Which are concepts undefined? which are symbols in the language of set theory?
$begingroup$
I know that in the most general terms, when we talk about a mathematical
theory, we have in mind a collection of axioms and Undefined Concepts in a certain language.
Now in axiomatic set theory ;
Which are concepts undefined?
which are symbols in the language of set theory?
first-order-logic
edited Dec 16 '18 at 16:35
Andrés E. Caicedo
65.1k8158247
asked Dec 16 '18 at 12:29
Almot1960Almot1960
2,536823
$endgroup$
|
show 11 more comments
$begingroup$
I know that in the most general terms, when we talk about a mathematical
theory, we have in mind a collection of axioms and Undefined Concepts in a certain language.
Now in axiomatic set theory ;
Which are concepts undefined?
which are symbols in the language of set theory?
first-order-logic
edited Dec 16 '18 at 16:35
Andrés E. Caicedo
65.1k8158247
asked Dec 16 '18 at 12:29
Almot1960Almot1960
2,536823
$endgroup$
$begingroup$
What do you mean byUndefined Concepts? and in the language of $sf ZFC$ the only symbol is $in$
$endgroup$
– Holo
Dec 16 '18 at 12:39
2
$begingroup$
In set theory "set" and $in$ are undefined.
$endgroup$
– drhab
Dec 16 '18 at 12:40
$begingroup$
There is no single answer to this. In any axiomatic theory, we may take concepts A and B to be "undefined" and define concept C in terms of those, or take B and C to be "undefined" and define A in terms of B and C.
$endgroup$
– user247327
Dec 16 '18 at 12:41
$begingroup$
@drhab there are set theories with Urelements en.wikipedia.org/wiki/Urelement#Urelements_in_set_theory
$endgroup$
– Holo
Dec 16 '18 at 12:46
$begingroup$
@Holo .for example in esuclidean geometry :undefined Concepts are point ,...
$endgroup$
– Almot1960
Dec 16 '18 at 12:49
|
show 11 more comments
$begingroup$
I know that in the most general terms, when we talk about a mathematical
theory, we have in mind a collection of axioms and Undefined Concepts in a certain language.
Now in axiomatic set theory ;
Which are concepts undefined?
which are symbols in the language of set theory?
first-order-logic
edited Dec 16 '18 at 16:35
Andrés E. Caicedo
65.1k8158247
asked Dec 16 '18 at 12:29
Almot1960Almot1960
2,536823
$endgroup$
I know that in the most general terms, when we talk about a mathematical
theory, we have in mind a collection of axioms and Undefined Concepts in a certain language.
Now in axiomatic set theory ;
Which are concepts undefined?
which are symbols in the language of set theory?
first-order-logic
first-order-logic
edited Dec 16 '18 at 16:35
Andrés E. Caicedo
65.1k8158247
asked Dec 16 '18 at 12:29
Almot1960Almot1960
2,536823
edited Dec 16 '18 at 16:35
Andrés E. Caicedo
65.1k8158247
asked Dec 16 '18 at 12:29
Almot1960Almot1960
2,536823
edited Dec 16 '18 at 16:35
Andrés E. Caicedo
65.1k8158247
edited Dec 16 '18 at 16:35
Andrés E. Caicedo
65.1k8158247
edited Dec 16 '18 at 16:35
Andrés E. Caicedo
65.1k8158247
65.1k8158247
asked Dec 16 '18 at 12:29
Almot1960Almot1960
2,536823
asked Dec 16 '18 at 12:29
Almot1960Almot1960
2,536823
asked Dec 16 '18 at 12:29
Almot1960Almot1960
2,536823
2,536823
$begingroup$
What do you mean byUndefined Concepts? and in the language of $sf ZFC$ the only symbol is $in$
$endgroup$
– Holo
Dec 16 '18 at 12:39
2
$begingroup$
In set theory "set" and $in$ are undefined.
$endgroup$
– drhab
Dec 16 '18 at 12:40
$begingroup$
There is no single answer to this. In any axiomatic theory, we may take concepts A and B to be "undefined" and define concept C in terms of those, or take B and C to be "undefined" and define A in terms of B and C.
$endgroup$
– user247327
Dec 16 '18 at 12:41
$begingroup$
@drhab there are set theories with Urelements en.wikipedia.org/wiki/Urelement#Urelements_in_set_theory
$endgroup$
– Holo
Dec 16 '18 at 12:46
$begingroup$
@Holo .for example in esuclidean geometry :undefined Concepts are point ,...
$endgroup$
– Almot1960
Dec 16 '18 at 12:49
|
show 11 more comments
$begingroup$
What do you mean byUndefined Concepts? and in the language of $sf ZFC$ the only symbol is $in$
$endgroup$
– Holo
Dec 16 '18 at 12:39
2
$begingroup$
In set theory "set" and $in$ are undefined.
