/7 are not numbers? Is that the position you're taking? So if we are in math class and we are learning to do long division, 1/2 is a number, but 1/3 simply can't be done? How does this work, exactly?

2) But isn't termination just a function of the base? For example 1/3 = .3333.... in base 10, so 1/3 is "not a number" in your mathematics.

But in base 3, 1/3 = .1. It terminates. So in base 3, is 1/3 now a number? How can the base, which is just a particular representation, determine whether or not something is a number.

Do you understand the difference between a number and its representation?

[Skepdick]

Infinity itself is defined using numbers, it simply means "keep adding numbers (or dividing), no end!"

That's a circular definition. How many times do you have to "keep adding numbers"? No end? That's infinity!

Your "definition" is circular.

So, if you somehow define numbers using infinity, you are still using the notion of numbers to define numbers. It's not getting you anywhere.

No, I am not. I am using my understanding of infinity a priori any definitions. It gets me everywhere.

Infinity is undefined. Just the same - I am going to use it.

You cannot use the notion of "size", when talking about infinity.

Sure I can. The set of natural numbers is infinite. The set of real numbers is infinite. One of those infinities is not like the other.

The word "size" by definition means it has some numerical measurement.

It does? In so far as I can tell it's not even a Mathematical notion.

It's common sense concept that any person off the street understands intuitively. It refers to magnitude or dimension of a thing.
You don't need any "numbers" to determine that the size of a truck is bigger than the size of an ant - it's intuitive. Even without any numbers.

That doesn't exist in infinity by definition.

Nobody cares about the definitions. Not computer scientists. Not physicists.

Intuition and use of concepts comes first. Definitions (codification in language) comes later; or never.

Also, you must understand the when you choose a number between 0 and 1, you are defining the "size", of the space between 0 and 1.

Sure. I am defining the unit-interval. So what? Do the exact same thing as the entire number line.

Pick any number in the (-∞, +∞) interval. Whatever you pick there's a bijetion from (-∞, x] to [x; -∞).
So no matter how hard you try you'll end up picking 0.

If I choose 0.6, it means 0 to 1 has a size of 10
If I choose 0.47, it means 0 to 1 has a size of 100
If I choose 0.838299, it means 0 to 1 has a size of 1,000,000
If I add a 0.001 to 0.47, I have now redefined the space to the size of 1000

Not sure what this has to do with anything I am saying.

If you choose pi 3.141592... Then you are defining the number 3,141,592 in a relative space that is the size of 10,000,000. Any additional pi numbers after the 2 are just new outputs of a function that continuously redefines the size of the relational space. pi is not a number until you put a limit on it. If you don't put any limit, then it's a function or a "potentiality", not a number.

Which is why I haven't assigned any numerical size to the space. I am only speaking about the relative sizes of (-∞, x] to [x; -∞).

SInce there's a bijection between the two - the size is identical. Whatever that size is.

The defining of the size of the space, is what gives a meaning to the number chosen in between, otherwise, the number chosen would not have any measurable meaning.

No idea what you are trying to say.

So no, a point chosen anywhere in an infinity, cannot define an "amount"

It doesn't define an amount. 0 is not an amount. It's just an arbitrary point. If you have some emotional/cultural baggage about "how much" 0 is - let it go.

Any point on (-∞, +∞) is as good as 0.

, all you are saying is that you choose an infinity "identity point" inside the infinity itself. It most certainly does not give you the number 0.5, which would mean it divides the infinity into two halves with a measurable amount of 5. You have gotten the whole conceptual framework wrong.

Not me. You are confused.

All I am saying is that the point is equidistant from 0 AND 1. So any point is as good as 0.5

That said, what you can say is that infinity is always symmetrical to itself. "size", and "middle", are the wrong words to use in the context of infinity.

Oh yeah? I think "wrong" is a wrong word to use in Mathematics. But that's just me.

Infinity simply = Infinity, it is the Law of Identity, all infinities are the same.

Uhhhh. So in your mind the size of the natural numbers is "the same" as as the size of the Real numbers?

Lets try that. 1 in N maps to 1 in R. 2 in N maps to 2 in R. .... x in N maps to x in R.
What in N maps to 0.5 in R ?

Despite what you may have learned, Cantor's diagonal proof is wrong, there is no notion of "size" or "cardinality", you can attach to infinity. It is simply endlessness itself.

*yawn*

Which set has more members? N or R? Are you really going to insist that there is a bijection from N to R?

Yes, of course! It's so easy, even a 2-year-old can do it! You start with 0.0, then you move to 00.00, and then to 000.000, and then so on, and so on.
In each box where the number of zeros in total is >2, you need to move the decimal place over so as to cover all the options: 000.000, 00.0000, 0.00000, 0000.00, 00000.0 yay!

And now we list!

1. 0.0
2. 0.1
3. 0.2
4. 0.3
5. 0.4
6. 0.5
7. 0.6
8. 0.7
9. 0.8
10. 0.9
11. 1.0
12. 1.1
13. 1.2
14. 1.3
15. 1.4
16. 1.5
17. 1.6
18. 1.7
19. 1.8
20. 1.9
21. 2.0
22. 2.1
23. 2.2
24. 2.3
25. 2.4
26. 2.5
27. 2.6
28. 2.7
29. 2.8
30. 2.9
31. 3.0
32. 3.1
33. 3.2
34. 3.3
35. 3.4
36. 3.5
37. 3.6
38. 3.7
39. 3.8
40. 3.9
41. 4.0
42. 4.1
43. 4.2
44. 4.3
45. 4.4
46. 4.5
47. 4.6
48. 4.7
49. 4.8
50. 4.9
51. 5.0
52. 5.1
53. 5.2
54. 5.3
55. 5.4
56. 5.5
57. 5.6
58. 5.7
59. 5.8
60. 5.9
61. 6.0
62. 6.1
63. 6.2
64. 6.3
65. 6.4
66. 6.5
67. 6.6
68. 6.7
69. 6.8
70. 6.9
71. 7.0
72. 7.1
73. 7.2
74. 7.3
75. 7.4
76. 7.5
77. 7.6
78. 7.7
79. 7.8
80. 7.9
81. 8.0
82. 8.1
83. 8.2
84. 8.3
85. 8.4
86. 8.5
87. 8.6
88. 8.7
89. 8.8
90. 8.9
91. 9.0
92. 9.1
93. 9.2
94. 9.3
95. 9.4
96. 9.5
97. 9.6
98. 9.7
99. 9.8
100. 9.9

After this, we go on to a larger box of 00.00, and we can continue that forever, by increasing the sizes of the boxes after each completion of a finite-sized box. Each number from N can be mapped to the Reals multiple times over.

Now, I know what you may be thinking, what about "numbers" that are irrational numbers like pi that go on infinitely? I am not listing the infinities! Will, I hate to break it to you, but infinities are not numbers! The infinity of pi is a function that keeps spitting out larger and larger numbers, and my listing method above can list each and every output of the function of pi, in fact, it already has! (32) on the list is: "3.1", fresh from the oven! it's a miracle!

