Lecture 8: Back-and-forth - to go back my main page.
... s’s at the end. The inductive condition ensures that g preserves the interpretation of all symbols in LA , and fixes I pointwise. Clearly, the inductive condition is initially satisfied. During the construction, we need to make sure g is total, surjective, and moves arbitrarily small points above I. ...
... s’s at the end. The inductive condition ensures that g preserves the interpretation of all symbols in LA , and fixes I pointwise. Clearly, the inductive condition is initially satisfied. During the construction, we need to make sure g is total, surjective, and moves arbitrarily small points above I. ...
Predicate Logic - Teaching-WIKI
... First-order logic is of great importance to the foundations of mathematics However it is not possible to formalize Arithmetic in a complete way in FOL Gödel’s (First) Incompleteness Theorem: There is no sound (aka consistent), complete proof system for Arithmetic in FOL – Either there are sentences ...
... First-order logic is of great importance to the foundations of mathematics However it is not possible to formalize Arithmetic in a complete way in FOL Gödel’s (First) Incompleteness Theorem: There is no sound (aka consistent), complete proof system for Arithmetic in FOL – Either there are sentences ...
04. Zeno (5th century B.C.)
... • And: This is true only with respect to the A-chariots. ! The B-chariots are moving at 2 s.u./t.u. with respect to the C-chariots. ! The C-chariots are moving at 2 s.u./t.u. with respect to the B-chariots. ...
... • And: This is true only with respect to the A-chariots. ! The B-chariots are moving at 2 s.u./t.u. with respect to the C-chariots. ! The C-chariots are moving at 2 s.u./t.u. with respect to the B-chariots. ...
Class notes for Thursday, 10/1
... except for itself and one, which is the definition of a prime number. Question: Will we get the same answer if we start with 48=(2)(24)? Answer: Yes. It is true in the integers that every integer can be factored into prime numbers uniquely (up to multiplication by a unit). **Remember, we said that o ...
... except for itself and one, which is the definition of a prime number. Question: Will we get the same answer if we start with 48=(2)(24)? Answer: Yes. It is true in the integers that every integer can be factored into prime numbers uniquely (up to multiplication by a unit). **Remember, we said that o ...
Mathematical Structures for Reachability Sets and Relations Summary
... reachability sets by formulæ in a decidable logic is a difficult task, which is also witnessed by the fact that Hack (1976), proved that the problem of checking whether two reachability sets are equal is undecidable (also known as the equivalence problem). Consequently, undecidability entails that i ...
... reachability sets by formulæ in a decidable logic is a difficult task, which is also witnessed by the fact that Hack (1976), proved that the problem of checking whether two reachability sets are equal is undecidable (also known as the equivalence problem). Consequently, undecidability entails that i ...
Full text
... No odd multiperfect numbers are known. In many papers concerned with odd perfect numbers (summarized in McDaniel & Hagis [5]), values have been obtained which cannot be taken by the even exponents on the prime factors of such numbers, if all those exponents are equal. McDaniel [4] has given results ...
... No odd multiperfect numbers are known. In many papers concerned with odd perfect numbers (summarized in McDaniel & Hagis [5]), values have been obtained which cannot be taken by the even exponents on the prime factors of such numbers, if all those exponents are equal. McDaniel [4] has given results ...
1 Natural numbers and integers
... of the negative integers −1, −2, −3, . . . in Z slightly complicates the concept of prime number. Since any integer n is divisible by 1, −1, n and −n, we have to define a prime in Z to be an integer p divisible only by ±1 (the so-called units of Z) and ±p. In general, however, it is simpler to work ...
... of the negative integers −1, −2, −3, . . . in Z slightly complicates the concept of prime number. Since any integer n is divisible by 1, −1, n and −n, we have to define a prime in Z to be an integer p divisible only by ±1 (the so-called units of Z) and ±p. In general, however, it is simpler to work ...
Arithmetic and Geometric Sequences
... Comment. From the examples above, one can see that, when p is prime, the length of the principal period in the decimal expansion of 1/p is p − 1 (p = 7, 17), or smaller (p = 3, 11, 13). It is an unsolved problem whether there are infinitely many primes p such that the principal period in the decimal ...
... Comment. From the examples above, one can see that, when p is prime, the length of the principal period in the decimal expansion of 1/p is p − 1 (p = 7, 17), or smaller (p = 3, 11, 13). It is an unsolved problem whether there are infinitely many primes p such that the principal period in the decimal ...
On Countable Chains Having Decidable Monadic Theory.
... an interval J ⊆ I is C -uniform if for every t ∈ T the set J ∩ C −1 (t) is either empty or dense in J . We shall use the following result (see [15]). Proposition 2.1. Let I be a dense ordering. For every finite set T and every coloring C : I → T , I contains an infinite C -uniform interval. 2.2. Log ...
... an interval J ⊆ I is C -uniform if for every t ∈ T the set J ∩ C −1 (t) is either empty or dense in J . We shall use the following result (see [15]). Proposition 2.1. Let I be a dense ordering. For every finite set T and every coloring C : I → T , I contains an infinite C -uniform interval. 2.2. Log ...
Progressions
... 25. The following construction leads to the object called the Koch snowflake. Begin with an equilateral triangle K1 having unit side lengths. Then divide each side into three congruent segments, and build an equilateral triangle on the middle segment as the base in the exterior of the original tria ...
... 25. The following construction leads to the object called the Koch snowflake. Begin with an equilateral triangle K1 having unit side lengths. Then divide each side into three congruent segments, and build an equilateral triangle on the middle segment as the base in the exterior of the original tria ...
Distance, Ruler Postulate and Plane Separation Postulate
... ● We have assumed the undefined terms and axioms of set theory (so, for example, "lie-on" just means "element of") ● Notation for a line (a double-headed arrow overbar, p36) ● Definitions of lie-on, incident with and external point ● Definition and notation for parallel lines, l || m ● Trichotomy of ...
... ● We have assumed the undefined terms and axioms of set theory (so, for example, "lie-on" just means "element of") ● Notation for a line (a double-headed arrow overbar, p36) ● Definitions of lie-on, incident with and external point ● Definition and notation for parallel lines, l || m ● Trichotomy of ...
