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

[Skepdick]

Given the interval (0,1) on ℝ choose any point you want!

For any point x the cardinality of [0,x) is always identical to the cardinality of (x,1]

Therefore it's impossible to choose anything other than 0.5.

[Magnolia5275]

Infinity is not an amount or "number". You cannot use infinity to define what a number is. That said, it's true that in absolute terms, numbers can be viewed as relative. I can say "1000" is the "new one", so 2000 = 2, because: "one can be divided by 1000". But that is not really how we think about numbers, so no, you are not actually correct.

[Impenitent]

not all red birds have cardinality ...

-Imp

[Harbal]

Given the interval (0,1) on ℝ choose any point you want!

For any point x the cardinality of [0,x) is always identical to the cardinality of (x,1]

Therefore it's impossible to choose anything other than 0.5.

That doesn't make sense.

[Skepdick]

Infinity is not an amount or "number".

Sure. I didn't say it is. It's a useful concept ready for exploitation.

You cannot use infinity to define what a number is.

That's not true. If infinity was a number; and we used it to define what a number is - that would be a circular definition. Right?
No different to Peano proclaiming that 0 (and its infinite successors) are numbers without ever attempting to define what a number is!

So, of course I can use infinity (a non-number!) and my basic geometric intuition to define what a number is. That way my definition won't be circular/vacuous.

That said, it's true that in absolute terms, numbers can be viewed as relative. I can say "1000" is the "new one", so 2000 = 2, because: "one can be divided by 1000". But that is not really how we think about numbers, so no, you are not actually correct.

OK, but it is actually how we think about infinities. They can be compared.

And since there's a bijection from [0,x) to (x,1] for any x in (0,1) - I am actually correct.

If any x always splits the interval [0,1] into two parts of equal size how could it be anything other than 0.5 ?

You can do this exact thought experiment with the entire Real number line. Pick any point on (-∞, +∞). Since there's a bijection from (-∞, x] to [x, ∞) for any x then every x must be exactly in the middle! So... x=0.

Quotent types is precisely how we think about dividing things into equal halves.

[Skepdick]

Given the interval (0,1) on ℝ choose any point you want!

For any point x the cardinality of [0,x) is always identical to the cardinality of (x,1]

Therefore it's impossible to choose anything other than 0.5.

That doesn't make sense.

Maybe your bad eyes are catching up with you. Need me to change the color/size of the font?

[Harbal]

That doesn't make sense.

Maybe your bad eyes are catching up with you. Need me to change the color/size of the font?

I still wouldn't understand it.

[Skepdick]

I still wouldn't understand it.

OK, I'll translate the formal stuff into plain English.

Take a piece of string.
Cut anywhere.
Now you have two strings.
Put the strings side by side.

Imagine your surprise when you discover the two strings are exactly the same size.
Imagine your further surprise if every time you did the experiment you ended up with two strings of identical size.
Imagine your ultimate surprise if despite your best efforts you are unable to cut the string such that the two parts are NOT identical in size.

That would be a magical piece of string, wouldn't it?

That piece of string is what Mathematicians call "The Real number line".

[Harbal]

I still wouldn't understand it.

OK, I'll translate the formal stuff into plain English.

Take a piece of string.
Cut anywhere.
Now you have two strings.

Imagine your surprise when you put the two strings side by side you discover that they are exactly the same size.
Imagine your surprise when you keep doing the experiment and exactly the same thing keeps happening every time.

That would be a magical piece of string, wouldn't it?

Yes, I agree, it would be unusual string. But string doesn't normally behave like that, so what is the equation supposed to show?

[Skepdick]

Yes, I agree, it would be unusual string. But string doesn't normally behave like that, so what is the equation supposed to show?

The object Mathematicians commonly refer to as "The Real number line" behaves exactly like that!

Despite your best efforts you are unable to cut the string such that the two parts are NOT identical in size.

[Harbal]

Despite your best efforts you are unable to cut the string such that the two parts are NOT identical in size.

You've lost me again. What does ∀x in R mean?

[Skepdick]

You've lost me again. What does ∀x in R mean?

For all points on the Real number line.

[Magnolia5275]

Infinity is not an amount or "number".

Sure. I didn't say it is. It's a useful concept ready for exploitation.

You cannot use infinity to define what a number is.

That's not true. If infinity was a number; and we used it to define what a number is - that would be a circular definition. Right?
No different to Peano proclaiming that 0 (and its infinite successors) are numbers without ever attempting to define what a number is!

So, of course I can use infinity (a non-number!) and my basic geometric intuition to define what a number is. That way my definition won't be circular/vacuous.

That said, it's true that in absolute terms, numbers can be viewed as relative. I can say "1000" is the "new one", so 2000 = 2, because: "one can be divided by 1000". But that is not really how we think about numbers, so no, you are not actually correct.

OK, but it is actually how we think about infinities. They can be compared.

And since there's a bijection from [0,x) to (x,1] for any x in (0,1) - I am actually correct.

If any x always splits the interval [0,1] into two parts of equal size how could it be anything other than 0.5 ?

You can do this exact thought experiment with the entire Real number line. Pick any point on (-∞, +∞). Since there's a bijection from (-∞, x] to [x, ∞) for any x then every x must be exactly in the middle! So... x=0.

Quotent types is precisely how we think about dividing things into equal halves.

That's not true. If infinity was a number; and we used it to define what a number is - that would be a circular definition. Right?
No different to Peano proclaiming that 0 (and its infinite successors) are numbers without ever attempting to define what a number is!

So, of course I can use infinity (a non-number!) and my basic geometric intuition to define what a number is. That way my definition won't be circular/vacuous.

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

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.

OK, but it is actually how we think about infinities. They can be compared.

And since there's a bijection from [0,x) to (x,1] for any x in (0,1) - I am actually correct.

If any x always splits the interval [0,1] into two parts of equal size how could it be anything other than 0.5 ?

You can do this exact thought experiment with the entire Real number line. Pick any point on (-∞, +∞). Since there's a bijection from (-∞, x] to [x, ∞) for any x then every x must be exactly in the middle! So... x=0.

Quotent types is precisely how we think about dividing things into equal halves.

You cannot use the notion of "size", when talking about infinity. The word "size" by definition means it has some numerical measurement. That doesn't exist in infinity by definition.

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.

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

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.

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.

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

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.

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

[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" of infinity depends on an infinite number of steps. That's 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 conceptual 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 larger than the other.

No numbers required - just intuition.

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, but an English notion.

It's common sense concept that any person off the street understands intuitively. It refers to magnitude or dimension of something.
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.

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.

I care about the semantic properties of an object. How it behaves in my head - not its definition.

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.

No, I am not. The interval (0, 1) is isomorphic to ℝ, so whether I am talking about (0,1) or (-∞, +∞) - there's no difference.

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. SO every object in N maps to an object in R.

And what does 0.5 in R map to in N?

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 between N and R?

[Magnolia5275]

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