What exactly is a formula in set theory?
$begingroup$
I've taken a look at this: Set theory formula
But I'm still a little confused. Is the formula supposed to take x, y, z as input and produce "TRUE" as an output? Are all set theory formula's TRUE?
set-theory first-order-logic
edited Dec 6 '18 at 16:25
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 11:22
l3utterflyl3utterfly
1224
$endgroup$
add a comment |
$begingroup$
I've taken a look at this: Set theory formula
But I'm still a little confused. Is the formula supposed to take x, y, z as input and produce "TRUE" as an output? Are all set theory formula's TRUE?
set-theory first-order-logic
edited Dec 6 '18 at 16:25
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 11:22
l3utterflyl3utterfly
1224
$endgroup$
add a comment |
$begingroup$
I've taken a look at this: Set theory formula
But I'm still a little confused. Is the formula supposed to take x, y, z as input and produce "TRUE" as an output? Are all set theory formula's TRUE?
set-theory first-order-logic
edited Dec 6 '18 at 16:25
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 11:22
l3utterflyl3utterfly
1224
$endgroup$
I've taken a look at this: Set theory formula
But I'm still a little confused. Is the formula supposed to take x, y, z as input and produce "TRUE" as an output? Are all set theory formula's TRUE?
set-theory first-order-logic
set-theory first-order-logic
edited Dec 6 '18 at 16:25
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 11:22
l3utterflyl3utterfly
1224
edited Dec 6 '18 at 16:25
Andrés E. Caicedo
65.4k8158248
asked Dec 6 '18 at 11:22
l3utterflyl3utterfly
1224
edited Dec 6 '18 at 16:25
Andrés E. Caicedo
65.4k8158248
edited Dec 6 '18 at 16:25
Andrés E. Caicedo
65.4k8158248
edited Dec 6 '18 at 16:25
Andrés E. Caicedo
65.4k8158248
65.4k8158248
asked Dec 6 '18 at 11:22
l3utterflyl3utterfly
1224
asked Dec 6 '18 at 11:22
l3utterflyl3utterfly
1224
asked Dec 6 '18 at 11:22
l3utterflyl3utterfly
1224
1224
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
A formula is an expression of the language of set theory built up acoording to the rules of the syntax.
Examples : $∃y ∀x ¬(x ∈ y), ∀x ¬(x ∈ emptyset)$.
A formula can be a sentence, i.e. without free variables (like the two previous examples) ore an open one, like e.g. : $(x∈y)$.
A sentence has a definite truth value : $∀x ¬(x ∈ emptyset)$ means "the empty set has no elements" and it is true in set theory.
An open formula, like $(x∈y)$ has not a definite truth value; its truth value depends on the "reference" assigned to the variables.
Consider some simple arithmetical examples : $forall n (n ge 0)$ is true in $mathbb N$.
Consider $(n > 0)$ instead : it is false if $n$ denotes $0$ and is true otherwise.
For a formal definition, see the post : In Mathematical Logic, What is a Language?
answered Dec 6 '18 at 11:44
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
$endgroup$
add a comment |
$begingroup$
Put simply in the language of set theory we start with atomic formulas
$$(xin y)$$
or
$$(x=y)$$
where $x$ and $y$ are variables.
Then we expand the definition of formulas to so that it is closed under $$neg(cdot)$$ and $$(cdot)wedge(cdot)$$ and $$(exists x)(cdot)$$ where $x$ is a variable.
Then we introduce the following notations.
$$(varphiveepsi)equivneg(negvarphiwedgenegpsi)$$
$$(varphiRightarrowpsi)equivnegvarphiveepsi$$
$$(varphiLeftrightarrowpsi)equiv(varphiRightarrowpsi)wedge(psiRightarrowvarphi)$$
$$(forall x)varphiequivneg(exists x)negvarphi$$
We take variables, punctuations, $=$, $in$, $neg$, $wedge$, $exists$ as primitive notions. That is to say, they are undefined symbols which are informally trying to capture the notion of variables, punctuations, equality, membership, negation, conjunction, existential quantifier. Establishing a collection of axioms is how we try to capture desired notions.
Let me speak in the language of set theory:
$$(exists x)(neg(x=x))$$
What I just said is false because of established axioms of set theory; most popular being $mathsf{ZFC}$.
Let $varphi$ be an arbitrary formula. Then certain variables within $varphi$ are "free." We typically denote those variables by writing $varphi(x_1,ldots,x_n)$ instead of simply $varphi$.
answered Dec 6 '18 at 13:49
Alberto TakaseAlberto Takase
2,025417
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
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active
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$begingroup$
A formula is an expression of the language of set theory built up acoording to the rules of the syntax.
Examples : $∃y ∀x ¬(x ∈ y), ∀x ¬(x ∈ emptyset)$.
A formula can be a sentence, i.e. without free variables (like the two previous examples) ore an open one, like e.g. : $(x∈y)$.
A sentence has a definite truth value : $∀x ¬(x ∈ emptyset)$ means "the empty set has no elements" and it is true in set theory.
