Chapter 1 Logic and Set Theory
... reasoning. The standard form of axiomatic set theory is the Zermelo-Fraenkel set theory, together with the axiom of choice. Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can ...
... reasoning. The standard form of axiomatic set theory is the Zermelo-Fraenkel set theory, together with the axiom of choice. Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can ...
How Big Is Infinity?
... We're going to use the idea of a one-toone correspondence to compare the sizes of sets, infinite as well as finite. If a one-to-one correspondence exists between two sets S and T then we say that S and T have the same cardinality, denoted S ~ T. Informally, the cardinality of a set is the ...
... We're going to use the idea of a one-toone correspondence to compare the sizes of sets, infinite as well as finite. If a one-to-one correspondence exists between two sets S and T then we say that S and T have the same cardinality, denoted S ~ T. Informally, the cardinality of a set is the ...
Chapter 1 Logic and Set Theory
... reasoning. The standard form of axiomatic set theory is the Zermelo-Fraenkel set theory, together with the axiom of choice. Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can ...
... reasoning. The standard form of axiomatic set theory is the Zermelo-Fraenkel set theory, together with the axiom of choice. Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can ...
study guide.
... on the value of these free variables. A domain of a predicate is a set from which the free variables can take their values (e.g., the domain of Even(n) can be integers). • Quantifiers For a predicate P (x), a quantified statement “for all” (“every”, “all”) ∀xP (x) is true iff P (x) is true for every ...
... on the value of these free variables. A domain of a predicate is a set from which the free variables can take their values (e.g., the domain of Even(n) can be integers). • Quantifiers For a predicate P (x), a quantified statement “for all” (“every”, “all”) ∀xP (x) is true iff P (x) is true for every ...
On the determination of sets by the sets of sums of a certain order
... assume that the sets X are real, and hence S2 > 0. This proves ^(16) ^ 3. On the other hand ^(16) ^ 2 as was shown in [4]. We do not know at present whether F2(16) = 2 or 3. This type of reasoning can probably be made to yield the estimate F2(2k) ^ α, where a is the least integer such that (k + l)α ...
... assume that the sets X are real, and hence S2 > 0. This proves ^(16) ^ 3. On the other hand ^(16) ^ 2 as was shown in [4]. We do not know at present whether F2(16) = 2 or 3. This type of reasoning can probably be made to yield the estimate F2(2k) ^ α, where a is the least integer such that (k + l)α ...
Chapter 3 Finite and infinite sets
... matches each element of A to just one element of B, and each element of B to just one element of A. (This is not a precise definition; we will see the definition later. But the idea is clear without worrying about how the definition goes.) We say that A and B can be matched if there is a matching be ...
... matches each element of A to just one element of B, and each element of B to just one element of A. (This is not a precise definition; we will see the definition later. But the idea is clear without worrying about how the definition goes.) We say that A and B can be matched if there is a matching be ...
On interpretations of arithmetic and set theory
... of writing interpretation-applications as superscripts is at odds with the usual mapson-left convention for morphisms.) As usual, two morphisms f : T → S and g : S → T are said to be inverse to each other if fg = 1S and gf = 1T . An interpretation is then a morphism in this category, i.e., a mapping ...
... of writing interpretation-applications as superscripts is at odds with the usual mapson-left convention for morphisms.) As usual, two morphisms f : T → S and g : S → T are said to be inverse to each other if fg = 1S and gf = 1T . An interpretation is then a morphism in this category, i.e., a mapping ...
Lecture 7. Model theory. Consistency, independence, completeness
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
