Notes on Infinite Sets
... It should be intuitively clear that we have a 1-1, onto correspondence. However, especially with the potentially elusive idea of infinity, we need to be able to follow this up with a precise function definition that permits a convincing demonstration of the desired properties of the correspondence. ...
... It should be intuitively clear that we have a 1-1, onto correspondence. However, especially with the potentially elusive idea of infinity, we need to be able to follow this up with a precise function definition that permits a convincing demonstration of the desired properties of the correspondence. ...
Godel`s Incompleteness Theorem
... • Formal proofs demonstrate consequence, but not non-consequence • Formal proof systems themselves aren’t systematic • But maybe a systematic method can nevertheless be created on the basis of formal logic? – Truth trees are systematic … and can demonstrate consequence as well as non-consequence. Co ...
... • Formal proofs demonstrate consequence, but not non-consequence • Formal proof systems themselves aren’t systematic • But maybe a systematic method can nevertheless be created on the basis of formal logic? – Truth trees are systematic … and can demonstrate consequence as well as non-consequence. Co ...
Lecture One: Overview and Fundamental Concepts
... aA -- “a is an element of A” or “a is in A” aA -- “a is not an element of A” or “a is not in A” S T --- “S is a subset of T”, i.e., every element of S is also an element of T AB -- “the union of A and B”, i.e., the set of objects that are in either A or B or both AB -- “the intersection of A an ...
... aA -- “a is an element of A” or “a is in A” aA -- “a is not an element of A” or “a is not in A” S T --- “S is a subset of T”, i.e., every element of S is also an element of T AB -- “the union of A and B”, i.e., the set of objects that are in either A or B or both AB -- “the intersection of A an ...
Infinity 1. Introduction
... The rigorous analytic theory of limits, infinite sums, differentiation and integration developed in the nineteenth century banished infinitesimals and infinite quantities in favour of arbitrarily small positive numbers ε and arbitrarily large natural numbers n. This certainly represents a supplantin ...
... The rigorous analytic theory of limits, infinite sums, differentiation and integration developed in the nineteenth century banished infinitesimals and infinite quantities in favour of arbitrarily small positive numbers ε and arbitrarily large natural numbers n. This certainly represents a supplantin ...
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... The diagonal lemma shows that in theories that can represent computability all formulas have a fixed point. Fixed point constructors, on the other hand, lead to inconsistencies, as they make it possible to define formulas that are equivalent to their own negation. Before we prove this, let us introd ...
... The diagonal lemma shows that in theories that can represent computability all formulas have a fixed point. Fixed point constructors, on the other hand, lead to inconsistencies, as they make it possible to define formulas that are equivalent to their own negation. Before we prove this, let us introd ...
Introduction to Theoretical Computer Science, lesson 3
... then determine the truth-values of atomic formulas, and finally, determine the truth-value of the (composed) formula Evaluation of terms: Let v be a valuation that associates each variable x with an element of the universe: v(x) U. By evaluation e of terms induced by v we obtain an element e(x) of ...
... then determine the truth-values of atomic formulas, and finally, determine the truth-value of the (composed) formula Evaluation of terms: Let v be a valuation that associates each variable x with an element of the universe: v(x) U. By evaluation e of terms induced by v we obtain an element e(x) of ...
Tools-Slides-3 - Michael Johnson`s Homepage
... In binary notation, 1 = 0.111... There exists a set S that can be paired one-to-one with its power set. The rational numbers are the same size as the power set of the natural numbers. According to standard set theory, the numerical size of the real numbers is the next highest number after the numeri ...
... In binary notation, 1 = 0.111... There exists a set S that can be paired one-to-one with its power set. The rational numbers are the same size as the power set of the natural numbers. According to standard set theory, the numerical size of the real numbers is the next highest number after the numeri ...
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... Corollary: Arithmetic is not axiomatizable. Gödel’s incompleteness theorem is often described as “any consistent and sufficiently strong formal theory of arithmetic is incomplete”, where a formal theory is viewed as one whose theorems are derivable from an axiom system. For such theories there will ...
... Corollary: Arithmetic is not axiomatizable. Gödel’s incompleteness theorem is often described as “any consistent and sufficiently strong formal theory of arithmetic is incomplete”, where a formal theory is viewed as one whose theorems are derivable from an axiom system. For such theories there will ...
Section 2.4 Countable Sets
... His argument was that for every perfect square n 2 , there is exactly one natural number n , and conversely, for every natural number n there is exactly one square n 2 . Galileo came to the conclusion that the concepts of “less than,” “equal”, and “greater than” applied only to finite sets and not i ...
... His argument was that for every perfect square n 2 , there is exactly one natural number n , and conversely, for every natural number n there is exactly one square n 2 . Galileo came to the conclusion that the concepts of “less than,” “equal”, and “greater than” applied only to finite sets and not i ...
