ONT Re: Model Theory
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Note 36
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| 1. Introduction
|
| 1.4. Theories and Examples of Theories (cont.)
|
| 1.4.9. Example. [Commutative Rings to Ordered Fields].
|
| Let $L$ = {+, ·, 0, 1},
|
| where + and · are 2-placed function symbols
| and 0 and 1 are constant symbols.
|
| The theory of 'commutative rings (with unit)' has
| the axioms (1, 2, 3, 4) listed above [copied here]
| plus the axioms (8, 9, 10, 11) given below:
|
| 1. Associativity (+)
|
| x + (y + z) = (x + y) + z
|
| 2. Identity (+)
|
| x + 0 = x
|
| 0 + x = x
|
| 3. Existence of Inverse (+)
|
| (`E`y) (x + y = 0 & y + x = 0)
|
| 4. Commutativity (+)
|
| x + y = y + x
|
| 8. Unit (·)
|
| 1 · x = x
|
| x · 1 = x
|
| 9. Associativity (·)
|
| x · (y · z) = (x · y) · z
|
| 10. Commutativity (·)
|
| x · y = y · x
|
| 11. Distributivity (· over +)
|
| x · (y + z) = (x · y) + (x · z)
|
| Adding one more axiom:
|
| 12. Absence of Zero Divisors
|
| x · y = 0 => x = 0 or y = 0
|
| gives us the theory of 'integral domains'.
|
| Adding the two axioms:
|
| 13. Distinct Identities
|
| 0 =/= 1
|
| 14. Existence of Inverse (·)
|
| x =/= 0 => (`E`y)(y · x = 1)
|
| gives us the important theory of 'fields'.
| For a fixed prime p, if we add the axiom:
|
| 15_p. Prime Characteristic
|
| p1 = 0
|
| we have the theory of 'fields of characteristic p'.
| On the other hand, if we add for all primes p the
| negation of (15_p), namely, all the axioms:
|
| 16. Characteristic Zero
|
| p1 =/= 0, with p a prime
|
| we have the theory of 'fields of characteristic zero'.
| We now introduce the abbreviation x^n for the expression:
|
| x · (x · ( ... (x · x) ... )), n times.
|
| The infinite list of axioms, one for each n >= 1, as follows:
|
| 17_n. (`E`y)
|
| (x_n · y^n + x_(n-1) · y^(n-1) + ... + x_1 · y + x_0 = 0)
|
| or x_n = 0
|
| when added to the theory of fields, gives us
| the theory of 'algebraically closed fields'.
|
| 1.4.10. Proposition.
|
| Any two uncountable algebraically closed fields of
| the same characteristic and power are isomorphic.
|
| Each axiom (17_n) says that every polynomial of degree n has a root.
| The theory of 'real closed fields' has as axioms all the axioms for
| fields plus the axiom:
|
| 18. (`A`x)(`E`y) (y^2 = x or y^2 + x = 0)
|
| and two infinite lists of axioms. One is the infinite list (17_n)
| for all odd n, and the other is the infinite list that says that 0
| is not a sum of nontrivial squares:
|
| 18_n. (x_0)^2 + (x_1)^2 + ... + (x_n)^2 = 0
|
| =>
|
| x_0 = 0 & x_1 = 0 & ... & x_n = 0
|
| The theory of 'ordered fields' is formulated in
| the language $L$ = {=<, +, ·, 0, 1}. It has all
| the field axioms, the linear order axioms, and the
| additional axioms:
|
| 19. x =< y => x + z =< y + z
|
| 20. x =< y & 0 =< z => x · z =< y · z
|
| The ordered fields of rational numbers and of real numbers are examples.
|
| Of the examples of theories we have discussed so far, the following are complete:
| dense order without endpoints, atomless Boolean algebras, infinite Abelian groups
| with all elements of order p, torsion-free divisible Abelian groups, algebraically
| closed fields of a given characteristic, and real closed fields. The various
| propositions show that each of these complete theories, except the last one,
| enjoys the unusual property that in some (sometimes all) infinite powers
| all models of the given theory of that power are isomorphic.
|
| Chang & Keisler, 'Model Theory', pages 40-42.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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