- #1

- 28

- 1

0.1 - 1

0.2 - 2

0.3 - 3

...

0.21 - 12

...

0.123 - 321

...

0.1245 - 5421

...

I think that is a one-to-one corresepondence. Any errors here?

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- I
- Thread starter emptyboat
- Start date

- #1

- 28

- 1

0.1 - 1

0.2 - 2

0.3 - 3

...

0.21 - 12

...

0.123 - 321

...

0.1245 - 5421

...

I think that is a one-to-one corresepondence. Any errors here?

- #2

fresh_42

Mentor

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The number you get, if you concatenate all changed digits is a number that was not in your list, i.e. whichever list you choose, there is a number that is not listed. Hence the real numbers are uncountable.

- #3

- 28

- 1

Your logic is like this.

The number you get, if you concatenate all changed digits is a number that was not in your list, i.e. whichever list you choose, there is a number that is not listed. Hence the real numbers are uncountable.

1 - 1

2 - 2

3 - 3

...

10 - 10

11 - 11

...

123-123

...

Assume that this list contains all natural numbers. Now add 1 to the ones place of the first number.(1+1=2), add 1 to the tens place of the second number(0+1=1), add 1 to the hundreds place of the third number(0+1=1), and so on. (If you do it third times, you get 112.)

The number you get, if you concatenate all changed digits is a number that was not in your list, i.e. whichever list you choose, there is a number that is not listed. Hence the natural numbers are uncountable.

- #4

fresh_42

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No. Natural numbers are not infinitely long. My argument goes like this:Your logic is like this.

1 - 1

2 - 2

3 - 3

...

10 - 10

11 - 11

...

123-123

...

Assume that this list contains all natural numbers. Now add 1 to the ones place of the first number.(1+1=2), add 1 to the tens place of the second number(0+1=1), add 1 to the hundreds place of the third number(0+1=1), and so on. (If you do it third times, you get 112.)

The number you get, if you concatenate all changed digits is a number that was not in your list, i.e. whichever list you choose, there is a number that is not listed. Hence the natural numbers are uncountable.

0.168435 ...

0.899958 ...

0.915476 ...

0.153798 ...

...

turns into

0.268435 ...

0.809958 ...

0.916476 ...

0.153898 ...

...

If we have e.g. 0.5, then we write it as 0.5000000 ...

You cannot do this with natural numbers.

- #5

- 28

- 1

If INo. Natural numbers are not infinitely long. My argument goes like this:

0.168435 ...

0.899958 ...

0.915476 ...

0.153798 ...

...

turns into

0.268435 ...

0.809958 ...

0.916476 ...

0.153898 ...

...

If we have e.g. 0.5, then we write it as 0.5000000 ...

You cannot do this with natural numbers.

1

2

3

4

...

turns into

2

12

103

1004

I can go on this process.

Natural numbers have the potential to be infinitely long.

- #6

jim mcnamara

Mentor

- 4,402

- 3,089

But naturals are countably infinite. The reals are uncountably infinite. Not the same.

I do not see where things got confused, but I think that @fresh_42 did not want to explicitly inflict Cantor's proof on you. I think he is using the proof in "example mode". He can correct me.

Further reading which I hope helps you:

https://en.wikipedia.org/wiki/Countable_set

- #7

FactChecker

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All the numbers you listed have finite length and end in infinite 0's. At what point would you count 0.1212121212...?

0.1 - 1

0.2 - 2

0.3 - 3

...

0.21 - 12

...

0.123 - 321

...

0.1245 - 5421

...

I think that is a one-to-one corresepondence. Any errors here?

And there are a lot more where that came from. I think that you can see that you have only counted a tiny, tiny subset of the real numbers.

EDIT: In fact, this method misses a lot of rational numbers, which are countable. 0.12121212.. = 12/99.

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- #8

- 31,819

- 8,668

Besides the actual proofs that the reals are uncountable, it seems clear that your construction fails for real numbers greater than 1. I also don’t think it works for irrational numbers and maybe not even some rational numbers.I think that is a one-to-one corresepondence. Any errors here?

- #9

TeethWhitener

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After we have the second list:No. Natural numbers are not infinitely long. My argument goes like this:

0.168435 ...

0.899958 ...

0.915476 ...

0.153798 ...

...

turns into

0.268435 ...

0.809958 ...

0.916476 ...

0.153898 ...

...

If we have e.g. 0.5, then we write it as 0.5000000 ...

You cannot do this with natural numbers.

0.

0.8

0.91

0.153

we take the changed (bolded) numbers and make a new number (0.2068…) that wasn’t in the previous list. We know it wasn’t because if it were the nth number, its nth digit would be 1 less than it is.

- #10

- 28

- 1

Every sigle-digit decimals correspond to every single-digit natural numbers(9 pieces).

(1-0.1, 2-0.2, 3-0.3, ...., 9-0.9)

Every two-digits decimals correspond to every two-digit natural numbers(90 pieces).

