What exactly is a formula in set theory?












1














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







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    1














    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







    share|cite|improve this question



























      1












      1








      1







      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







      share|cite|improve this question















      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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 6 at 16:25









      Andrés E. Caicedo

      64.7k8158246




      64.7k8158246










      asked Dec 6 at 11:22









      l3utterfly

      1204




      1204






















          2 Answers
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          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 $varphi$.






          share|cite|improve this answer































            1














            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?






            share|cite|improve this answer























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              2 Answers
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              2 Answers
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              1














              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 $varphi$.






              share|cite|improve this answer




























                1














                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 $varphi$.






                share|cite|improve this answer


























                  1












                  1








                  1






                  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 $varphi$.






                  share|cite|improve this answer














                  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 $varphi$.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Dec 6 at 14:09

























                  answered Dec 6 at 13:49









                  Alberto Takase

                  1,738414




                  1,738414























                      1














                      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?






                      share|cite|improve this answer




























                        1














                        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?






                        share|cite|improve this answer


























                          1












                          1








                          1






                          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?






                          share|cite|improve this answer














                          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?







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited Dec 16 at 12:53

























                          answered Dec 6 at 11:44









                          Mauro ALLEGRANZA

                          64.1k448111




                          64.1k448111






























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