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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7b945f66-390e-417d-9fc3-363b543734f1%2F4a66ec95-b015-4cc7-adc4-4196668357e9%2Fhgxqsrt_processed.png&w=3840&q=75)
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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