$endgroup$
– drhab
Dec 16 '18 at 12:40
$begingroup$
There is no single answer to this. In any axiomatic theory, we may take concepts A and B to be "undefined" and define concept C in terms of those, or take B and C to be "undefined" and define A in terms of B and C.
$endgroup$
– user247327
Dec 16 '18 at 12:41
$begingroup$
@drhab there are set theories with Urelements en.wikipedia.org/wiki/Urelement#Urelements_in_set_theory
$endgroup$
– Holo
Dec 16 '18 at 12:46
$begingroup$
@Holo .for example in esuclidean geometry :undefined Concepts are point ,...
$endgroup$
– Almot1960
Dec 16 '18 at 12:49
$begingroup$
What do you mean by
Undefined Concepts? and in the language of $sf ZFC$ the only symbol is $in$$endgroup$
– Holo
Dec 16 '18 at 12:39
$begingroup$
What do you mean by
Undefined Concepts? and in the language of $sf ZFC$ the only symbol is $in$$endgroup$
– Holo
Dec 16 '18 at 12:39
2
2
$begingroup$
In set theory "set" and $in$ are undefined.
$endgroup$
– drhab
Dec 16 '18 at 12:40
$begingroup$
In set theory "set" and $in$ are undefined.
$endgroup$
– drhab
Dec 16 '18 at 12:40
$begingroup$
There is no single answer to this. In any axiomatic theory, we may take concepts A and B to be "undefined" and define concept C in terms of those, or take B and C to be "undefined" and define A in terms of B and C.
$endgroup$
– user247327
Dec 16 '18 at 12:41
$begingroup$
There is no single answer to this. In any axiomatic theory, we may take concepts A and B to be "undefined" and define concept C in terms of those, or take B and C to be "undefined" and define A in terms of B and C.
$endgroup$
– user247327
Dec 16 '18 at 12:41
$begingroup$
@drhab there are set theories with Urelements en.wikipedia.org/wiki/Urelement#Urelements_in_set_theory
$endgroup$
– Holo
Dec 16 '18 at 12:46
$begingroup$
@drhab there are set theories with Urelements en.wikipedia.org/wiki/Urelement#Urelements_in_set_theory
$endgroup$
– Holo
Dec 16 '18 at 12:46
$begingroup$
@Holo .for example in esuclidean geometry :undefined Concepts are point ,...
$endgroup$
– Almot1960
Dec 16 '18 at 12:49
$begingroup$
@Holo .for example in esuclidean geometry :undefined Concepts are point ,...
$endgroup$
– Almot1960
Dec 16 '18 at 12:49
|
show 11 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Zermelo-Fraenkel set theory ($mathsf {ZF}$) :
is an axiom system formulated in first-order logic with equality and with only one binary relation symbol $in$ for membership.
Thus, its language is that of first-order logic, with sentential connective symbols: $→,¬, lor,land$; individual variables: $v_1, v_2, ldots$; the equality symbol $=$; and the quantifier symbols: $forall, exists$.
In addition to them, the theory has only one (binary) predicate : $in$ to mean "membership". It is the only "undefined" concept of the theory.
With them we can formulate the axioms of $mathsf {ZF}$ set theory.
One of the axioms is the Null Set axiom :
$∃x forall y lnot (y ∈ x)$.
Using this axiom and the Extensionality Axiom it is provable that the set satisfying the axiom is unique.
Thus, we may introduce the defined term "$emptyset$" (a new symbol) to denote it.
Note
The most common formulation of $mathsf {ZF}$ set theory is based on FOL with equality.
Thus "$=$" is part of the background logic, that means that the usual axioms like $∀x(x=x)$ are assumed.
There are versions where the background logic does not include equality "$=$"; in that case $x=y$ may be defined as an abbreviation for :
$∀z[z∈x ↔ z∈y] ∧ ∀w[x∈w ↔ y∈w]$,
with a suitable reformulation of the Extensionalty Axiom. In that case, the usual properties of = must be proved from the above definition.
answered Dec 17 '18 at 14:20
Mauro ALLEGRANZAMauro ALLEGRANZA
64.9k448112
$endgroup$
$begingroup$
Should "=" be ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:29
$begingroup$
@Almot1960 Should "$=$" be what?
$endgroup$
– Noah Schweber
Dec 17 '18 at 14:42
$begingroup$
In language of the set should be equality symbol ? or defined equality symbol by $∈$ ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:51
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%2f3042559%2fwhich-are-concepts-undefined-which-are-symbols-in-the-language-of-set-theory%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Zermelo-Fraenkel set theory ($mathsf {ZF}$) :
is an axiom system formulated in first-order logic with equality and with only one binary relation symbol $in$ for membership.