So not only did I list the Reals, my method in its simple form will even list the Reals multiple times over: 0.9 <-> 00000.9 <-> 00.9000. This is easy to fix by simply not listing what has already been listed. So yes, in principle a one-to-one correspondence can exist. See? Was it really that hard? Even my dog could have figured this out.

If you still insisted on having the infinite functions themselves listed, fear not! The numbers that my method lists can be converted into binary machine code. We can have it list one box normally and then another box, in binary. Eventually, my listing method will list every description of every possible function. It will list the wikipedia article about pi, it will list any program that can be programmed, it will list future events and the DNA of Abraham Lincoln, it will even list this post itself.

So now that I have proven the reals and naturals are the same, can you just admit that infinities don't have any size, and it was all just a big confusion? Come on, you can't seriously believe that: "non-ending > non-ending", even goats know that is not true! lol

Well, you left out something important I think.

Do you have a total ordering on your list? e.g can you arrange the number from smallest to largest?

Take any two adjacent numbers. Divide by 2... And you have yourself a number that's not in your list.

Surely you recognise this as a general property of the Real numbers? Given any three real number such that x < y < z
There's always a bijection between the interval (x,y) and the interval (y,z).

It's the exact same patter in the OP recursively applied.

[Magnolia5275]

That's a circular definition. How many times do you have to "keep adding numbers"? No end? That's infinity!

Your "definition" is circular.


No, I am not. I am using my understanding of infinity a priori any definitions. It gets me everywhere.

Infinity is undefined. Just the same - I am going to use it.


Sure I can. The set of natural numbers is infinite. The set of real numbers is infinite. One of those infinities is not like the other.


It does? In so far as I can tell it's not even a Mathematical notion.

It's common sense concept that any person off the street understands intuitively. It refers to magnitude or dimension of a thing.
You don't need any "numbers" to determine that the size of a truck is bigger than the size of an ant - it's intuitive. Even without any numbers.


Nobody cares about the definitions. Not computer scientists. Not physicists.

Intuition and use of concepts comes first. Definitions (codification in language) comes later; or never.


Sure. I am defining the unit-interval. So what? Do the exact same thing as the entire number line.

Pick any number in the (-∞, +∞) interval. Whatever you pick there's a bijetion from (-∞, x] to [x; -∞).
So no matter how hard you try you'll end up picking 0.


Not sure what this has to do with anything I am saying.


Which is why I haven't assigned any numerical size to the space. I am only speaking about the relative sizes of (-∞, x] to [x; -∞).

SInce there's a bijection between the two - the size is identical. Whatever that size is.


No idea what you are trying to say.


It doesn't define an amount. 0 is not an amount. It's just an arbitrary point. If you have some emotional/cultural baggage about "how much" 0 is - let it go.

Any point on (-∞, +∞) is as good as 0.


Not me. You are confused.

All I am saying is that the point is equidistant from 0 AND 1. So any point is as good as 0.5


Oh yeah? I think "wrong" is a wrong word to use in Mathematics. But that's just me.


Uhhhh. So in your mind the size of the natural numbers is "the same" as as the size of the Real numbers?

Lets try that. 1 in N maps to 1 in R. 2 in N maps to 2 in R. .... x in N maps to x in R.
What in N maps to 0.5 in R ?


*yawn*

Which set has more members? N or R? Are you really going to insist that there is a bijection from N to R?

Yes, of course! It's so easy, even a 2-year-old can do it! You start with 0.0, then you move to 00.00, and then to 000.000, and then so on, and so on.
In each box where the number of zeros in total is >2, you need to move the decimal place over so as to cover all the options: 000.000, 00.0000, 0.00000, 0000.00, 00000.0 yay!

And now we list!

1. 0.0
2. 0.1
3. 0.2
4. 0.3
5. 0.4
6. 0.5
7. 0.6
8. 0.7
9. 0.8
10. 0.9
11. 1.0
12. 1.1
13. 1.2
14. 1.3
15. 1.4
16. 1.5
17. 1.6
18. 1.7
19. 1.8
20. 1.9
21. 2.0
22. 2.1
23. 2.2
24. 2.3
25. 2.4
26. 2.5
27. 2.6
28. 2.7
29. 2.8
30. 2.9
31. 3.0
32. 3.1
33. 3.2
34. 3.3
35. 3.4
36. 3.5
37. 3.6
38. 3.7
39. 3.8
40. 3.9
41. 4.0
42. 4.1
43. 4.2
44. 4.3
45. 4.4
46. 4.5
47. 4.6
48. 4.7
49. 4.8
50. 4.9
51. 5.0
52. 5.1
53. 5.2
54. 5.3
55. 5.4
56. 5.5
57. 5.6
58. 5.7
59. 5.8
60. 5.9
61. 6.0
62. 6.1
63. 6.2
64. 6.3
65. 6.4
66. 6.5
67. 6.6
68. 6.7
69. 6.8
70. 6.9
71. 7.0
72. 7.1
73. 7.2
74. 7.3
75. 7.4
76. 7.5
77. 7.6
78. 7.7
79. 7.8
80. 7.9
81. 8.0
82. 8.1
83. 8.2
84. 8.3
85. 8.4
86. 8.5
87. 8.6
88. 8.7
89. 8.8
90. 8.9
91. 9.0
92. 9.1
93. 9.2
94. 9.3
95. 9.4
96. 9.5
97. 9.6
98. 9.7
99. 9.8
100. 9.9

After this, we go on to a larger box of 00.00, and we can continue that forever, by increasing the sizes of the boxes after each completion of a finite-sized box. Each number from N can be mapped to the Reals multiple times over.

Now, I know what you may be thinking, what about "numbers" that are irrational numbers like pi that go on infinitely? I am not listing the infinities! Will, I hate to break it to you, but infinities are not numbers! The infinity of pi is a function that keeps spitting out larger and larger numbers, and my listing method above can list each and every output of the function of pi, in fact, it already has! (32) on the list is: "3.1", fresh from the oven! it's a miracle!

So not only did I list the Reals, my method in its simple form will even list the Reals multiple times over: 0.9 <-> 00000.9 <-> 00.9000. This is easy to fix by simply not listing what has already been listed. So yes, in principle a one-to-one correspondence can exist. See? Was it really that hard? Even my dog could have figured this out.

If you still insisted on having the infinite functions themselves listed, fear not! The numbers that my method lists can be converted into binary machine code. We can have it list one box normally and then another box, in binary. Eventually, my listing method will list every description of every possible function. It will list the wikipedia article about pi, it will list any program that can be programmed, it will list future events and the DNA of Abraham Lincoln, it will even list this post itself.

So now that I have proven the reals and naturals are the same, can you just admit that infinities don't have any size, and it was all just a big confusion? Come on, you can't seriously believe that: "non-ending > non-ending", even goats know that is not true! lol

Well, you left out something important I think.

Do you have a total ordering on your list? e.g can you arrange the number from smallest to largest?

Take any two adjacent numbers. Divide by 2... And you have yourself a number that's not in your list.

Surely you recognise this as a general property of the Real numbers? Given any three real number such that x < y < z
There's always a bijection between the interval (x,y) and the interval (y,z).

