Cardinal of parts of N

[Edgard Neuman]

Hi,

 

I read an article about infinities, and as always, I don't get it.
The writer says : "℘(ℕ)" and "ℕ" are not in bijection..

but, it seems easy to me to create a bijection :


You take the binary writing of a number, and you take the rank integer that correspond to each 1

0 <=> {}

1 <=> {0}

2 <=> {1 }
3 <=>  {0 ; 1 }
4 <=> { 2 }

...
259 <=> { 0; 1 ; 8  }
..

etc and so on

you have an integer for each set of integer and vice-versa, isn't it a bijection ?

So what did I got wrong ?

 

 

Edited September 26, 2019 by Edgard Neuman

[studiot]

1 hour ago, Edgard Neuman said:

The writer says : "℘(ℕ)" and "ℕ" are not in bijection..

If you mean the power set of N and N are not in bijection that is not suprising.

The power set is always larger than the original set.
It is a set of subsets and contains individual subsets, one for every member of the original set (N) along with other sets such as N itself.

N can be put in bijection with R.

 

https://www.google.co.uk/search?ei=Y7aMXePVKO2G1fAPztqtUA&q=power+set+of+N&oq=power+set+of+N&gs_l=psy-ab.3..0l5j0i30l5.1166.1838..3256...0.2..0.1118.1570.4-1j7-1......0....1..gws-wiz.......0i71.Aa6pQ8WKKpI&ved=0ahUKEwijk6nQxe7kAhVtQxUIHU5tCwoQ4dUDCAo&uact=5

Edited September 26, 2019 by studiot

[uncool]

1 hour ago, Edgard Neuman said:

Hi,

 

I read an article about infinities, and as always, I don't get it.
The writer says : "℘(ℕ)" and "ℕ" are not in bijection..

but, it seems easy to me to create a bijection :


You take the binary writing of a number, and you take the rank integer that correspond to each 1

0 <=> {}

1 <=> {0}

2 <=> {1 }
3 <=>  {0 ; 1 }
4 <=> { 2 }

...
259 <=> { 0; 1 ; 8  }
..

etc and so on

you have an integer for each set of integer and vice-versa, isn't it a bijection ?

So what did I got wrong ?

 

 

Because for any integer n, the corresponding subset of N will be finite.

Which integer corresponds to the set of even integers?

[Edgard Neuman]

5 hours ago, studiot said:

If you mean the power set of N and N are not in bijection that is not suprising.

The power set is always larger than the original set.
It is a set of subsets and contains individual subsets, one for every member of the original set (N) along with other sets such as N itself.

N can be put in bijection with R.

 

https://www.google.co.uk/search?ei=Y7aMXePVKO2G1fAPztqtUA&q=power+set+of+N&oq=power+set+of+N&gs_l=psy-ab.3..0l5j0i30l5.1166.1838..3256...0.2..0.1118.1570.4-1j7-1......0....1..gws-wiz.......0i71.Aa6pQ8WKKpI&ved=0ahUKEwijk6nQxe7kAhVtQxUIHU5tCwoQ4dUDCAo&uact=5

No I understand that.. the set of subsets.. but isn't my construction a bijection between integers and sets of integer ??
You say the set of subset is bigger meaning you shouldn't be able to build a bijection ?
Each integer give a subset of integer and each subset of integer gives back the integer.. 

 

5 hours ago, uncool said:

Because for any integer n, the corresponding subset of N will be finite.

Which integer corresponds to the set of even integers?

 ok I get it ! It gives me a lot to think about.. (seems still fishy to me that we consider the set of infinite subset and the set of finite subset on the same level.. isn't it strange ? does card(ℕ) belong to ℕ ? shouldn't we make the distinction ?)
Thanks a lot !

Edited September 26, 2019 by Edgard Neuman

[studiot]

10 minutes ago, Edgard Neuman said:

No I understand that.. the set of subsets.. but isn't my construction a bijection between integers and sets of integer ??
You say the set of subset is bigger meaning you shouldn't be able to build a bijection ?
Each integer give a subset of integer and each subset of integer gives back the integer.. 

 

But you didn't answer my question so how can you understand that?

Did you mean the power set of N ?

 

 

[Edgard Neuman]

2 hours ago, studiot said:

 

But you didn't answer my question so how can you understand that?

Did you mean the power set of N ?

 

 

I wasn't sure exactly what the symbole mean, i copied pasted it from the article.. the article was indeed talking about the set of subsets, so I supposed that was what the symbol meant..   So I wrote that "I understand that" the power set is the set of subsets... (I just didn't think about infinite subsets)
thanks  

Edited September 26, 2019 by Edgard Neuman

[studiot]

2 hours ago, Edgard Neuman said:

I wasn't sure exactly what the symbole mean, i copied pasted it from the article.. the article was indeed talking about the set of subsets, so I supposed that was what the symbol meant..   So I wrote that "I understand that" the power set is the set of subsets... (I just didn't think about infinite subsets)
thanks  

Glad you got it now though I did originally point to at least one infinite subset of N - The set N itself.

In fact it is just awesome how many subsets there are and how many are infinite

This is what put up the cardinality of the power set to that of the continuum.