Thus, its language is that of first-order logic, with sentential connective symbols: $→,¬, lor,land$; individual variables: $v_1, v_2, ldots$; the equality symbol $=$; and the quantifier symbols: $forall, exists$.
In addition to them, the theory has only one (binary) predicate : $in$ to mean "membership". It is the only "undefined" concept of the theory.
With them we can formulate the axioms of $mathsf {ZF}$ set theory.
One of the axioms is the Null Set axiom :
$∃x forall y lnot (y ∈ x)$.
Using this axiom and the Extensionality Axiom it is provable that the set satisfying the axiom is unique.
Thus, we may introduce the defined term "$emptyset$" (a new symbol) to denote it.
Note
The most common formulation of $mathsf {ZF}$ set theory is based on FOL with equality.
Thus "$=$" is part of the background logic, that means that the usual axioms like $∀x(x=x)$ are assumed.
There are versions where the background logic does not include equality "$=$"; in that case $x=y$ may be defined as an abbreviation for :
$∀z[z∈x ↔ z∈y] ∧ ∀w[x∈w ↔ y∈w]$,
with a suitable reformulation of the Extensionalty Axiom. In that case, the usual properties of = must be proved from the above definition.
answered Dec 17 '18 at 14:20
Mauro ALLEGRANZAMauro ALLEGRANZA
64.9k448112
$endgroup$
$begingroup$
Should "=" be ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:29
$begingroup$
@Almot1960 Should "$=$" be what?
$endgroup$
– Noah Schweber
Dec 17 '18 at 14:42
$begingroup$
In language of the set should be equality symbol ? or defined equality symbol by $∈$ ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:51
add a comment |
$begingroup$
Zermelo-Fraenkel set theory ($mathsf {ZF}$) :
is an axiom system formulated in first-order logic with equality and with only one binary relation symbol $in$ for membership.
Thus, its language is that of first-order logic, with sentential connective symbols: $→,¬, lor,land$; individual variables: $v_1, v_2, ldots$; the equality symbol $=$; and the quantifier symbols: $forall, exists$.
In addition to them, the theory has only one (binary) predicate : $in$ to mean "membership". It is the only "undefined" concept of the theory.
With them we can formulate the axioms of $mathsf {ZF}$ set theory.
One of the axioms is the Null Set axiom :
$∃x forall y lnot (y ∈ x)$.
Using this axiom and the Extensionality Axiom it is provable that the set satisfying the axiom is unique.
Thus, we may introduce the defined term "$emptyset$" (a new symbol) to denote it.
Note
The most common formulation of $mathsf {ZF}$ set theory is based on FOL with equality.
Thus "$=$" is part of the background logic, that means that the usual axioms like $∀x(x=x)$ are assumed.
There are versions where the background logic does not include equality "$=$"; in that case $x=y$ may be defined as an abbreviation for :
$∀z[z∈x ↔ z∈y] ∧ ∀w[x∈w ↔ y∈w]$,
with a suitable reformulation of the Extensionalty Axiom. In that case, the usual properties of = must be proved from the above definition.
answered Dec 17 '18 at 14:20
Mauro ALLEGRANZAMauro ALLEGRANZA
64.9k448112
$endgroup$
$begingroup$
Should "=" be ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:29
$begingroup$
@Almot1960 Should "$=$" be what?
$endgroup$
– Noah Schweber
Dec 17 '18 at 14:42
$begingroup$
In language of the set should be equality symbol ? or defined equality symbol by $∈$ ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:51
add a comment |
$begingroup$
Zermelo-Fraenkel set theory ($mathsf {ZF}$) :
is an axiom system formulated in first-order logic with equality and with only one binary relation symbol $in$ for membership.
Thus, its language is that of first-order logic, with sentential connective symbols: $→,¬, lor,land$; individual variables: $v_1, v_2, ldots$; the equality symbol $=$; and the quantifier symbols: $forall, exists$.
In addition to them, the theory has only one (binary) predicate : $in$ to mean "membership". It is the only "undefined" concept of the theory.
With them we can formulate the axioms of $mathsf {ZF}$ set theory.
One of the axioms is the Null Set axiom :
$∃x forall y lnot (y ∈ x)$.
Using this axiom and the Extensionality Axiom it is provable that the set satisfying the axiom is unique.
Thus, we may introduce the defined term "$emptyset$" (a new symbol) to denote it.
Note
The most common formulation of $mathsf {ZF}$ set theory is based on FOL with equality.
Thus "$=$" is part of the background logic, that means that the usual axioms like $∀x(x=x)$ are assumed.