It's the exact same patter in the OP recursively applied.

But there is no "smallest" number after the decimal point. The numbers are completely relative. 0.1, is seen as the same as 0.1000, what is 1 in one relative space is 1000 in the other.

You cannot list units that have no defined size, what you can do is list them up to a certain precision, and then once you finish listing one precision space, you move up to the next precision space.

Again, just to clarify, you cannot list an infinitely precise unit. An "infinitely precise unit" is nothing at all, and doesn't mean anything. Only finite things can be listed, because only finites are "things".

Take any two adjacent numbers. Divide by 2... And you have yourself a number that's not in your list.

Yes, it is, say I take 4.5 and 4.6, that gives me 4.55, and indeed that would be on the list once the 00.00 precision box is listed. If you want, you can have some algorithm go back (after completing each precision box) and rearrange all the listed numbers in any way you wish. The point is, all the Reals can easily fit into the Naturals, one cannot be said to be "larger" than the other.

Also, don't forget what I said about listing the boxes in machine code. If you do that, you will be listing every possible, number, function, description, and anything that you could possibly think of would be on that list.

[Skepdick]

But there is no "smallest" number after the decimal point.

Of course there is. 0.000...

The numbers are completely relative. 0.1, is seen as the same as 0.1000, what is 1 in one relative space is 1000 in the other.

That's not true in general. You can't do this powers-of-ten trickery with 0. 0*10^1 and 0*10^10000 is still 0.

You cannot list units that have no defined size, what you can do is list them up to a certain precision, and then once you finish listing one precision space, you move up to the next precision space.

Well, when are you "finished" with the precision-space? How many precision-spaces are there?

Again, just to clarify, you cannot list an infinitely precise unit. An "infinitely precise unit" is nothing at all, and doesn't mean anything. Only finite things can be listed, because only finites are "things".

Can you try rewording this sentence in English? It's perfectly possible to present a finite algorithm for an infinite object.
For example the expression [0..] in Haskell generates ALL Natural numbers. Obviously - you don't have infinite time to read all of them but there's your example for a finite representation of an infinite object.

We simply exploit the regularity in structure to compress it down to its fundamental property. In this case that property is induction.

Take any two adjacent numbers. Divide by 2... And you have yourself a number that's not in your list.

Yes, it is, say I take 4.5 and 4.6, that gives me 4.55, and indeed that would be on the list once the 00.00 precision box is listed. If you want, you can have some algorithm go back (after completing each precision box) and rearrange all the listed numbers in any way you wish. The point is, all the Reals can easily fit into the Naturals, one cannot be said to be "larger" than the other.

You are contradicting yourself. If your list of real numbers is ordered then 4.5 and 4.6 are not "adjacent". By our own admission 4.55 (which is apparently on your list) is necessarily between 4.5 and 4.6.

Also, don't forget what I said about listing the boxes in machine code. If you do that, you will be listing every possible, number, function, description, and anything that you could possibly think of would be on that list.

But your list will necessarily be discrete! And the real numbers aren't...

[Magnolia5275]

But there is no "smallest" number after the decimal point.

Of course there is. 0.000...

The numbers are completely relative. 0.1, is seen as the same as 0.1000, what is 1 in one relative space is 1000 in the other.

That's not true in general. You can't do this powers-of-ten trickery with 0. 0*10^1 and 0*10^10000 is still 0.

You cannot list units that have no defined size, what you can do is list them up to a certain precision, and then once you finish listing one precision space, you move up to the next precision space.

Well, when are you "finished" with the precision-space? How many precision-spaces are there?

Again, just to clarify, you cannot list an infinitely precise unit. An "infinitely precise unit" is nothing at all, and doesn't mean anything. Only finite things can be listed, because only finites are "things".

Can you try rewording this sentence in English? It's perfectly possible to present a finite algorithm for an infinite object.
For example the expression [0..] in Haskell generates ALL Natural numbers. Obviously - you don't have infinite time to read all of them but there's your example for a finite representation of an infinite object.

We simply exploit the regularity in structure to compress it down to its fundamental property. In this case that property is induction.

Take any two adjacent numbers. Divide by 2... And you have yourself a number that's not in your list.

Yes, it is, say I take 4.5 and 4.6, that gives me 4.55, and indeed that would be on the list once the 00.00 precision box is listed. If you want, you can have some algorithm go back (after completing each precision box) and rearrange all the listed numbers in any way you wish. The point is, all the Reals can easily fit into the Naturals, one cannot be said to be "larger" than the other.

You are contradicting yourself. If your list of real numbers is ordered then 4.5 and 4.6 are not "adjacent". By our own admission 4.55 (which is apparently on your list) is necessarily between 4.5 and 4.6.

Also, don't forget what I said about listing the boxes in machine code. If you do that, you will be listing every possible, number, function, description, and anything that you could possibly think of would be on that list.

But your list will necessarily be discrete! And the real numbers aren't...

Listen, what you are saying I need to achieve is impossible and stupid. Saying the Reals are not listable (in their infinity), is like saying a square circle is not a shape that can be drawn, but that is because it's not a fricking shape at all!! It's the same here, of course I cannot list undefined numbers, because what the heck is the undefined number? That is not a number! How can I list "things" that are undefined? What is it exactly that I am supposed to be listing???

An infinite "number", is not a number at all. To say it is, would be to go against the law of identity, and the law of non-contradiction. An infinite number never concludes to a "thing", a "something", you cannot tell me to make a list of non-things. It's ridiculous!

It's perfectly possible to present a finite algorithm for an infinite object.
For example the expression [0..] in Haskell generates ALL Natural numbers. Obviously - you don't have infinite time to read all of them but there's your example for a finite representation of an infinite object.

Yes, I understand that, that is exactly what can be listed. You can list the finite representation of an infinite algorithm, but not the infinity itself. By asking me to list one Real number, and then another Real after that in such a way that there is no Real in between, is asking me to list the infinities themselves. This is like asking me to paint a picture without paint; you are asking me to make a list without defined number units!

But your list will necessarily be discrete! And the real numbers aren't...

You are contradicting yourself, the word "number" by definition is discrete. When you say Pi is an irrational number, it's not actually a number in itself, it's a collection of numbers that can be created by a function based on finite rules and starting values.

The fact is, any number you can think of, will be on my list at some point. Any level of precision you can define will be included, and you can also use the numbers themselves to represent functions and algorithms that output an infinite stream of numbers and precision. Anything that is a "thing" can be listed, but I cannot list undefinable non-things.

Infinities are not things, get that in your head. You have never experienced or known an infinity in your life. When you tell me to list the infinitely precise units themselves, one after the other, you are not talking about anything at all, you are speaking gibberish. Saying the Reals in their infinite precision are not listable like saying that the set of non-shapes shapes is not listable, Well yeah, but that is because "non-shapes shapes" is nonsense! it's not real!

When we say "Real numbers" we can only be referring to actual defined numbers or a function, and my list will list all that is real by increasing the size of each informational space after the compilation of the previous informational space.

[Skepdick]

Saying the Reals are not listable (in their infinity), is like saying a square circle is not a shape that can be drawn, but that is because it's not a fricking shape at all!!