An open formula, like $(x∈y)$ has not a definite truth value; its truth value depends on the "reference" assigned to the variables.
Consider some simple arithmetical examples : $forall n (n ge 0)$ is true in $mathbb N$.
Consider $(n > 0)$ instead : it is false if $n$ denotes $0$ and is true otherwise.
For a formal definition, see the post : In Mathematical Logic, What is a Language?
answered Dec 6 '18 at 11:44
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
$endgroup$
add a comment |
$begingroup$
A formula is an expression of the language of set theory built up acoording to the rules of the syntax.
Examples : $∃y ∀x ¬(x ∈ y), ∀x ¬(x ∈ emptyset)$.
A formula can be a sentence, i.e. without free variables (like the two previous examples) ore an open one, like e.g. : $(x∈y)$.
A sentence has a definite truth value : $∀x ¬(x ∈ emptyset)$ means "the empty set has no elements" and it is true in set theory.
An open formula, like $(x∈y)$ has not a definite truth value; its truth value depends on the "reference" assigned to the variables.
Consider some simple arithmetical examples : $forall n (n ge 0)$ is true in $mathbb N$.
Consider $(n > 0)$ instead : it is false if $n$ denotes $0$ and is true otherwise.
For a formal definition, see the post : In Mathematical Logic, What is a Language?
answered Dec 6 '18 at 11:44
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
$endgroup$
add a comment |
$begingroup$
A formula is an expression of the language of set theory built up acoording to the rules of the syntax.
Examples : $∃y ∀x ¬(x ∈ y), ∀x ¬(x ∈ emptyset)$.
A formula can be a sentence, i.e. without free variables (like the two previous examples) ore an open one, like e.g. : $(x∈y)$.
A sentence has a definite truth value : $∀x ¬(x ∈ emptyset)$ means "the empty set has no elements" and it is true in set theory.
An open formula, like $(x∈y)$ has not a definite truth value; its truth value depends on the "reference" assigned to the variables.
Consider some simple arithmetical examples : $forall n (n ge 0)$ is true in $mathbb N$.
Consider $(n > 0)$ instead : it is false if $n$ denotes $0$ and is true otherwise.
For a formal definition, see the post : In Mathematical Logic, What is a Language?
answered Dec 6 '18 at 11:44
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
$endgroup$
A formula is an expression of the language of set theory built up acoording to the rules of the syntax.
Examples : $∃y ∀x ¬(x ∈ y), ∀x ¬(x ∈ emptyset)$.
A formula can be a sentence, i.e. without free variables (like the two previous examples) ore an open one, like e.g. : $(x∈y)$.
A sentence has a definite truth value : $∀x ¬(x ∈ emptyset)$ means "the empty set has no elements" and it is true in set theory.
An open formula, like $(x∈y)$ has not a definite truth value; its truth value depends on the "reference" assigned to the variables.
Consider some simple arithmetical examples : $forall n (n ge 0)$ is true in $mathbb N$.
Consider $(n > 0)$ instead : it is false if $n$ denotes $0$ and is true otherwise.
For a formal definition, see the post : In Mathematical Logic, What is a Language?
answered Dec 6 '18 at 11:44
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
edited Dec 16 '18 at 12:53
answered Dec 6 '18 at 11:44
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
answered Dec 6 '18 at 11:44
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
answered Dec 6 '18 at 11:44
Mauro ALLEGRANZAMauro ALLEGRANZA
65.8k449114
65.8k449114
add a comment |
add a comment |
$begingroup$
Put simply in the language of set theory we start with atomic formulas
$$(xin y)$$
or
$$(x=y)$$
where $x$ and $y$ are variables.
Then we expand the definition of formulas to so that it is closed under $$neg(cdot)$$ and $$(cdot)wedge(cdot)$$ and $$(exists x)(cdot)$$ where $x$ is a variable.
Then we introduce the following notations.
$$(varphiveepsi)equivneg(negvarphiwedgenegpsi)$$
$$(varphiRightarrowpsi)equivnegvarphiveepsi$$
$$(varphiLeftrightarrowpsi)equiv(varphiRightarrowpsi)wedge(psiRightarrowvarphi)$$
$$(forall x)varphiequivneg(exists x)negvarphi$$
We take variables, punctuations, $=$, $in$, $neg$, $wedge$, $exists$ as primitive notions. That is to say, they are undefined symbols which are informally trying to capture the notion of variables, punctuations, equality, membership, negation, conjunction, existential quantifier. Establishing a collection of axioms is how we try to capture desired notions.
Let me speak in the language of set theory:
$$(exists x)(neg(x=x))$$
What I just said is false because of established axioms of set theory; most popular being $mathsf{ZFC}$.
Let $varphi$ be an arbitrary formula. Then certain variables within $varphi$ are "free." We typically denote those variables by writing $varphi(x_1,ldots,x_n)$ instead of simply $varphi$.
answered Dec 6 '18 at 13:49
Alberto TakaseAlberto Takase
2,025417
$endgroup$
add a comment |
$begingroup$
Put simply in the language of set theory we start with atomic formulas
$$(xin y)$$
or
$$(x=y)$$
where $x$ and $y$ are variables.
