Definition of the Number Two in ZF Set Theory
0
$begingroup$
This page shows how natural numbers may be defined as unions of sets.
What is a straightforward way to appeal to intuitive notions to dispel the misconception that
{Ø} + {Ø} = {{Ø}, {Ø}, Ø} and instead show that
{Ø} + {Ø} = {Ø, {Ø}}
?
elementary-set-theory natural-numbers
asked Dec 14 '18 at 21:01
bblohowiakbblohowiak
1099
$endgroup$
add a comment |
0
$begingroup$
This page shows how natural numbers may be defined as unions of sets.
What is a straightforward way to appeal to intuitive notions to dispel the misconception that
{Ø} + {Ø} = {{Ø}, {Ø}, Ø} and instead show that
{Ø} + {Ø} = {Ø, {Ø}}
?
elementary-set-theory natural-numbers
asked Dec 14 '18 at 21:01
bblohowiakbblohowiak
1099
$endgroup$
1
$begingroup$
But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
$endgroup$
– Saucy O'Path
Dec 14 '18 at 21:04
1
$begingroup$
Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
$endgroup$
– John Hughes
Dec 14 '18 at 21:07
add a comment |
0
0
0
$begingroup$
This page shows how natural numbers may be defined as unions of sets.
What is a straightforward way to appeal to intuitive notions to dispel the misconception that
{Ø} + {Ø} = {{Ø}, {Ø}, Ø} and instead show that
{Ø} + {Ø} = {Ø, {Ø}}
?
elementary-set-theory natural-numbers
asked Dec 14 '18 at 21:01
bblohowiakbblohowiak
1099
$endgroup$
This page shows how natural numbers may be defined as unions of sets.
What is a straightforward way to appeal to intuitive notions to dispel the misconception that
{Ø} + {Ø} = {{Ø}, {Ø}, Ø} and instead show that
{Ø} + {Ø} = {Ø, {Ø}}
?
elementary-set-theory natural-numbers
elementary-set-theory natural-numbers
asked Dec 14 '18 at 21:01
bblohowiakbblohowiak
1099
asked Dec 14 '18 at 21:01
bblohowiakbblohowiak
1099
edited Dec 15 '18 at 20:01
bblohowiak
asked Dec 14 '18 at 21:01
bblohowiakbblohowiak
1099
asked Dec 14 '18 at 21:01
bblohowiakbblohowiak
1099
asked Dec 14 '18 at 21:01
bblohowiakbblohowiak
1099
1099
1
$begingroup$
But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
$endgroup$
– Saucy O'Path
Dec 14 '18 at 21:04
1
$begingroup$
Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
$endgroup$
– John Hughes
Dec 14 '18 at 21:07
add a comment |
1
$begingroup$
But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
$endgroup$
– Saucy O'Path
Dec 14 '18 at 21:04
1
$begingroup$
Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
$endgroup$
– John Hughes
Dec 14 '18 at 21:07
1
1
$begingroup$
But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
$endgroup$
– Saucy O'Path
Dec 14 '18 at 21:04
$begingroup$
But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
$endgroup$
– Saucy O'Path
Dec 14 '18 at 21:04
1
1
$begingroup$
Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
$endgroup$
– John Hughes
Dec 14 '18 at 21:07
$begingroup$
Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
$endgroup$
– John Hughes
Dec 14 '18 at 21:07
add a comment |
1 Answer
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4
$begingroup$
Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.
answered Dec 14 '18 at 21:04
Asaf Karagila♦Asaf Karagila
306k33437767
$endgroup$
add a comment |
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4
$begingroup$
Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.
answered Dec 14 '18 at 21:04
Asaf Karagila♦Asaf Karagila
306k33437767
$endgroup$
add a comment |
4
$begingroup$
Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.
answered Dec 14 '18 at 21:04
Asaf Karagila♦Asaf Karagila
306k33437767
$endgroup$
add a comment |
4
4
4
$begingroup$
Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.
answered Dec 14 '18 at 21:04
Asaf Karagila♦Asaf Karagila
306k33437767
$endgroup$
Note that ${x,x,y}={x,y}$. So the two sets you suggest are in fact the same.
answered Dec 14 '18 at 21:04
Asaf Karagila♦Asaf Karagila
306k33437767
answered Dec 14 '18 at 21:04
Asaf Karagila♦Asaf Karagila
306k33437767
answered Dec 14 '18 at 21:04
Asaf Karagila♦Asaf Karagila
306k33437767
answered Dec 14 '18 at 21:04
Asaf Karagila♦Asaf Karagila
306k33437767
306k33437767
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1
$begingroup$
But ${{emptyset},{emptyset},emptyset}$ is ${emptyset,{emptyset}}$ !
$endgroup$
– Saucy O'Path
Dec 14 '18 at 21:04
1
$begingroup$
Furthermore, there's no particular reason why $0$ has to be defined as the empty set, $1$ has to be ${emptyset}$, and so on. We could equally well define $0$ to $ {emptyset }$, and $1$ to be ${emptyset, {emptyset} }$, etc. Of course, we'd then have to modify the definition of successor, and of addition, etc. --- but all we're looking for is a collection of distinct things that we can use to represent the natural numbers (in some orderly way to make other steps easier).
$endgroup$
– John Hughes
Dec 14 '18 at 21:07