Set Theory and Logic by Robert R. Stoll

Author:Robert R. Stoll
Language: eng
Format: epub, pdf
Publisher: Dover Publications
Published: 1961-06-14T16:00:00+00:00

EXERCISES

4.1. (a) Establish the consistency of the theory of partially ordered sets by way of a model.

(b) Show that this is a noncategorical theory.

(c) Show that the set of axioms {O1, O2, O3} for partially ordered sets is independent.

4.2. (a) Show that the theory of groups is noncategorical.

(b) Defining a group as an ordered triple 〈G, ·, e〉 such that G1, G2, and G3 of Example 2.2 hold, establish the independence of {G1, G2, G3}. (Suggestion: Use a multiplication table for displaying the operation which you introduce into any set.)

4.3. Consider the axiomatic theory having as its primitive notions two sets A and and having as axioms the following.

(i) Each element of is a two-element subset of A.

(ii) If a, a′ is a pair of distinct elements of A, then {a, a′} .

(iii) A .

(iv) If B, B′ is a pair of distinct elements of , then B ∩ B′ ⊆ A.

Show that this theory is consistent. Is it categorical?

4.4. Consider the axiomatic theory whose primitive notions are a nonempty set A and a binary operation 〈x, y〉 x – y (that is, we write the image of 〈x, y〉 as x – y) in A, which satisfies the identity

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