Then we expand the definition of formulas to so that it is closed under $$neg(cdot)$$ and $$(cdot)wedge(cdot)$$ and $$(exists x)(cdot)$$ where $x$ is a variable.
Then we introduce the following notations.
$$(varphiveepsi)equivneg(negvarphiwedgenegpsi)$$
$$(varphiRightarrowpsi)equivnegvarphiveepsi$$
$$(varphiLeftrightarrowpsi)equiv(varphiRightarrowpsi)wedge(psiRightarrowvarphi)$$
$$(forall x)varphiequivneg(exists x)negvarphi$$
We take variables, punctuations, $=$, $in$, $neg$, $wedge$, $exists$ as primitive notions. That is to say, they are undefined symbols which are informally trying to capture the notion of variables, punctuations, equality, membership, negation, conjunction, existential quantifier. Establishing a collection of axioms is how we try to capture desired notions.
Let me speak in the language of set theory:
$$(exists x)(neg(x=x))$$
What I just said is false because of established axioms of set theory; most popular being $mathsf{ZFC}$.
Let $varphi$ be an arbitrary formula. Then certain variables within $varphi$ are "free." We typically denote those variables by writing $varphi(x_1,ldots,x_n)$ instead of simply $varphi$.
answered Dec 6 '18 at 13:49
Alberto TakaseAlberto Takase
2,025417
$endgroup$
add a comment |
$begingroup$
Put simply in the language of set theory we start with atomic formulas
$$(xin y)$$
or
$$(x=y)$$
where $x$ and $y$ are variables.
Then we expand the definition of formulas to so that it is closed under $$neg(cdot)$$ and $$(cdot)wedge(cdot)$$ and $$(exists x)(cdot)$$ where $x$ is a variable.
Then we introduce the following notations.
$$(varphiveepsi)equivneg(negvarphiwedgenegpsi)$$
$$(varphiRightarrowpsi)equivnegvarphiveepsi$$
$$(varphiLeftrightarrowpsi)equiv(varphiRightarrowpsi)wedge(psiRightarrowvarphi)$$
$$(forall x)varphiequivneg(exists x)negvarphi$$
We take variables, punctuations, $=$, $in$, $neg$, $wedge$, $exists$ as primitive notions. That is to say, they are undefined symbols which are informally trying to capture the notion of variables, punctuations, equality, membership, negation, conjunction, existential quantifier. Establishing a collection of axioms is how we try to capture desired notions.
Let me speak in the language of set theory:
$$(exists x)(neg(x=x))$$
What I just said is false because of established axioms of set theory; most popular being $mathsf{ZFC}$.
Let $varphi$ be an arbitrary formula. Then certain variables within $varphi$ are "free." We typically denote those variables by writing $varphi(x_1,ldots,x_n)$ instead of simply $varphi$.
answered Dec 6 '18 at 13:49
Alberto TakaseAlberto Takase
2,025417
$endgroup$
Put simply in the language of set theory we start with atomic formulas
$$(xin y)$$
or
$$(x=y)$$
where $x$ and $y$ are variables.
Then we expand the definition of formulas to so that it is closed under $$neg(cdot)$$ and $$(cdot)wedge(cdot)$$ and $$(exists x)(cdot)$$ where $x$ is a variable.
Then we introduce the following notations.
$$(varphiveepsi)equivneg(negvarphiwedgenegpsi)$$
$$(varphiRightarrowpsi)equivnegvarphiveepsi$$
$$(varphiLeftrightarrowpsi)equiv(varphiRightarrowpsi)wedge(psiRightarrowvarphi)$$
$$(forall x)varphiequivneg(exists x)negvarphi$$
We take variables, punctuations, $=$, $in$, $neg$, $wedge$, $exists$ as primitive notions. That is to say, they are undefined symbols which are informally trying to capture the notion of variables, punctuations, equality, membership, negation, conjunction, existential quantifier. Establishing a collection of axioms is how we try to capture desired notions.
Let me speak in the language of set theory:
$$(exists x)(neg(x=x))$$
What I just said is false because of established axioms of set theory; most popular being $mathsf{ZFC}$.
Let $varphi$ be an arbitrary formula. Then certain variables within $varphi$ are "free." We typically denote those variables by writing $varphi(x_1,ldots,x_n)$ instead of simply $varphi$.
answered Dec 6 '18 at 13:49
Alberto TakaseAlberto Takase
2,025417
edited Dec 6 '18 at 14:09
answered Dec 6 '18 at 13:49
Alberto TakaseAlberto Takase
2,025417
answered Dec 6 '18 at 13:49
Alberto TakaseAlberto Takase
2,025417
answered Dec 6 '18 at 13:49
Alberto TakaseAlberto Takase
2,025417
2,025417
add a comment |
add a comment |
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