Which are concepts undefined? which are symbols in the language of set theory?












0












$begingroup$



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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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
















0












$begingroup$



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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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














0












0








0





$begingroup$



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







share|cite|improve this question











$endgroup$





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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 16 '18 at 16:35









Andrés E. Caicedo

65.1k8158247




65.1k8158247










asked Dec 16 '18 at 12:29









Almot1960Almot1960

2,536823




2,536823












  • $begingroup$
    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


















  • $begingroup$
    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
















$begingroup$
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










1 Answer
1






active

oldest

votes


















1












$begingroup$

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.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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











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1 Answer
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active

oldest

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









1












$begingroup$

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.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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
















1












$begingroup$

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.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    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














1












1








1





$begingroup$

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.






share|cite|improve this answer











$endgroup$



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.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 17 '18 at 15:36

























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


















  • $begingroup$
    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


















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