Idiot. Do you understand that the natural numbers are in fact listable (in their infinity)?

List 0.
List the next number.
And the next.
And the next...

Now, obviously you don't have infinite time but IF you had infinite time you would be able to list all natural numbers. So then in principle - you can.

Your procedure for listing "all the reals" doesn't work. Because while you are busy listing 0.0 and 0.00 and 0.000 and 0.0000 you NEVER actually get to listing 1.0 and 1.00 and 1.000. So even IF you had infinite time - your procedure fails.

Oh... and here is your square circle..

It's the same here, of course I cannot list undefined numbers, because what the heck is the undefined number? That is not a number! How can I list "things" that are undefined? What is it exactly that I am supposed to be listing???

List the definition? For each number?

An infinite "number", is not a number at all. To say it is, would be to go against the law of identity, and the law of non-contradiction. An infinite number never concludes to a "thing", a "something", you cannot tell me to make a list of non-things. It's ridiculous!

Nobody is saying anything like what you are misunderstanding.

Yes, I understand that, that is exactly what can be listed. You can list the finite representation of an infinite algorithm, but not the infinity itself.

What do you mean? The finite representation of infinity itself is the symbol "∞". Obviously I can't show you infinity itself because the concept is in my head. So you'll have to settle for the representation.

By asking me to list one Real number, and then another Real after that in such a way that there is no Real in between, is asking me to list the infinities themselves. This is like asking me to paint a picture without paint; you are asking me to make a list without defined number units!

It's nothing like that.

I can run the algorithm for genrating the Natural numbers in my head.
I can't run your algorithm for producing the Real numbers in my head.
Your algorithm never even gets to 1.

You are contradicting yourself, the word "number" by definition is discrete. When you say Pi is an irrational number, it's not actually a number in itself, it's a collection of numbers that can be created by a function based on finite rules and starting values.

You are contradicting yourself. Functions don't generate numbers. They generate representations of numbers. Any function which generates pi to an arbitrary precision is a representation of pi.

The fact is, any number you can think of, will be on my list at some point.

The number 1 will never be on your list. And 2. And 3. etc. etc. Your algorithm only enumerates

Any level of precision you can define will be included, and you can also use the numbers themselves to represent functions and algorithms that output an infinite stream of numbers and precision. Anything that is a "thing" can be listed, but I cannot list undefinable non-things.

And all the stuff you generate will have smaller cardinality than the Real numbers.

Infinities are not things, get that in your head.

Nobody says they are things? They are concepts. And they are already in my head...

You have never experienced or known an infinity in your life.

I have experienced; and know every single concept in my head. Including the concept of infinity.

Perhaps the word really triggers your sensibilities? If you don't like the word "infinite", then use the phrase "not finite".

When you tell me to list the infinitely precise units themselves, one after the other, you are not talking about anything at all, you are speaking gibberish.

Inability to understanding on your part does not imply gibberish on my part.

Saying the Reals in their infinite precision are not listable like saying that the set of non-shapes shapes is not listable, Well yeah, but that is because "non-shapes shapes" is nonsense! it's not real!

That's a terrible analogy. The set of Reals is not empty. The set of non-shape shapes is empty.

When we say "Real numbers" we can only be referring to actual defined numbers or a function, , and my list will list all that is real by increasing the size of each informational space after the compilation of the previous informational space.

We who? How many voices are there in your head?

When I say "Real numbers" I am talking about the concept of the Real numbers. Nothing to do with definitions.

Of course, there's a couple of different ways to characterise the relevant structure on R, but once you figure out which properties you care about - you can automatically compute a description of its computational representaton.

https://math.andrej.com/2008/02/06/repr ... able-sets/

[Magnolia5275]

Saying the Reals are not listable (in their infinity), is like saying a square circle is not a shape that can be drawn, but that is because it's not a fricking shape at all!!

Idiot. Do you understand that the natural numbers are in fact listable (in their infinity)?

List 0.
List the next number.
And the next.
And the next...

Now, obviously you don't have infinite time but IF you had infinite time you would be able to list all natural numbers. So then in principle - you can.

Your procedure for listing "all the reals" doesn't work. Because while you are busy listing 0.0 and 0.00 and 0.000 and 0.0000 you NEVER actually get to listing 1.0 and 1.00 and 1.000. So even IF you had infinite time - your procedure fails.

Oh... and here is your square circle..

It's the same here, of course I cannot list undefined numbers, because what the heck is the undefined number? That is not a number! How can I list "things" that are undefined? What is it exactly that I am supposed to be listing???

List the definition? For each number?

An infinite "number", is not a number at all. To say it is, would be to go against the law of identity, and the law of non-contradiction. An infinite number never concludes to a "thing", a "something", you cannot tell me to make a list of non-things. It's ridiculous!

Nobody is saying anything like what you are misunderstanding.

Yes, I understand that, that is exactly what can be listed. You can list the finite representation of an infinite algorithm, but not the infinity itself.

What do you mean? The finite representation of infinity itself is the symbol "∞". Obviously I can't show you infinity itself because the concept is in my head. So you'll have to settle for the representation.

By asking me to list one Real number, and then another Real after that in such a way that there is no Real in between, is asking me to list the infinities themselves. This is like asking me to paint a picture without paint; you are asking me to make a list without defined number units!

It's nothing like that.

I can run the algorithm for genrating the Natural numbers in my head.
I can't run your algorithm for producing the Real numbers in my head.
Your algorithm never even gets to 1.

You are contradicting yourself, the word "number" by definition is discrete. When you say Pi is an irrational number, it's not actually a number in itself, it's a collection of numbers that can be created by a function based on finite rules and starting values.

You are contradicting yourself. Functions don't generate numbers. They generate representations of numbers. Any function which generates pi to an arbitrary precision is a representation of pi.

The fact is, any number you can think of, will be on my list at some point.

The number 1 will never be on your list. And 2. And 3. etc. etc. Your algorithm only enumerates

Any level of precision you can define will be included, and you can also use the numbers themselves to represent functions and algorithms that output an infinite stream of numbers and precision. Anything that is a "thing" can be listed, but I cannot list undefinable non-things.

And all the stuff you generate will have smaller cardinality than the Real numbers.

Infinities are not things, get that in your head.

Nobody says they are things? They are concepts. And they are already in my head...

You have never experienced or known an infinity in your life.

I have experienced; and know every single concept in my head. Including the concept of infinity.

Perhaps the word really triggers your sensibilities? If you don't like the word "infinite", then use the phrase "not finite".

When you tell me to list the infinitely precise units themselves, one after the other, you are not talking about anything at all, you are speaking gibberish.

Inability to understanding on your part does not imply gibberish on my part.

Saying the Reals in their infinite precision are not listable like saying that the set of non-shapes shapes is not listable, Well yeah, but that is because "non-shapes shapes" is nonsense! it's not real!

That's a terrible analogy. The set of Reals is not empty. The set of non-shape shapes is empty.

When we say "Real numbers" we can only be referring to actual defined numbers or a function, , and my list will list all that is real by increasing the size of each informational space after the compilation of the previous informational space.

