Mixing
What does: “for all free variables shown” mean in Set Theory.
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I am reading the definition of what it means for a class $A$ to model a formula of the Language of Set Theory. It begin,
Let $A$ be a class and $phi(x_1,ldots, x_n)$ be a formula of the Language of Set Theory with all free variables shown.
What does this mean? Does this mean that in the formula $phi$, the free variables are exactly $x_1ldots x_n$. Or are the free variables among $x_1, ldots x_n$, i.e. some may be free and some may not be free?
For example if $phi$ were the formula $forall x(x = x_1)$ would we write $phi(x_1)$ or $phi(x,x_1)$? The first only shows the free variables while the second shows all variables.
logic set-theory
edited Jan 20 at 14:12
Scientifica
6,79641335
asked Jan 20 at 14:05
foshofosho
4,7661033
$endgroup$
add a comment |
$begingroup$
I am reading the definition of what it means for a class $A$ to model a formula of the Language of Set Theory. It begin,
Let $A$ be a class and $phi(x_1,ldots, x_n)$ be a formula of the Language of Set Theory with all free variables shown.
What does this mean? Does this mean that in the formula $phi$, the free variables are exactly $x_1ldots x_n$. Or are the free variables among $x_1, ldots x_n$, i.e. some may be free and some may not be free?
For example if $phi$ were the formula $forall x(x = x_1)$ would we write $phi(x_1)$ or $phi(x,x_1)$? The first only shows the free variables while the second shows all variables.
logic set-theory
edited Jan 20 at 14:12
Scientifica
6,79641335
asked Jan 20 at 14:05
foshofosho
4,7661033
$endgroup$
$begingroup$
"all free variables shown" means exactly that $x_1, ldots, x_n$ are all and only the free vars occurring in $phi$.
$endgroup$
– Mauro ALLEGRANZA
Jan 20 at 14:10
add a comment |
$begingroup$
I am reading the definition of what it means for a class $A$ to model a formula of the Language of Set Theory. It begin,
Let $A$ be a class and $phi(x_1,ldots, x_n)$ be a formula of the Language of Set Theory with all free variables shown.
What does this mean? Does this mean that in the formula $phi$, the free variables are exactly $x_1ldots x_n$. Or are the free variables among $x_1, ldots x_n$, i.e. some may be free and some may not be free?
For example if $phi$ were the formula $forall x(x = x_1)$ would we write $phi(x_1)$ or $phi(x,x_1)$? The first only shows the free variables while the second shows all variables.
logic set-theory
edited Jan 20 at 14:12
Scientifica
6,79641335
asked Jan 20 at 14:05
foshofosho
4,7661033
$endgroup$
I am reading the definition of what it means for a class $A$ to model a formula of the Language of Set Theory. It begin,
Let $A$ be a class and $phi(x_1,ldots, x_n)$ be a formula of the Language of Set Theory with all free variables shown.
What does this mean? Does this mean that in the formula $phi$, the free variables are exactly $x_1ldots x_n$. Or are the free variables among $x_1, ldots x_n$, i.e. some may be free and some may not be free?
For example if $phi$ were the formula $forall x(x = x_1)$ would we write $phi(x_1)$ or $phi(x,x_1)$? The first only shows the free variables while the second shows all variables.
logic set-theory
logic set-theory
edited Jan 20 at 14:12
Scientifica
6,79641335
asked Jan 20 at 14:05
foshofosho
4,7661033
edited Jan 20 at 14:12
Scientifica
6,79641335
asked Jan 20 at 14:05
foshofosho
4,7661033
edited Jan 20 at 14:12
Scientifica
6,79641335
edited Jan 20 at 14:12
Scientifica
6,79641335
edited Jan 20 at 14:12
Scientifica
6,79641335
6,79641335
asked Jan 20 at 14:05
foshofosho
4,7661033
asked Jan 20 at 14:05
foshofosho
4,7661033
asked Jan 20 at 14:05
foshofosho
4,7661033
4,7661033
$begingroup$
"all free variables shown" means exactly that $x_1, ldots, x_n$ are all and only the free vars occurring in $phi$.
$endgroup$
– Mauro ALLEGRANZA
Jan 20 at 14:10
add a comment |
$begingroup$
"all free variables shown" means exactly that $x_1, ldots, x_n$ are all and only the free vars occurring in $phi$.
$endgroup$
– Mauro ALLEGRANZA
Jan 20 at 14:10
$begingroup$
"all free variables shown" means exactly that $x_1, ldots, x_n$ are all and only the free vars occurring in $phi$.
$endgroup$
– Mauro ALLEGRANZA
Jan 20 at 14:10
$begingroup$
"all free variables shown" means exactly that $x_1, ldots, x_n$ are all and only the free vars occurring in $phi$.