(10 - 0.01, 11 - 0.11, 12-0.21, ..., 97 - 0.79, 98 - 0.89, 99 - 0.99)

Every three-digits decimals correspond to every three-digit natural numbers(900 pieces).

(100 - 0.001, 101 - 0.101, 102 - 0.201, ..., 997 - 0.799, 998 - 0.899, 999 - 0.999)

And so on....

Every dicimals have countable digit decimals, aren't they?

If you let go of the mindset that Real numbers are uncountable, it's very simple and easy.

Aren't they?

- #11

TeethWhitener

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Prove it.Every dicimals have countable digit decimals, aren't they?

- #12

- 28

- 1

If natural numbers have countable digits, and every dicimal also has countable digits.Prove it.

(Because it's mirror refection with zero in center).

Natural numbers are countable and natural numbers are greater than digits of natural numbers, so digits of natural numbers are countable, and also every dicimal has countable digits.

- #13

FactChecker

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What would you call ##12/99 = 0.1212121212...= 0.121212\overline{12}##? Are you going to ignore all the rational numbers like that?If natural numbers have countable digits, and every dicimal also has countable digits.

(Because it's mirror refection with zero in center).

Natural numbers are countable and natural numbers are greater than digits of natural numbers, so digits of natural numbers are countable, and also every dicimal has countable digits.

- #14

TeethWhitener

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You can’t just assert the part I asked you to prove and expect anyone to buy it. If you want to assert that every infinite string can be mapped to the reversed infinite string, then yes, you’ve just effectively shown that you can map the real numbers to the real numbers. But you have to put “every dicimal” in 1-1 correspondence with every natural number first. You’ve been given a number of counterexamples at this point. I suggest you study them with a little bit of seriousness.and every dicimal also has countable digits.

- #15

- 28

- 1

12/99 is correspond to .....1212121212, but there is not such a natural number.What would you call ##12/99 = 0.1212121212...= 0.121212\overline{12}##? Are you going to ignore all the rational numbers like that?

But I think if your logic is true, rational numbers are also uncountable.

In the case of rational numbers, countable means like that, it shows a possibility of one-to-one correspondence. But in the case of real numbers, the perspective changes.

- #16

mathwonk

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- #17

FactChecker

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No. It just shows that the natural numbers are not countable12/99 is correspond to .....1212121212, but there is not such a natural number.

But I think if your logic is true, rational numbers are also uncountable.

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- #18

- 28

- 1

No. It just shows that the natural numbers are not countableby your method. They are countable but your method is grossly flawed.

Yes. I understand countable meaning and subtle difference between rational numbers and real numbers.

I pick a random rational number as 12/99 and if order can be given it's countable.

But if I pick π, I cannot give order to that number. So it's uncountable.

Thanks!!

- #19

FactChecker

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I think it is worthwhile to realize how easy it is to create irrational numbers. Just make sequences that never start repeating, like 0.101001000100001000000100000001000000001...

And for any such number, you can add it to a list of rational numbers to get a list of irrational numbers.

IMHO, that makes it easier to accept how the irrational numbers are such a huge set compared to the rationals (although it is not a proof by any means).

Cantor's proof is genius and is worth studying. Mathematicians of his time hated and resisted his conclusion that there are higher, uncountable, orders of infinity, but it was undeniable.

And for any such number, you can add it to a list of rational numbers to get a list of irrational numbers.

IMHO, that makes it easier to accept how the irrational numbers are such a huge set compared to the rationals (although it is not a proof by any means).

Cantor's proof is genius and is worth studying. Mathematicians of his time hated and resisted his conclusion that there are higher, uncountable, orders of infinity, but it was undeniable.

Last edited by a moderator:

- #20

mathman

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You list has only numbers with finite decimal expansions - not enough.

0.1 - 1

0.2 - 2

0.3 - 3

...

0.21 - 12

...

0.123 - 321

...

0.1245 - 5421

...

I think that is a one-to-one corresepondence. Any errors here?

- #21

- 2,120

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- #22

Svein

Science Advisor

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And the set of all subsets of the real numbers are even more uncountable and so on...

- #23

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I trust that you understand that 'uncountable' is a 'yes-or-no' condition, and that therefore nothing can possibly be "even more uncountable" than something else that is 'uncountable' is; however, I think that it's worth pointing out that even the infinitesimal is exactly as 'uncountably infinite' in its fractional expansion as the entirety of ##\mathbb{R}## (the reals) in its fractional expansions is (the concept of divisibility may be regarded as inconsistent with the concept of infinitesimality; i.e., it could be not unrightly said that the infinitesimal is not divisible, wherefore, there could be no 'fractional expansion' of the infinitesimal).And the set of all subsets of the real numbers are even more uncountable and so on...

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- #24

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That reminds me of Russel's ParadoxAnd the set of all subsets of the real numbers are even more uncountable and so on...

- #25

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- 1,019

The reals on [0,1] are easily countable, just paste them in Excel and click 'sort ascending'

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