We who? How many voices are there in your head?

When I say "Real numbers" I am talking about the concept of the Real numbers. Nothing to do with definitions.

Of course, there's a couple of different ways to characterise the relevant structure on R, but once you figure out which properties you care about - you can automatically compute a description of its computational representaton.

https://math.andrej.com/2008/02/06/repr ... able-sets/

Your procedure for listing "all the reals" doesn't work. Because while you are busy listing 0.0 and 0.00 and 0.000 and 0.0000 you NEVER actually get to listing 1.0 and 1.00 and 1.000. So even IF you had infinite time - your procedure fails.

You completely didn't understand how my procedure works. When I say you start with a box of 0.0, I mean you first list all the options in the box of 0.0, before going on to the next box of 00.00. So if we start with 0.0, we will get to 1.0:

0.0
0.1
0.2
.
...
0.9
*1.0*
1.1
1.2
1.3
.
...
1.9
2.0
2.1
.
...
9.9

Box of 0.0 finished!

And now we move on to the box of [00.00]

00.00
00.01
.
...
00.99
...
99.99

(Done! And now we move the decimal point...)
0.000
0.001
.
...
9.999

(and again we move the decimal point...)
000.0
000.9
.
...
999.9

And that is it! We finished all the finite possibilities of box: [00.00]! Now we move on to the next box of [000.000] and after that, we go to [0000.0000], and so on, and so on.

As you can see, my procedure covers all the possibilities. Any number you can think of will be listed at some point. When you say I don't get to 1.0, you are simply lying, it does! It reaches that in the first informational box of two zeros: [0.0]

I am aware that my list has repetitions of the same numbers like 000.9 and 0.9, but you can simply not list the repetitions, and in principle, you can have a one-to-one correspondence.

So let's play a really simple game. You are claiming that my method doesn't list all the Reals. So give me any Real you can think of, and I will tell you which information box contains the number you chose. I guarantee you I will have an answer for any number you can think of. This will prove that my list contains all the numbers of any kind!!

Idiot. Do you understand that the natural numbers are in fact listable (in their infinity)?

Idiot, you don't understand what I was referring to. I'm not talking about listing a non-ending list of finite numbers. What I am saying is that you cannot make a list of infinite numbers, which is: 0.3333333333......, I can list 0.3 and 0.33, but not 0.3333333.... because that is not anything real at all! Only the finite fraction "1/3" which can be represented with finite numbers is listable but not 0.333333333...., but yes: "0.33333333333..." as a finite text that references the finite concept of dividing a decimal unit by 3.

Again, in case you have missed the point, The assertion that I must list the Reals (in their infinite precision) one after the other, Is the assertion that I must actually list the infinities themselves. There is no "infinite precision" that comes after another "infinite precision" because "infinite precision" does not reference anything in its totality, it's not units, it's nothing at all. The "Reals", cannot be the collection of "nothingnesses"; nothingness does not exist, only "somethingness" exists, and I can list all the "somethingnesses".

That said, my list does go through every single precision space, and since it is never ending, the list itself will contain every level of precision all the way to infinity. So ALL the Reals or contained in my list. Go ahead test me! Tall me which Real is not on my list! I don't care that you call me an idiot, but to say my list doesn't work is just rude! I feel personally insulted when you say that!

"But there is no "smallest" number after the decimal point."
_______________________
Of course there is. 0.000...

When do you reach the 1 at the end of infinity? If you reach it, is that the end of infinity? A calculation that never ends outputs ZERO results. If A never = B, then A does NOT = B. Who is the idiot now? Even the mountain sheep in Alaska know that! lol

----------------------------------------
(edit)

Here is an example of an irrational number whose representation is listed in my method:

√5

Binary number representation:

11100010 10001000 10011010 00110101

Conversion to decimal:

3800603189

Box_20_zeros >>> [0000000000.0000000000] >> [3800603189.0000000000]

Even my goldfish knows how to do that!

(edit2)

OMG! Even this whole response itself (not including this edit) is on my list! I am on the list!

010110010110111101110101001000000110001101101111011011010111000001101100011001010111010001100101011011000111100100100000011001000110100101100100011011100010011101110100001000000111010101101110011001000110010101110010011100110111010001100001011011100110010000100000011010000110111101110111001000000110110101111001001000000111000001110010011011110110001101100101011001000111010101... 111001100100000011010000110111101110111001000000111010001101111001000000110010001101111001000000111010001101000011000010111010000100001

Converted to decimal:
1452971614802563820723241451379142052817369169460541620201783505577485737832... 4504777469147428680109345599553975462573093365001835072831179566584567145857370765128676085163699312144275258569761

Number of digits: "9183"*2 = 18366

Box_18366_zeros >> [00000... 00.00 ...00000] >>> [145297161480256382072324145137914... 75258569761.00 ...00000]

It has been destined all along! The universe agrees with me! I know I was right!

[Skepdick]

00.00
00.01
.
...
00.99
...
99.99

And that is it! We finished all the finite possibilities of box: [00.00]! Now we move on to the next box of [000.000] and after that, we go to [0000.0000], and so on, and so on.

Ah! I understand now! So your list doesn't contain any irrational numbers.

There's no √2, no 1/3, No π. No e.

As you can see, my procedure covers all the possibilities.

You have a very weird notion of "all". Perhaps you meant to say "All except the irrational real numbers"?

Any number you can think of will be listed at some point.

Any number except the ones not listed. Right?

I am aware that my list has repetitions of the same numbers like 000.9 and 0.9, but you can simply not list the repetitions, and in principle, you can have a one-to-one correspondence.

Well yeah! It's easy to get a 1:1 corrspondence when you leave stuff out of R!

So let's play a really simple game. You are claiming that my method doesn't list all the Reals. So give me any Real you can think of, and I will tell you which information box contains the number you chose. I guarantee you I will have an answer for any number you can think of. This will prove that my list contains all the numbers of any kind!!

OK! Which box will contain √2 ?

Idiot, you don't understand what I was referring to. I'm not talking about listing a non-ending list of finite numbers.

So why did you choose an infinite representation of the number? Choose a finite representation!

What I am saying is that you cannot make a list of infinite numbers, which is: 0.3333333333......, I can list 0.3 and 0.33, but not 0.3333333....

So choose a finite representation! Why can't you?

I can list 0.3 and 0.33, but not 0.3333333.... because that is not anything real at all! only the finite fraction "1/3" which can be represented with finite numbers is listable but not 0.333333333...., but yes: "0.33333333333..." as a finite text that references the finite concept of dividing a decimal unit by 3.

1/3 is just another way of saying "0.333....". It's the same number.

Again, in case you have missed the point, The assertion that I must list the Reals (in their infinite precision)

Nobody said anything about precision? There is no "infinite precision" in 1/3 unless you attempt to compute the division. Don't compute the division!

one after the other, Is the assertion that I must actually list the infinities themselves.

So whose problem is it that your algorithm can't list 1/3 ? Yours or mine?

1/3 is a Real number. Your procedure doesn't list it. Fail.

