DLO ⊢ ∀ u̅ ( φ ↔ ψ ).
Prove that there is no theory Σ such that every model of Σ is a wellordering and Σ has an infinite model.
Assume that ( | A | , R^{A} ) is a wellordering.
A = ( ℝ , 0^{A} , 1^{A}, | ⋅ | ^{A}, +^{A} , -^{A} , ×^{A} , ÷^{A} , <^{A} )
be a structure with the standard zero, one, absolute value, addition, subtraction, multiplication and division (made total) and less than.
Let π : A ≈ A^{*} ≼ B be as in § 4.4.
Define a relation ≡ on | B | by setting x ≡ y iff y -^{B} x ∈ Finite ( B ).
(Here we write [ x ] for the equivalence class of x.)
Let A be as in Exercise 4.7.
Let π : A ≈ A^{*} ≼ B be as in § 4.4.
Let y ∈ Finite ( B ) - | A^{*} |.
Prove that there is a unique way to write y as x +^{B} δ where x ∈ | A^{*} | and δ is infinitesimal.
(x is called the standard part of y.)
Follow this hint:
Let S = { x ∈ | A^{*} | ∣ x <^{B} y }.
Say why S is nonempty and S has an upper bound in A^{*}.
We know that A has the least upper bound property. (Do not include a proof of this fact!)
Explain why A^{*} has the least upper bound property too.
Therefore, S has a least upper bound in A^{*}.
Let x = lub ( S ).
Let δ = y -^{B} x.
Prove that δ is an infinitesimal.
Prove that if y = x' +^{B} δ' where x' ∈ | A^{*} | and δ' is an infinitesimal, then x' = x and δ' = δ.
Use the language from Exercise 4.8 expanded by a unary function symbol F.
Use A from Exercise 4.8 expanded so that F^{A} = f.
Let π : A ≈ A^{*} ≼ B be as in § 4.4.
Prove that the following are equivalent.
For every real number s > 0, there exists a real number r > 0 such that for every real number x, if | x | < r, then | f ( x ) - f ( 0 ) | < s.
Let A_{i} for i ∈ I be structures with a common language.
Let B = Ult ( < A_{i} ∣ i ∈ I > , U ).
Assume that U is principal.
In other words, there exists i ∈ I such that U = { X ⊆ I ∣ i ∈ X }
Prove that B is isomorphic to A_{i}.
Let U be a nonprincipal ultrafilter over ω.
Let B = Ult ( A , U ).
Draw as accurate a picture of B as you can and justify the elements of your picture with proofs.
Say A_{Δ} ⊨ Δ for all finite Δ ⊆ Σ.
Let I = { Δ ⊆ Σ ∣ Δ is finite }.
For Δ ∈ I, let X_{Δ} = { Γ ∈ I ∣ Δ ⊆ Γ }.
Let F = { Y ⊆ I ∣ Y ⊇ X_{Δ} for some Δ ∈ I }.
Let B = Ult ( < A_{Δ} ∣ Δ ∈ I > , U ).
Prove that B ⊨ Σ.
A = ( ℝ , 0^{A} , 1^{A}, | ⋅ | ^{A}, +^{A} , -^{A} , ×^{A} , ÷^{A} , <^{A} )
be a structure with the standard zero, one, absolute value, addition, subtraction, multiplication and division (made total) and less than.
Let U be a nonprincipal ultrafilter over ω.
Let B = Ult ( A , U ) and π : A ≈ A^{*} ≼ B be as in § 4.7.
As a reminder, by definition, π ( x ) = [ n ↦ x ]_{U} where n ↦ x is the function from ω to ℝ with constant value x.
Prove that [ n ↦ n ]_{U} is an infinite element of | B | in the sense that for every x ∈ ℝ,
π ( x ) <^{B} [ n ↦ n ]_{U}.
Prove that [ n ↦ ( 1 / n ) ]_{U} is an infinitesimal element of | B | in the sense that for every positive x ∈ ℝ,
π ( 0 ) <^{B} [ n ↦ ( 1 / n ) ]_{U} <^{B} π ( x ).
The upward and downward Löwenheim-Skolem theorems in Chapter 4 only distinguish between finite, countable and uncountable languages, structures and sets. More generally, the cardinality of a structure is the cardinality of its universe.
Then there exists an elementary substructure A of B with X ⊆ | A | such that A has the same cardinality as X.
Say A has cardinality κ and λ > κ.
Then there exists an elementary extension B of A such that the cardinality of B is λ.