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Let N = (IN, 0, S, +,-) be the usual structure in the signature darthm= (0, S, +, .), i.e. N is the standard model of PA. (a) Show that for every model M of PA, there is a unique homomorphism h: N → M and this hy is one-to-one, so hy is an embedding. (b) Call the image hm [N] the standard part of M and denote it by NM. Show that if M is nonstandard then its standard part NM is not definable in M. (c) (Overspill) Let M be a nonstandard model of PA, let p(x,y) be an extended ♂arthm formula, where |7] = k, and let e Mk. Show that if M = o(n,a) for infinitely many n E INM, then there is w € M \ NM such that M = (w,a). In other words, if a statement is true about infinitely many standard elements, then it is true about a nonstandard element.

Let N = (IN, 0, S, +,-) be the usual structure in the signature darthm= (0, S, +, .), i.e. N is the standard model of PA. (a) Show that for every model M of PA, there is a unique homomorphism h: N → M and this hy is one-to-one, so hy is an embedding. (b) Call the image hm [N] the standard part of M and denote it by NM. Show that if M is nonstandard then its standard part NM is not definable in M. (c) (Overspill) Let M be a nonstandard model of PA, let p(x,y) be an extended ♂arthm formula, where |7] = k, and let e Mk. Show that if M = o(n,a) for infinitely many n E INM, then there is w € M \ NM such that M = (w,a). In other words, if a statement is true about infinitely many standard elements, then it is true about a nonstandard element.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.5: Isomorphisms
Problem 4E: Let G=1,i,1,i under multiplication, and let G=4=[ 0 ],[ 1 ],[ 2 ],[ 3 ] under addition. Find an...
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Let N = (IN, 0, S, +, .) be the usual structure in the signature garthm := (0, S, +,-), i.e. N is the standard model of PA. (a) Show that for every model M of PA, there is a unique homomorphism hm : N → M and this hy is one-to-one, so hy is an embedding. (b) Call the image hå [N] the standard part of M and denote it by NM. Show that if M is nonstandard then its standard part NM is not definable in M. (c) (Overspill) Let M be a nonstandard model of PA, let p(x,y) be an extended arthm formula, where |ỹ] = k, and let de Mk. Show that if M = p(n, a) for infinitely many ne NM, then there is w EM \ NM such that M = p(w,a). In other words, if a statement is true about infinitely many standard elements, then it is true about a nonstandard element.
Transcribed Image Text:Let N = (IN, 0, S, +, .) be the usual structure in the signature garthm := (0, S, +,-), i.e. N is the standard model of PA. (a) Show that for every model M of PA, there is a unique homomorphism hm : N → M and this hy is one-to-one, so hy is an embedding. (b) Call the image hå [N] the standard part of M and denote it by NM. Show that if M is nonstandard then its standard part NM is not definable in M. (c) (Overspill) Let M be a nonstandard model of PA, let p(x,y) be an extended arthm formula, where |ỹ] = k, and let de Mk. Show that if M = p(n, a) for infinitely many ne NM, then there is w EM \ NM such that M = p(w,a). In other words, if a statement is true about infinitely many standard elements, then it is true about a nonstandard element.
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