PLATONISM IN MODERN MATHEMATICS A University Thesis
... Finally, set theorists describe all mathematical objects as sets. Cantor developed this school with his studies in the infinite. When he showed that there are certain infinite sets that can not map to infinite numbers, this led to the analysis of infinity in modern mathematics. All the schools of ma ...
... Finally, set theorists describe all mathematical objects as sets. Cantor developed this school with his studies in the infinite. When he showed that there are certain infinite sets that can not map to infinite numbers, this led to the analysis of infinity in modern mathematics. All the schools of ma ...
equivalence relation notes
... And now here is a ”problem” for you to ponder: it appears as though that sentence just defined, in 20 words or less, a number that can’t be defined in 20 words or less! So it seems we have a connundrum on our hands. Example 2. A teacher announces to her class that there will be a surprise exam next ...
... And now here is a ”problem” for you to ponder: it appears as though that sentence just defined, in 20 words or less, a number that can’t be defined in 20 words or less! So it seems we have a connundrum on our hands. Example 2. A teacher announces to her class that there will be a surprise exam next ...
A Basis Theorem for Perfect Sets
... A.r.1. Let n be minimal so that n is greater than t, Gx (n) 6= S(r), and Gx n_ hS(n)i, the concatenation of hGx (0), . . . , Gx (n − 1)i with hS(n)i, is in T ; we replace Gx with the least Gj such that Gj extends Gx n_ hS(n)i, Gj ∈ [T ], and Gj is not eventually constant in T . A.r.2. We let s b ...
... A.r.1. Let n be minimal so that n is greater than t, Gx (n) 6= S(r), and Gx n_ hS(n)i, the concatenation of hGx (0), . . . , Gx (n − 1)i with hS(n)i, is in T ; we replace Gx with the least Gj such that Gj extends Gx n_ hS(n)i, Gj ∈ [T ], and Gj is not eventually constant in T . A.r.2. We let s b ...
Scharp on Replacing Truth
... that governs it is radically incomplete. For example, although the axioms of Scharp’s theory are provably safe, if you conjoin some of those axioms together the resulting conjunction is not provably safe. The theory on its own does not tell us whether these conjunctions are safe (see section 3 for m ...
... that governs it is radically incomplete. For example, although the axioms of Scharp’s theory are provably safe, if you conjoin some of those axioms together the resulting conjunction is not provably safe. The theory on its own does not tell us whether these conjunctions are safe (see section 3 for m ...
equivalents of the compactness theorem for locally finite sets of
... Thus R is dense. Since every R–consistent choice on A is also an R∗ –consistent choice on A∗ , we get an R∗ –consistent choice S on the family A∗ . Then we easily see that {π(x) : x ∈ S} is an R–consistent choice on A. 2 As it is known (see [2]) Ff in is equivalent to some statement about propositio ...
... Thus R is dense. Since every R–consistent choice on A is also an R∗ –consistent choice on A∗ , we get an R∗ –consistent choice S on the family A∗ . Then we easily see that {π(x) : x ∈ S} is an R–consistent choice on A. 2 As it is known (see [2]) Ff in is equivalent to some statement about propositio ...
Appendix A Infinite Sets
... Imagine for a moment that you are a member of an ancient civilization, one that has not yet developed a counting system. Further imagine that you are a shepherd with a collection of sheep and you want to take them out in the countryside to graze then bring them all back home. How would you be sure t ...
... Imagine for a moment that you are a member of an ancient civilization, one that has not yet developed a counting system. Further imagine that you are a shepherd with a collection of sheep and you want to take them out in the countryside to graze then bring them all back home. How would you be sure t ...
The disjunction introduction rule: Syntactic and semantics
... Obviously, this fact could be interpreted as evidence that the mental models theory holds, since it appears to show that people only reason considering semantic models, and not formal or syntactic rules. However, this problem does not really affect theories such as the mental logic theory. As indica ...
... Obviously, this fact could be interpreted as evidence that the mental models theory holds, since it appears to show that people only reason considering semantic models, and not formal or syntactic rules. However, this problem does not really affect theories such as the mental logic theory. As indica ...
COMPLETENESS OF THE RANDOM GRAPH
... We switch now to a basic discussion of the random graph. We remind the reader of the following: Definition 3.1. A graph G is a pair of sets (V, E) where E ⊆ V × V . The elements of V we call vertices, and the elments of E we call edges. Two vertices v1 , v2 ∈ V are adjacent to each other if (v1 , v2 ...
... We switch now to a basic discussion of the random graph. We remind the reader of the following: Definition 3.1. A graph G is a pair of sets (V, E) where E ⊆ V × V . The elements of V we call vertices, and the elments of E we call edges. Two vertices v1 , v2 ∈ V are adjacent to each other if (v1 , v2 ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.