There is no "infinite precision" that comes after another "infinite precision" because "infinite precision" does not reference anything in its totality, it's not units, it's nothing at all. The "Reals", cannot be the collection of "nothingnesses"; nothingness does not exist, only "somethingness" exists, and I can list all the "somethingnesses".

That's a word salad.

That said, my list does go through every single precision space, and since it is never ending, the list itself will contain every level of precision all the way to infinity.

Precision doesn't matter to me. I'll settle for a finite representation that can compute up to infinite precision in principle.

e.g I'll settle for an algorithm which approximates √2. Even if you don't list √2 to infinite precision.

So ALL the Reals or contained in my list. Go ahead test me! Tall me which Real is not on my list!

√2.

I don't care that you call me an idiot, but to say my list doesn't work is just rude! I feel personally insulted when you say that!

It's not rude when you keep refusing to accept that your list doesn't work

At some point I must assume you are an idiot unable to accept critical feedback.

When do you reach the 1 at the end of infinity? If you reach it, is that the end of infinity? A calculation that never ends outputs ZERO results. If A never = B, then A does NOT = B. Who is the idiot now? Even the mountain sheep in Alaska know that! lol

Of course it does! Have you heard of the Maybe monad?

Either it returns Just(the result of the computation); or it returns Nothing.

https://en.wikipedia.org/wiki/Monad_(fu ... ple:_Maybe

I guess the mountain sheep in Alaska don't understand Monads.

Here is an example of an irrational number whose representation is listed in my method:

√5

Binary number representation:

11100010 10001000 10011010 00110101

Conversion to decimal:

3800603189

Box_20_zeros >>> [0000000000.0000000000] >> [3800603189.0000000000]

Even my goldfish knows how to do that!

That's not √5. Even for an approximation that's super terrible! √5 ≈ 2.23606797749979

You can't represent √5 with finitely many symbols using the representation schema you've chosen.

[Magnolia5275]

00.00
00.01
.
...
00.99
...
99.99

And that is it! We finished all the finite possibilities of box: [00.00]! Now we move on to the next box of [000.000] and after that, we go to [0000.0000], and so on, and so on.

Ah! I understand now! So your list doesn't contain any irrational numbers.

There's no √2, no 1/3, No π. No e.

As you can see, my procedure covers all the possibilities.

You have a very weird notion of "all". Perhaps you meant to say "All except the irrational real numbers"?

Any number you can think of will be listed at some point.

Any number except the ones not listed. Right?

I am aware that my list has repetitions of the same numbers like 000.9 and 0.9, but you can simply not list the repetitions, and in principle, you can have a one-to-one correspondence.

Well yeah! It's easy to get a 1:1 corrspondence when you leave stuff out of R!

So let's play a really simple game. You are claiming that my method doesn't list all the Reals. So give me any Real you can think of, and I will tell you which information box contains the number you chose. I guarantee you I will have an answer for any number you can think of. This will prove that my list contains all the numbers of any kind!!

OK! Which box will contain √2 ?

Idiot, you don't understand what I was referring to. I'm not talking about listing a non-ending list of finite numbers.

So why did you choose an infinite representation of the number? Choose a finite representation!

What I am saying is that you cannot make a list of infinite numbers, which is: 0.3333333333......, I can list 0.3 and 0.33, but not 0.3333333....

So choose a finite representation! Why can't you?

I can list 0.3 and 0.33, but not 0.3333333.... because that is not anything real at all! only the finite fraction "1/3" which can be represented with finite numbers is listable but not 0.333333333...., but yes: "0.33333333333..." as a finite text that references the finite concept of dividing a decimal unit by 3.

1/3 is just another way of saying "0.333....". It's the same number.

Again, in case you have missed the point, The assertion that I must list the Reals (in their infinite precision)

Nobody said anything about precision? There is no "infinite precision" in 1/3 unless you attempt to compute the division. Don't compute the division!

one after the other, Is the assertion that I must actually list the infinities themselves.

So whose problem is it that your algorithm can't list 1/3 ? Yours or mine?

1/3 is a Real number. Your procedure doesn't list it. Fail.

There is no "infinite precision" that comes after another "infinite precision" because "infinite precision" does not reference anything in its totality, it's not units, it's nothing at all. The "Reals", cannot be the collection of "nothingnesses"; nothingness does not exist, only "somethingness" exists, and I can list all the "somethingnesses".

That's a word salad.

That said, my list does go through every single precision space, and since it is never ending, the list itself will contain every level of precision all the way to infinity.

Precision doesn't matter to me. I'll settle for a finite representation that can compute up to infinite precision in principle.

e.g I'll settle for an algorithm which approximates √2. Even if you don't list √2 to infinite precision.

So ALL the Reals or contained in my list. Go ahead test me! Tall me which Real is not on my list!

√2.

I don't care that you call me an idiot, but to say my list doesn't work is just rude! I feel personally insulted when you say that!

It's not rude when you keep refusing to accept that your list doesn't work

At some point I must assume you are an idiot unable to accept critical feedback.

When do you reach the 1 at the end of infinity? If you reach it, is that the end of infinity? A calculation that never ends outputs ZERO results. If A never = B, then A does NOT = B. Who is the idiot now? Even the mountain sheep in Alaska know that! lol

Of course it does! Have you heard of the Maybe monad?

Either it returns Just(the result of the computation); or it returns Nothing.

https://en.wikipedia.org/wiki/Monad_(fu ... ple:_Maybe

I guess the mountain sheep in Alaska don't understand Monads.

Here is an example of an irrational number whose representation is listed in my method:

√5

Binary number representation:

11100010 10001000 10011010 00110101

Conversion to decimal:

3800603189

Box_20_zeros >>> [0000000000.0000000000] >> [3800603189.0000000000]

Even my goldfish knows how to do that!

That's not √5. Even for an approximation that's super terrible! √5 ≈ 2.23606797749979

You can't represent √5 with finitely many symbols using the representation schema you've chosen.

That's not √5. Even for an approximation that's super terrible! √5 ≈ 2.23606797749979

You can't represent √5 with finitely many symbols using the representation schema you've chosen.

Yes, it is!

Take the number: 3800603189 that is in box_20_zeros: [3800603189.0000000000], and put it into a decimal-to-binary converter here:
https://www.rapidtables.com/convert/num ... inary.html

Then put the binary result into this binary-to-text converter:
https://www.binaryhexconverter.com/bina ... -converter

You will get: "√5" That's it! What more do you want?


My listing method can list each of the completed boxes twice. The first time just to list all the numbers (which means finite), the second time the box will be to represent binary machine code in decimal that can then be converted into binary, which can then be converted into text that can then represent any irrational number that can be written in Ascii Text >> "√5"

The truth is, this is not even really necessary. Every number can just be shown in "Ascii Text", so you can have a list that is made entirely of anything that can be written, which would also include all possible representations of Real numbers. What more could you possibly want? What is missing? I am starting to get really upset that you don't think my list is perfection. It is!