$endgroup$
– Mauro ALLEGRANZA
Jan 20 at 14:10
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Your first interpretation is correct. In general, when we write "$varphi(x_1,...,x_n)$" we are indicating that each of the variables $x_1,...,x_n$ is free in $varphi$. (This is very similar to function notation: the idea is that $varphi$ can take inputs corresponding to the $x_i$s. The weird bit is that sometimes we don't display all the free variables.)
answered Jan 20 at 14:10
Noah SchweberNoah Schweber
126k10151290
$endgroup$
add a comment |
$begingroup$
My guess is that it means that the free variables are precisely $x_1,dots,x_n$, i.e, each of them is a free variable and all of them appear somewhere in the formula. For example, you wouldn't write $forall x (x=x_1)$ as $phi(x_1,x_2)$ but rather $phi(x_1)$ because $x_2$ doesn't appear in it.
answered Jan 20 at 14:09
ScientificaScientifica
6,79641335
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your first interpretation is correct. In general, when we write "$varphi(x_1,...,x_n)$" we are indicating that each of the variables $x_1,...,x_n$ is free in $varphi$. (This is very similar to function notation: the idea is that $varphi$ can take inputs corresponding to the $x_i$s. The weird bit is that sometimes we don't display all the free variables.)
answered Jan 20 at 14:10
Noah SchweberNoah Schweber
126k10151290
$endgroup$
add a comment |
$begingroup$
Your first interpretation is correct. In general, when we write "$varphi(x_1,...,x_n)$" we are indicating that each of the variables $x_1,...,x_n$ is free in $varphi$. (This is very similar to function notation: the idea is that $varphi$ can take inputs corresponding to the $x_i$s. The weird bit is that sometimes we don't display all the free variables.)
answered Jan 20 at 14:10
Noah SchweberNoah Schweber
126k10151290
$endgroup$
add a comment |
$begingroup$
Your first interpretation is correct. In general, when we write "$varphi(x_1,...,x_n)$" we are indicating that each of the variables $x_1,...,x_n$ is free in $varphi$. (This is very similar to function notation: the idea is that $varphi$ can take inputs corresponding to the $x_i$s. The weird bit is that sometimes we don't display all the free variables.)
answered Jan 20 at 14:10
Noah SchweberNoah Schweber
126k10151290
$endgroup$
Your first interpretation is correct. In general, when we write "$varphi(x_1,...,x_n)$" we are indicating that each of the variables $x_1,...,x_n$ is free in $varphi$. (This is very similar to function notation: the idea is that $varphi$ can take inputs corresponding to the $x_i$s. The weird bit is that sometimes we don't display all the free variables.)
answered Jan 20 at 14:10
Noah SchweberNoah Schweber
126k10151290
answered Jan 20 at 14:10
Noah SchweberNoah Schweber
126k10151290
answered Jan 20 at 14:10
Noah SchweberNoah Schweber
126k10151290
answered Jan 20 at 14:10
Noah SchweberNoah Schweber
126k10151290
126k10151290
add a comment |
add a comment |
$begingroup$
My guess is that it means that the free variables are precisely $x_1,dots,x_n$, i.e, each of them is a free variable and all of them appear somewhere in the formula. For example, you wouldn't write $forall x (x=x_1)$ as $phi(x_1,x_2)$ but rather $phi(x_1)$ because $x_2$ doesn't appear in it.
answered Jan 20 at 14:09
ScientificaScientifica
6,79641335
$endgroup$
add a comment |
$begingroup$
My guess is that it means that the free variables are precisely $x_1,dots,x_n$, i.e, each of them is a free variable and all of them appear somewhere in the formula. For example, you wouldn't write $forall x (x=x_1)$ as $phi(x_1,x_2)$ but rather $phi(x_1)$ because $x_2$ doesn't appear in it.
answered Jan 20 at 14:09
ScientificaScientifica
6,79641335
$endgroup$
add a comment |
$begingroup$
My guess is that it means that the free variables are precisely $x_1,dots,x_n$, i.e, each of them is a free variable and all of them appear somewhere in the formula. For example, you wouldn't write $forall x (x=x_1)$ as $phi(x_1,x_2)$ but rather $phi(x_1)$ because $x_2$ doesn't appear in it.
answered Jan 20 at 14:09
ScientificaScientifica
6,79641335
$endgroup$
My guess is that it means that the free variables are precisely $x_1,dots,x_n$, i.e, each of them is a free variable and all of them appear somewhere in the formula. For example, you wouldn't write $forall x (x=x_1)$ as $phi(x_1,x_2)$ but rather $phi(x_1)$ because $x_2$ doesn't appear in it.
answered Jan 20 at 14:09
ScientificaScientifica
6,79641335
answered Jan 20 at 14:09
ScientificaScientifica
6,79641335
answered Jan 20 at 14:09
ScientificaScientifica
6,79641335
answered Jan 20 at 14:09
ScientificaScientifica
6,79641335
6,79641335
add a comment |
add a comment |
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$begingroup$
"all free variables shown" means exactly that $x_1, ldots, x_n$ are all and only the free vars occurring in $phi$.
$endgroup$
– Mauro ALLEGRANZA
Jan 20 at 14:10