What exactly is a formula in set theory?
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 at 16:25
Andrés E. Caicedo
64.7k8158246
asked Dec 6 at 11:22
l3utterfly
1204
add a comment |
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 at 16:25
Andrés E. Caicedo
64.7k8158246
asked Dec 6 at 11:22
l3utterfly
1204
add a comment |
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 at 16:25
Andrés E. Caicedo
64.7k8158246
asked Dec 6 at 11:22
l3utterfly
1204
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 at 16:25
Andrés E. Caicedo
64.7k8158246
asked Dec 6 at 11:22
l3utterfly
1204
edited Dec 6 at 16:25
Andrés E. Caicedo
64.7k8158246
asked Dec 6 at 11:22
l3utterfly
1204
edited Dec 6 at 16:25
Andrés E. Caicedo
64.7k8158246
edited Dec 6 at 16:25
Andrés E. Caicedo
64.7k8158246
edited Dec 6 at 16:25
Andrés E. Caicedo
64.7k8158246
64.7k8158246
asked Dec 6 at 11:22
l3utterfly
1204
asked Dec 6 at 11:22
l3utterfly
1204
asked Dec 6 at 11:22
l3utterfly
1204
1204
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
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 at 13:49
Alberto Takase
1,738414
add a comment |
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 at 11:44
Mauro ALLEGRANZA
64.1k448111
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028366%2fwhat-exactly-is-a-formula-in-set-theory%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 at 13:49
Alberto Takase
1,738414
add a comment |
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 at 13:49
Alberto Takase
1,738414
add a comment |
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 at 13:49
Alberto Takase
1,738414
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 at 13:49
Alberto Takase
1,738414
edited Dec 6 at 14:09
answered Dec 6 at 13:49
Alberto Takase
1,738414
answered Dec 6 at 13:49
Alberto Takase
1,738414
answered Dec 6 at 13:49
Alberto Takase
1,738414
1,738414
add a comment |
add a comment |
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 at 11:44
Mauro ALLEGRANZA
64.1k448111
add a comment |
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 at 11:44
Mauro ALLEGRANZA
64.1k448111
add a comment |
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 at 11:44
Mauro ALLEGRANZA
64.1k448111
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 at 11:44
Mauro ALLEGRANZA
64.1k448111
edited Dec 16 at 12:53
answered Dec 6 at 11:44
Mauro ALLEGRANZA
64.1k448111
answered Dec 6 at 11:44
Mauro ALLEGRANZA
64.1k448111
answered Dec 6 at 11:44
Mauro ALLEGRANZA
64.1k448111
64.1k448111
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028366%2fwhat-exactly-is-a-formula-in-set-theory%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