Even the part of your response that I put in quotes above is the number (when converted from text to> binary to> decimal):

3104385642726441179741060634727152565936061285193997139443635168960330358567868766787750422108695473006590514601458512104222439704223845730443407944500934194617236084100809307817589367594756806768072627057439846207130410989035472704918673580111497989951697638874589148042066193449819748621311978793492447555438977790137838777316619637964392498451805372571667163588073107254318303424934966730494651118007284030308623022844535883622084696046927305889868486635054

This number has 460 digits which means it will fit into box_920_zeros: [31043856427264411... ...35054.00000000000000... ...00000000]

So literally anything you write as a response, that you say is not on my list, is in fact on my list! Anything you say, is on my list!

Check it for yourself: https://www.online-toolz.com/tools/text ... vertor.php

And then from the binary into the number: https://www.binaryhexconverter.com/bina ... -converter

[Skepdick]

Yes, it is!

Take the number: 3800603189 that is in box_20_zeros: [3800603189.0000000000], and put it into a decimal-to-binary converter here:
https://www.rapidtables.com/convert/num ... inary.html

Then put the binary result into this binary-to-text converter:
https://www.binaryhexconverter.com/bina ... -converter

You will get: "√5" That's it! What more do you want?

Ah, well that's super peculiar!

If 3800603189.0000000000 means √5, then where is the number 3800603189 in your list?

But you know, I got really curious. I figured "hey! Why don't I do the exact same transformation for 3800603188"

And you know what? It said that 3800603188 is the number √4. So 3800603188 = 2. Is that about right?

And where are all the numbers between √4 and √5 ?

My listing method can list each of the completed boxes twice. The first time just to list all the numbers (which means finite), the second time the box will be to represent binary machine code in decimal that can then be converted into binary, which can then be converted into text that can then represent any irrational number that can be written in Ascii Text >> "√5"

So where's your encoding/decoding specified?

How do you distinguish between 3800603189 which means √5 and 3800603189 which means 3800603189?

The truth is, this is not even really necessary. Every number can just be shown in "Ascii Text", so you can have a list that is made entirely of anything that can be written, which would also include all possible representations of Real numbers. What more could you possibly want? What is missing? I am starting to get really upset that you don't think my list is perfection. It is!

What you are missing is an implicit encoding. It's no good telling me that the bitstring 01011001010100100100100101 represents √10 if you don't tell me how to perform the transformation.

Even the part of your response that I put in quotes above is the number (when converted from text to> binary to> decimal):

Conversion really doesn't help your cause. You are converting the syntax, not the semantics.

So literally anything you write as a response, that you say is not on my list, is in fact on my list! Anything you say, is on my list!

pi isn't on your list. Not the string "pi". The number...

Do you understand the difference between strings and numbers? Doesn't look like it.

string(1) + string(1) = string(11)
number(1) + number(1) = number(2)

[Magnolia5275]

Yes, it is!

Take the number: 3800603189 that is in box_20_zeros: [3800603189.0000000000], and put it into a decimal-to-binary converter here:
https://www.rapidtables.com/convert/num ... inary.html

Then put the binary result into this binary-to-text converter:
https://www.binaryhexconverter.com/bina ... -converter

You will get: "√5" That's it! What more do you want?

Ah, well that's super peculiar!

If 3800603189.0000000000 means √5, then where is the number 3800603189 in your list?

But you know, I got really curious. I figured "hey! Why don't I do the exact same transformation for 3800603188"

And you know what? It said that 3800603188 is the number √4. So 3800603188 = 2. Is that about right?

And where are all the numbers between √4 and √5 ?

My listing method can list each of the completed boxes twice. The first time just to list all the numbers (which means finite), the second time the box will be to represent binary machine code in decimal that can then be converted into binary, which can then be converted into text that can then represent any irrational number that can be written in Ascii Text >> "√5"

So where's your encoding/decoding specified?

How do you distinguish between 3800603189 which means √5 and 3800603189 which means 3800603189?

The truth is, this is not even really necessary. Every number can just be shown in "Ascii Text", so you can have a list that is made entirely of anything that can be written, which would also include all possible representations of Real numbers. What more could you possibly want? What is missing? I am starting to get really upset that you don't think my list is perfection. It is!

What you are missing is an implicit encoding. It's no good telling me that the bitstring 01011001010100100100100101 represents √10 if you don't tell me how to perform the transformation.

Even the part of your response that I put in quotes above is the number (when converted from text to> binary to> decimal):

Conversion really doesn't help your cause. You are converting the syntax, not the semantics.

So literally anything you write as a response, that you say is not on my list, is in fact on my list! Anything you say, is on my list!

pi isn't on your list. Not the string "pi". The number...

Do you understand the difference between strings and numbers? Doesn't look like it.

string(1) + string(1) = string(11)
number(1) + number(1) = number(2)

And where are all the numbers between √4 and √5 ?

Any function/algorithm/representation that can possibly exist, that will give you an infinite stream of numbers between √4 and √5, will necessarily have a finite representation in itself, and would therefore be on the list. Our language is rich enough, in principle, to describe any function that can exist. Anything that can be described in language will be on the list. All possible encoding/decoding methods are also listed, so yes, it includes everything.

What you are missing is an implicit encoding. It's no good telling me that the bitstring 01011001010100100100100101 represents √10 if you don't tell me how to perform the transformation.

If you list the naturals, where is the encoding for our western number digits? What if the person reading your naturals list doesn't know the numeral system? What if he only understands Roman Numerals or Babylonian Numerals? Would you say the list fails then?

Your point here is completely irrelevant, the point is that in principle it is possible to represent all the Reals in a one-to-one correspondence with the naturals. The representation of a number is entirely in our minds. The numeral system in itself does not correspond to the mathematical truth.

Didn't Kurt Godel do something similar to this idea with his Godel numbering system that he used for his famous proofs? In fact, he did exactly what I am doing here which is representing functions using numbers. Would you say his proof is wrong because you are "not allowed" to use numbers in that way?

pi isn't on your list. Not the string "pi". The number...

Do you understand the difference between strings and numbers? Doesn't look like it.

string(1) + string(1) = string(11)
number(1) + number(1) = number(2)

Irrelevant, your entire responses to me are strings. If you, in a post here, show how the naturals are listable, do I get to dismiss it by saying, that technically in computer language you are just listing strings and not numbers that can be plopped into computer code? I honestly don't understand what point it is you are making. It's just so completely irrelevant to the mathematical truth.

Do you know the difference between representation and the underlying mathematical truth? Let me help you: 5 = [ * * * * * ]

Just in case I misunderstood, you are not asking me again to list an infinity? Like actually list the infinity of pi, right? Not even god can list the infinity of pi, it is by definition impossible, there is no such thing as the pi function actualized in infinity. Not only is that not a number, It's not real at all! It cannot possibly exist! You want me to list non-existence?

[Skepdick]

If you list the naturals, where is the encoding for our western number digits? What if the person reading your naturals list doesn't know the numeral system? What if he only understands Roman Numerals or Babylonian Numerals? Would you say the list fails then?

Why are you answering the question with a question?

The issue isn't the encoding (which is a matter of convention) the issue is that you are using multiple encodings in the same list.

So you claim that 1.0 is unencoded, bubt then 3800603189.0 is encoded.

