Set theory, by Thomas Jech, Academic Press, New York, 1978, xii +
... perhaps Cohen's proof which had the greater influence on mathematics; Gödel's proof produced just one consistency result (or one model), whereas Cohen's method of forcing, as later expanded by Solovay and others, was seen to apply to produce a wide variety of consistency results (or, many different ...
... perhaps Cohen's proof which had the greater influence on mathematics; Gödel's proof produced just one consistency result (or one model), whereas Cohen's method of forcing, as later expanded by Solovay and others, was seen to apply to produce a wide variety of consistency results (or, many different ...
Induction
... Let P(x) be a predicate statement, whose universe of discourse is the natural numbers. Suppose the following are true statements. 1) Base Case: P(1) : the statement is true for n = 1 2) Induction Hypothesis: P(n) implies P(n+1): the statement being true for n implies the statement is true for n +1 I ...
... Let P(x) be a predicate statement, whose universe of discourse is the natural numbers. Suppose the following are true statements. 1) Base Case: P(1) : the statement is true for n = 1 2) Induction Hypothesis: P(n) implies P(n+1): the statement being true for n implies the statement is true for n +1 I ...
Axiomatic Systems
... The Hilbert Program In 1900, a famous mathematician named David Hilbert set out a list of 23 unsolved mathematical problems to focus the direction of research in the 20th Century. Many of these problems remain unsolved to this day. Hilbert’s Second Problem challenged mathematicians to prove that mat ...
... The Hilbert Program In 1900, a famous mathematician named David Hilbert set out a list of 23 unsolved mathematical problems to focus the direction of research in the 20th Century. Many of these problems remain unsolved to this day. Hilbert’s Second Problem challenged mathematicians to prove that mat ...
Tautologies Arguments Logical Implication
... A formula A logically implies B if A ⇒ B is a tautology. Theorem: An argument is valid iff the conjunction of its premises logically implies the conclusion. Proof: Suppose the argument is valid. We want to show (A1 ∧ . . . ∧ An) ⇒ B is a tautology. • Do we have to try all 2k truth assignments (where ...
... A formula A logically implies B if A ⇒ B is a tautology. Theorem: An argument is valid iff the conjunction of its premises logically implies the conclusion. Proof: Suppose the argument is valid. We want to show (A1 ∧ . . . ∧ An) ⇒ B is a tautology. • Do we have to try all 2k truth assignments (where ...
Lesson 1.2
... • The cities Lubbock, Tulsa, and St. Louis lie approximately in a straight line on a map. Tulsa is between Lubbock and St. Louis, 380 miles from Lubbock and 360 miles from St. Louis. Find the distance from Lubbock, Texas to St. Louis, Missouri ...
... • The cities Lubbock, Tulsa, and St. Louis lie approximately in a straight line on a map. Tulsa is between Lubbock and St. Louis, 380 miles from Lubbock and 360 miles from St. Louis. Find the distance from Lubbock, Texas to St. Louis, Missouri ...
2015Khan-What is Math-anOverview-IJMCS-2015
... assumptions and used them to prove deductively most of the geometric facts we know today. Our high school geometry is an excellent example of a deductive system. Recall that in the study of geometry, we began with a set of undefined terms, such as point, line etc. We then made some definitions, for ...
... assumptions and used them to prove deductively most of the geometric facts we know today. Our high school geometry is an excellent example of a deductive system. Recall that in the study of geometry, we began with a set of undefined terms, such as point, line etc. We then made some definitions, for ...
Lecture 7. Model theory. Consistency, independence, completeness
... of the sentences in ∆ hold in the model.) And if the answer is NO, usually the easiest way to show it is by deriving a contradiction, i.e. by showing that ∆ ├ ⊥. See homework problems 5-8. 2.4. Independence. The notion of independence is less crucial than some of the other notions we have studied; i ...
... of the sentences in ∆ hold in the model.) And if the answer is NO, usually the easiest way to show it is by deriving a contradiction, i.e. by showing that ∆ ├ ⊥. See homework problems 5-8. 2.4. Independence. The notion of independence is less crucial than some of the other notions we have studied; i ...
KRIPKE-PLATEK SET THEORY AND THE ANTI
... By a ∆0 formula we mean a formula of set theory in which all the quantifiers appear restricted, that is have one of the forms (∀x ∈ b) or (∃x ∈ b). KP arises from ZF by completely omitting the Power Set Axiom and restricting separation and collection to absolute predicates (cf. [3]), i.e. ∆0 formula ...
... By a ∆0 formula we mean a formula of set theory in which all the quantifiers appear restricted, that is have one of the forms (∀x ∈ b) or (∃x ∈ b). KP arises from ZF by completely omitting the Power Set Axiom and restricting separation and collection to absolute predicates (cf. [3]), i.e. ∆0 formula ...
