How to prove $mathbb{R}$x$mathbb{N}$ is not enumerable
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How to prove $mathbb{R}$x$mathbb{N}$ is not enumerable?
I dont know how to prove these, i know $mathbb{R}$ is not enumarable so $mathbb{R}$x$mathbb{N}$ can be enumareble. Please i need help.
first-order-logic
asked Nov 16 at 0:16
aderey
31
add a comment |
up vote
0
down vote
favorite
How to prove $mathbb{R}$x$mathbb{N}$ is not enumerable?
I dont know how to prove these, i know $mathbb{R}$ is not enumarable so $mathbb{R}$x$mathbb{N}$ can be enumareble. Please i need help.
first-order-logic
asked Nov 16 at 0:16
aderey
31
1
What do you know about comparing cardinalities? How would one show that one cardinality is at least as big as another, for instance?
– Arthur
Nov 16 at 0:19
Well, you have an injective function $mathbb{R} hookrightarrow mathbb{R} times mathbb{N}$ and you know that the reals are not countable. What if you could countably list all of $mathbb{R} times mathbb{N}$, couldn't you backtrack and countably list all the reals?
– Guido A.
Nov 16 at 0:19
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
How to prove $mathbb{R}$x$mathbb{N}$ is not enumerable?
I dont know how to prove these, i know $mathbb{R}$ is not enumarable so $mathbb{R}$x$mathbb{N}$ can be enumareble. Please i need help.
first-order-logic
asked Nov 16 at 0:16
aderey
31
How to prove $mathbb{R}$x$mathbb{N}$ is not enumerable?
I dont know how to prove these, i know $mathbb{R}$ is not enumarable so $mathbb{R}$x$mathbb{N}$ can be enumareble. Please i need help.
first-order-logic
first-order-logic
asked Nov 16 at 0:16
aderey
31
asked Nov 16 at 0:16
aderey
31
asked Nov 16 at 0:16
aderey
31
asked Nov 16 at 0:16
aderey
31
asked Nov 16 at 0:16
aderey
31
31
1
What do you know about comparing cardinalities? How would one show that one cardinality is at least as big as another, for instance?
– Arthur
Nov 16 at 0:19
Well, you have an injective function $mathbb{R} hookrightarrow mathbb{R} times mathbb{N}$ and you know that the reals are not countable. What if you could countably list all of $mathbb{R} times mathbb{N}$, couldn't you backtrack and countably list all the reals?
– Guido A.
Nov 16 at 0:19
add a comment |
1
What do you know about comparing cardinalities? How would one show that one cardinality is at least as big as another, for instance?
– Arthur
Nov 16 at 0:19
Well, you have an injective function $mathbb{R} hookrightarrow mathbb{R} times mathbb{N}$ and you know that the reals are not countable. What if you could countably list all of $mathbb{R} times mathbb{N}$, couldn't you backtrack and countably list all the reals?
– Guido A.
Nov 16 at 0:19
1
1
What do you know about comparing cardinalities? How would one show that one cardinality is at least as big as another, for instance?
– Arthur
Nov 16 at 0:19
What do you know about comparing cardinalities? How would one show that one cardinality is at least as big as another, for instance?
– Arthur
Nov 16 at 0:19
Well, you have an injective function $mathbb{R} hookrightarrow mathbb{R} times mathbb{N}$ and you know that the reals are not countable. What if you could countably list all of $mathbb{R} times mathbb{N}$, couldn't you backtrack and countably list all the reals?
– Guido A.
Nov 16 at 0:19
Well, you have an injective function $mathbb{R} hookrightarrow mathbb{R} times mathbb{N}$ and you know that the reals are not countable. What if you could countably list all of $mathbb{R} times mathbb{N}$, couldn't you backtrack and countably list all the reals?
– Guido A.
Nov 16 at 0:19
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
If you know $Bbb R$ is not enumerable, you can inject it into $Bbb {R times N}$ by $x to (x,1)$ to show $Bbb {R times N}$ is not enumerable.
answered Nov 16 at 0:20
Ross Millikan
288k23195365
Thanks, i forgot about injections to compare cardinality.
– aderey
Nov 16 at 0:32
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
If you know $Bbb R$ is not enumerable, you can inject it into $Bbb {R times N}$ by $x to (x,1)$ to show $Bbb {R times N}$ is not enumerable.
answered Nov 16 at 0:20
Ross Millikan
288k23195365
Thanks, i forgot about injections to compare cardinality.
– aderey
Nov 16 at 0:32
add a comment |
up vote
2
down vote
accepted
If you know $Bbb R$ is not enumerable, you can inject it into $Bbb {R times N}$ by $x to (x,1)$ to show $Bbb {R times N}$ is not enumerable.
answered Nov 16 at 0:20
Ross Millikan
288k23195365
Thanks, i forgot about injections to compare cardinality.
– aderey
Nov 16 at 0:32
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
If you know $Bbb R$ is not enumerable, you can inject it into $Bbb {R times N}$ by $x to (x,1)$ to show $Bbb {R times N}$ is not enumerable.
answered Nov 16 at 0:20
Ross Millikan
288k23195365
If you know $Bbb R$ is not enumerable, you can inject it into $Bbb {R times N}$ by $x to (x,1)$ to show $Bbb {R times N}$ is not enumerable.
answered Nov 16 at 0:20
Ross Millikan
288k23195365
answered Nov 16 at 0:20
Ross Millikan
288k23195365
answered Nov 16 at 0:20
Ross Millikan
288k23195365
answered Nov 16 at 0:20
Ross Millikan
288k23195365
288k23195365
Thanks, i forgot about injections to compare cardinality.
– aderey
Nov 16 at 0:32
add a comment |
Thanks, i forgot about injections to compare cardinality.
– aderey
Nov 16 at 0:32
Thanks, i forgot about injections to compare cardinality.
– aderey
Nov 16 at 0:32
Thanks, i forgot about injections to compare cardinality.
– aderey
Nov 16 at 0:32
add a comment |
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What do you know about comparing cardinalities? How would one show that one cardinality is at least as big as another, for instance?
– Arthur
Nov 16 at 0:19
Well, you have an injective function $mathbb{R} hookrightarrow mathbb{R} times mathbb{N}$ and you know that the reals are not countable. What if you could countably list all of $mathbb{R} times mathbb{N}$, couldn't you backtrack and countably list all the reals?
– Guido A.
Nov 16 at 0:19