Fine. Where is the unencoded 3800603189.0 ?

Your point here is completely irrelevant, the point is that in principle it is possible to represent all the Reals in a one-to-one correspondence with the naturals. The representation of a number is entirely in our minds.

No, you have that confused. The representation is in front of you. That which the representation is refering to is in your mind.

The numeral system in itself does not correspond to the mathematical truth.

No idea what you are talking about. Mathematics isn't about "truth" - it's about structure.

Didn't Kurt Godel do something similar to this idea with his Godel numbering system that he used for his famous proofs? In fact, he did exactly what I am doing here which is representing functions using numbers. Would you say his proof is wrong because you are "not allowed" to use numbers in that way?

No. I am saying that your proof is wrong because you have namespace conflicts. Does 3800603189 encode the number 3800603189 or √5 ?

Irrelevant, your entire responses to me are strings.

No they aren't. They are numbers. If you are interpreting them as strings then you are misinterpreting them.

If you, in a post here, show how the naturals are listable, do I get to dismiss it by saying, that technically in computer language you are just listing strings and not numbers that can be plopped into computer code? I honestly don't understand what point it is you are making. It's just so completely irrelevant to the mathematical truth.

No. You are dismissing yourself. What do you mean by "the natuals". Don't show me the representations of the naturals - show me the naturals.

Do you know the difference between representation and the underlying mathematical truth? Let me help you: 5 = [ * * * * * ]

Dude. If you want to play stupid I will outstupid your stupid.

0 = [*]
1 = [**]
2 = [***]
3 = [****]
4 = [*****]
5 = [******]

Off-by-1 error. Oops!

Or maybe you are just representing each number with a random character, so the length of the string is sufficient to determine which number you intended to encode?

Code: Select all

In [1]: len("[ * * * * * ]")
Out[1]: 13

You don't even know what "=" means.

Just in case I misunderstood, you are not asking me again to list an infinity? Like actually list the infinity of pi, right?

I am asking you to produce a list of computable representations (in your favourite model of computation) such that if/when evaluated the representation produces the digits of pi.

Not even god can list the infinity of pi, it is by definition impossible, there is no such thing as the pi function actualized in infinity. Not only is that not a number, It's not real at all! It cannot possibly exist! You want me to list non-existence?

All I am seeing on the screen is some characters and strings. I guess none of that stuff you wrote relates to; or represents anything in your head then?
Like what is this "cannot" string mean? It's just an array [c,a,n,n,o,t]. And this "exists" - another array! [e,x,i,s,t,s]. What am I to make of these character arrays?

Gotcha! The contents of your mind don't exist.

Why am I even talking to an empty mind?

[Magnolia5275]

If you list the naturals, where is the encoding for our western number digits? What if the person reading your naturals list doesn't know the numeral system? What if he only understands Roman Numerals or Babylonian Numerals? Would you say the list fails then?

Why are you answering the question with a question?

The issue isn't the encoding (which is a matter of convention) the issue is that you are using multiple encodings in the same list.

So you claim that 1.0 is unencoded, bubt then 3800603189.0 is encoded.

Fine. Where is the unencoded 3800603189.0 ?

Your point here is completely irrelevant, the point is that in principle it is possible to represent all the Reals in a one-to-one correspondence with the naturals. The representation of a number is entirely in our minds.

No, you have that confused. The representation is in front of you. That which the representation is refering to is in your mind.

The numeral system in itself does not correspond to the mathematical truth.

No idea what you are talking about. Mathematics isn't about "truth" - it's about structure.

Didn't Kurt Godel do something similar to this idea with his Godel numbering system that he used for his famous proofs? In fact, he did exactly what I am doing here which is representing functions using numbers. Would you say his proof is wrong because you are "not allowed" to use numbers in that way?

No. I am saying that your proof is wrong because you have namespace conflicts. Does 3800603189 encode the number 3800603189 or √5 ?

Irrelevant, your entire responses to me are strings.

No they aren't. They are numbers. If you are interpreting them as strings then you are misinterpreting them.

If you, in a post here, show how the naturals are listable, do I get to dismiss it by saying, that technically in computer language you are just listing strings and not numbers that can be plopped into computer code? I honestly don't understand what point it is you are making. It's just so completely irrelevant to the mathematical truth.

No. You are dismissing yourself. What do you mean by "the natuals". Don't show me the representations of the naturals - show me the naturals.

Do you know the difference between representation and the underlying mathematical truth? Let me help you: 5 = [ * * * * * ]

Dude. If you want to play stupid I will outstupid your stupid.

0 = [*]
1 = [**]
2 = [***]
3 = [****]
4 = [*****]
5 = [******]

Off-by-1 error. Oops!

Or maybe you are just representing each number with a random character, so the length of the string is sufficient to determine which number you intended to encode?

Code: Select all

In [1]: len("[ * * * * * ]")
Out[1]: 13

You don't even know what "=" means.

Just in case I misunderstood, you are not asking me again to list an infinity? Like actually list the infinity of pi, right?

I am asking you to produce a list of computable representations (in your favourite model of computation) such that if/when evaluated the representation produces the digits of pi.

Not even god can list the infinity of pi, it is by definition impossible, there is no such thing as the pi function actualized in infinity. Not only is that not a number, It's not real at all! It cannot possibly exist! You want me to list non-existence?

All I am seeing on the screen is some characters and strings. I guess none of that stuff you wrote relates to; or represents anything in your head then?
Like what is this "cannot" string mean? It's just an array [c,a,n,n,o,t]. And this "exists" - another array! [e,x,i,s,t,s]. What am I to make of these character arrays?

Gotcha! The contents of your mind don't exist.

Why am I even talking to an empty mind?

No. I am saying that your proof is wrong because you have namespace conflicts. Does 3800603189 encode the number 3800603189 or √5 ?

It's like you didn't read my previous explanations. I said you can list each box twice, one after the other. Once for numbers as seen on the list, and then for the next copy of the same box listed after that, as numbers for conversion into text that can then represent anything.

If that is too complicated for you, and you cannot tolerate having two types of representations in one list, then you can just have the conversion representation. All the numbers on the list will be decimal representations of text. For example, the number 55 will be on my list as: [13621.00000], box_10_zeros. So now we have ONE encoding for the whole list, happy?

I think I have explained everything there is to explain. If you cannot see how this shows, that in principle, you can have an arrow, pointing from each natural number to a corresponding real number, in such a way that all Reals are in the list, then the problem is with your comprehension. I can only explain so much.

I am asking you to produce a list of computable representations (in your favourite model of computation) such that if/when evaluated the representation produces the digits of pi.

Why do I need to be the one to show you it's on the list? Can't you tell for yourself? If any text can be listed, then a whole book, that is just about how to calculate the digits of Pi will be on the list! Here take a look at this web page: https://www.craig-wood.com/nick/articles/pi-chudnovsky/

The whole text there can be converted into a decimal representation. And that is what Pi is! What else is it? It's a function of a Mathematical constant; a way to calculate the ratio of a circle's circumference to its diameter.

No idea what you are talking about. Mathematics isn't about "truth" - it's about structure.

Is the "structure" a truth?