loading

Logout succeed

Logout succeed. See you again!

search
Advance search
xAdvance search
submenu
search
search
xAdvance search
Students access

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

ebook img

Standard Systems of Nonstandard Models of Peano Arithmetic PDF

pages84 Pages
release year2008
file size0.66 MB
languageEnglish

Preview Standard Systems of Nonstandard Models of Peano Arithmetic

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

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.