There are versions where the background logic does not include equality "$=$"; in that case $x=y$ may be defined as an abbreviation for :
$∀z[z∈x ↔ z∈y] ∧ ∀w[x∈w ↔ y∈w]$,
with a suitable reformulation of the Extensionalty Axiom. In that case, the usual properties of = must be proved from the above definition.
answered Dec 17 '18 at 14:20
Mauro ALLEGRANZAMauro ALLEGRANZA
64.9k448112
$endgroup$
Zermelo-Fraenkel set theory ($mathsf {ZF}$) :
is an axiom system formulated in first-order logic with equality and with only one binary relation symbol $in$ for membership.
Thus, its language is that of first-order logic, with sentential connective symbols: $→,¬, lor,land$; individual variables: $v_1, v_2, ldots$; the equality symbol $=$; and the quantifier symbols: $forall, exists$.
In addition to them, the theory has only one (binary) predicate : $in$ to mean "membership". It is the only "undefined" concept of the theory.
With them we can formulate the axioms of $mathsf {ZF}$ set theory.
One of the axioms is the Null Set axiom :
$∃x forall y lnot (y ∈ x)$.
Using this axiom and the Extensionality Axiom it is provable that the set satisfying the axiom is unique.
Thus, we may introduce the defined term "$emptyset$" (a new symbol) to denote it.
Note
The most common formulation of $mathsf {ZF}$ set theory is based on FOL with equality.
Thus "$=$" is part of the background logic, that means that the usual axioms like $∀x(x=x)$ are assumed.
There are versions where the background logic does not include equality "$=$"; in that case $x=y$ may be defined as an abbreviation for :
$∀z[z∈x ↔ z∈y] ∧ ∀w[x∈w ↔ y∈w]$,
with a suitable reformulation of the Extensionalty Axiom. In that case, the usual properties of = must be proved from the above definition.
answered Dec 17 '18 at 14:20
Mauro ALLEGRANZAMauro ALLEGRANZA
64.9k448112
edited Dec 17 '18 at 15:36
answered Dec 17 '18 at 14:20
Mauro ALLEGRANZAMauro ALLEGRANZA
64.9k448112
answered Dec 17 '18 at 14:20
Mauro ALLEGRANZAMauro ALLEGRANZA
64.9k448112
answered Dec 17 '18 at 14:20
Mauro ALLEGRANZAMauro ALLEGRANZA
64.9k448112
64.9k448112
$begingroup$
Should "=" be ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:29
$begingroup$
@Almot1960 Should "$=$" be what?
$endgroup$
– Noah Schweber
Dec 17 '18 at 14:42
$begingroup$
In language of the set should be equality symbol ? or defined equality symbol by $∈$ ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:51
add a comment |
$begingroup$
Should "=" be ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:29
$begingroup$
@Almot1960 Should "$=$" be what?
$endgroup$
– Noah Schweber
Dec 17 '18 at 14:42
$begingroup$
In language of the set should be equality symbol ? or defined equality symbol by $∈$ ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:51
$begingroup$
Should "=" be ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:29
$begingroup$
Should "=" be ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:29
$begingroup$
@Almot1960 Should "$=$" be what?
$endgroup$
– Noah Schweber
Dec 17 '18 at 14:42
$begingroup$
@Almot1960 Should "$=$" be what?
$endgroup$
– Noah Schweber
Dec 17 '18 at 14:42
$begingroup$
In language of the set should be equality symbol ? or defined equality symbol by $∈$ ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:51
$begingroup$
In language of the set should be equality symbol ? or defined equality symbol by $∈$ ?
$endgroup$
– Almot1960
Dec 17 '18 at 14:51
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.
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%2f3042559%2fwhich-are-concepts-undefined-which-are-symbols-in-the-language-of-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
$begingroup$
What do you mean by
Undefined Concepts? and in the language of $sf ZFC$ the only symbol is $in$$endgroup$
– Holo
Dec 16 '18 at 12:39
2
$begingroup$
In set theory "set" and $in$ are undefined.
$endgroup$
– drhab
Dec 16 '18 at 12:40
$begingroup$
There is no single answer to this. In any axiomatic theory, we may take concepts A and B to be "undefined" and define concept C in terms of those, or take B and C to be "undefined" and define A in terms of B and C.
$endgroup$
– user247327
Dec 16 '18 at 12:41
$begingroup$
@drhab there are set theories with Urelements en.wikipedia.org/wiki/Urelement#Urelements_in_set_theory
$endgroup$
– Holo
Dec 16 '18 at 12:46
$begingroup$
@Holo .for example in esuclidean geometry :undefined Concepts are point ,...
$endgroup$
– Almot1960
Dec 16 '18 at 12:49