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Standard Systems of Nonstandard Models of Peano Arithmetic VictoriaGitman NewYorkCityCollegeofTechnology,CUNY [email protected] http://websupport1.citytech.cuny.edu/faculty/vgitman May13,2008 VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 1/17 History PeanoArithmetic History (19th century) Inthe19thcentury,motivatedbyCantor’sproofthattheunitintervalhasthesame cardinalityastheunitsquare,GiuseppePeanoshowedthatthereexistsacontinuous functionfromtheunitintervalontotheunitsquare,thePeanospacefillingcurve but thisisnotitsstory.... Peanoalsoproposedanaxiomatizationofnumbertheory,i.e. ofthestructure (N,+,×,<,0,1)... Anaxiomatizationofastructureisacollectionofstatementsfromwhicheverytrue statementaboutthatstructurecanbederived. VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 2/17 History PeanoArithmetic History (19th century) Inthe19thcentury,motivatedbyCantor’sproofthattheunitintervalhasthesame cardinalityastheunitsquare,GiuseppePeanoshowedthatthereexistsacontinuous functionfromtheunitintervalontotheunitsquare,thePeanospacefillingcurve but thisisnotitsstory.... Peanoalsoproposedanaxiomatizationofnumbertheory,i.e. ofthestructure (N,+,×,<,0,1)... Anaxiomatizationofastructureisacollectionofstatementsfromwhicheverytrue statementaboutthatstructurecanbederived. VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 2/17 History PeanoArithmetic History (19th century) Inthe19thcentury,motivatedbyCantor’sproofthattheunitintervalhasthesame cardinalityastheunitsquare,GiuseppePeanoshowedthatthereexistsacontinuous functionfromtheunitintervalontotheunitsquare,thePeanospacefillingcurve but thisisnotitsstory.... Peanoalsoproposedanaxiomatizationofnumbertheory,i.e. ofthestructure (N,+,×,<,0,1)... Anaxiomatizationofastructureisacollectionofstatementsfromwhicheverytrue statementaboutthatstructurecanbederived. VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 2/17 History PeanoArithmetic History (19th century) Inthe19thcentury,motivatedbyCantor’sproofthattheunitintervalhasthesame cardinalityastheunitsquare,GiuseppePeanoshowedthatthereexistsacontinuous functionfromtheunitintervalontotheunitsquare,thePeanospacefillingcurve but thisisnotitsstory.... Peanoalsoproposedanaxiomatizationofnumbertheory,i.e. ofthestructure (N,+,×,<,0,1)... Anaxiomatizationofastructureisacollectionofstatementsfromwhicheverytrue statementaboutthatstructurecanbederived. VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 2/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17

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Standard Systems of Nonstandard Models of Peano Arithmetic Victoria Gitman New York City College of Technology, CUNY [email protected]
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Standard Systems of Nonstandard Models of Peano Arithmetic VictoriaGitman NewYorkCityCollegeofTechnology,CUNY [email protected] http://websupport1.citytech.cuny.edu/faculty/vgitman May13,2008 VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 1/17 History PeanoArithmetic History (19th century) Inthe19thcentury,motivatedbyCantor’sproofthattheunitintervalhasthesame cardinalityastheunitsquare,GiuseppePeanoshowedthatthereexistsacontinuous functionfromtheunitintervalontotheunitsquare,thePeanospacefillingcurve but thisisnotitsstory.... Peanoalsoproposedanaxiomatizationofnumbertheory,i.e. ofthestructure (N,+,×,<,0,1)... Anaxiomatizationofastructureisacollectionofstatementsfromwhicheverytrue statementaboutthatstructurecanbederived. VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 2/17 History PeanoArithmetic History (19th century) Inthe19thcentury,motivatedbyCantor’sproofthattheunitintervalhasthesame cardinalityastheunitsquare,GiuseppePeanoshowedthatthereexistsacontinuous functionfromtheunitintervalontotheunitsquare,thePeanospacefillingcurve but thisisnotitsstory.... Peanoalsoproposedanaxiomatizationofnumbertheory,i.e. ofthestructure (N,+,×,<,0,1)... Anaxiomatizationofastructureisacollectionofstatementsfromwhicheverytrue statementaboutthatstructurecanbederived. VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 2/17 History PeanoArithmetic History (19th century) Inthe19thcentury,motivatedbyCantor’sproofthattheunitintervalhasthesame cardinalityastheunitsquare,GiuseppePeanoshowedthatthereexistsacontinuous functionfromtheunitintervalontotheunitsquare,thePeanospacefillingcurve but thisisnotitsstory.... Peanoalsoproposedanaxiomatizationofnumbertheory,i.e. ofthestructure (N,+,×,<,0,1)... Anaxiomatizationofastructureisacollectionofstatementsfromwhicheverytrue statementaboutthatstructurecanbederived. VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 2/17 History PeanoArithmetic History (19th century) Inthe19thcentury,motivatedbyCantor’sproofthattheunitintervalhasthesame cardinalityastheunitsquare,GiuseppePeanoshowedthatthereexistsacontinuous functionfromtheunitintervalontotheunitsquare,thePeanospacefillingcurve but thisisnotitsstory.... Peanoalsoproposedanaxiomatizationofnumbertheory,i.e. ofthestructure (N,+,×,<,0,1)... Anaxiomatizationofastructureisacollectionofstatementsfromwhicheverytrue statementaboutthatstructurecanbederived. VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 2/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17 History PeanoArithmetic Peano’s Axioms PeanoArithmetic(PA) AdditionandMultiplication ∀x,y,z (x+y)+z =x+(y+z) (associativityofaddition) ∀x,y x+y =y+x (commutativityofaddition) ∀x,y,z (xy)z =x(yz) (associativityofmultiplication) ∀x,y xy =yx (commutativityofmultiplication) ∀x,y,z x(y+z)=xy+xz. (distributivelaw) ∀x (x+0=x)∧(x1=x) (existenceofadditiveandmultiplicativeidentity) VictoriaGitman(CUNY) NonstandardModelsofPA May13,2008 3/17

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Standard Systems of Nonstandard Models of Peano Arithmetic Victoria Gitman New York City College of Technology, CUNY [email protected]
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