Bridge to Abstract Mathematics: Mathematical Proof and

... techniques, including induction, indirect proof, specialization, division into cases, and counterexample, are also studied. Solved examples and exercises calling for the writing of proofs are selected from set theory, intermediate algebra, trigonometry, elementary calculus, matrix algebra, and eleme ...

... techniques, including induction, indirect proof, specialization, division into cases, and counterexample, are also studied. Solved examples and exercises calling for the writing of proofs are selected from set theory, intermediate algebra, trigonometry, elementary calculus, matrix algebra, and eleme ...

HOW TO DEFINE A MEREOLOGICAL (COLLECTIVE) SET

... or what are its essential features. First, we can see that every single one of objects a, b, c and d is part of x. Second, whatever part of x we take (any its fragment), it overlaps one of the four objects in question. On the other hand we can notice as well that any object which is exterior to ever ...

... or what are its essential features. First, we can see that every single one of objects a, b, c and d is part of x. Second, whatever part of x we take (any its fragment), it overlaps one of the four objects in question. On the other hand we can notice as well that any object which is exterior to ever ...

Completeness or Incompleteness of Basic Mathematical Concepts

... Whether I am right or wrong about Gödel’s use, I will always use “implied be the concept” in a non-epistemic sense.56 (4) The concept of the natural numbers is first-order complete: it determines truth values for all sentences of first-order arithmetic. That is, it implies each first-order sentence ...

... Whether I am right or wrong about Gödel’s use, I will always use “implied be the concept” in a non-epistemic sense.56 (4) The concept of the natural numbers is first-order complete: it determines truth values for all sentences of first-order arithmetic. That is, it implies each first-order sentence ...

week6-recursion

... recursive call gets closer to a base case. • Recursion can always be used instead of a loop. (This is a mathematical fact.) In declarative programming languages, like Prolog, there are no loops. There is only recursion. • Recursion is elegant and sometimes very handy, but it is marginally less effic ...

... recursive call gets closer to a base case. • Recursion can always be used instead of a loop. (This is a mathematical fact.) In declarative programming languages, like Prolog, there are no loops. There is only recursion. • Recursion is elegant and sometimes very handy, but it is marginally less effic ...

Incompleteness

... Note that the definition of a language is purely “syntactical,” that is, no meaning is in anyway assigned to these symbols of the language (e.g., there is nothing that says ` somehow really represents what we think of a addition on the natural numbers, whatever that means). For any first-order langu ...

... Note that the definition of a language is purely “syntactical,” that is, no meaning is in anyway assigned to these symbols of the language (e.g., there is nothing that says ` somehow really represents what we think of a addition on the natural numbers, whatever that means). For any first-order langu ...

Proof, Sets, and Logic - Boise State University

... Some comments: I may want to do some forcing and some FrankelMostowski methods in type theory. It is useful to be able to do basic consistency and independence results in the basic foundational theory, and it may make it easier to follow the development in NFU or NF. Definitely introduce TNTU (and T ...

... Some comments: I may want to do some forcing and some FrankelMostowski methods in type theory. It is useful to be able to do basic consistency and independence results in the basic foundational theory, and it may make it easier to follow the development in NFU or NF. Definitely introduce TNTU (and T ...

Proof, Sets, and Logic - Department of Mathematics

... Added material on forcing in type theory. Note that I have basic constructions for consistency and independence set up in the outline in to be in the first instance in type theory rather than in either untyped or stratified theories. My foundation for mathematics as far as possible is in TSTU here, ...

... Added material on forcing in type theory. Note that I have basic constructions for consistency and independence set up in the outline in to be in the first instance in type theory rather than in either untyped or stratified theories. My foundation for mathematics as far as possible is in TSTU here, ...

Comparing sizes of sets

... Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A to B iff |A| = |B|. Let’s do it. . . Take arbitrary finite sets A and B. LR: Assume f : A → B is bijective. Then f is inje ...

... Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A to B iff |A| = |B|. Let’s do it. . . Take arbitrary finite sets A and B. LR: Assume f : A → B is bijective. Then f is inje ...

slides - University of Edinburgh

... Now, the number is getting larger - what's going on? The point is that the DIFFERENCE between m and n is getting smaller. It's essential that something gets smaller - otherwise the recursion will never terminate. ...

... Now, the number is getting larger - what's going on? The point is that the DIFFERENCE between m and n is getting smaller. It's essential that something gets smaller - otherwise the recursion will never terminate. ...

Recursive call to factorial(1)

... The loop computes each Fibonacci number by starting at 2 and working its way upward Clearly, the number of iterations is bounded above by n The amount of space required is constant ...

... The loop computes each Fibonacci number by starting at 2 and working its way upward Clearly, the number of iterations is bounded above by n The amount of space required is constant ...

REVERSE MATHEMATICS, WELL-QUASI

... A(Pf[ (Q)) and U(Pf[ (Q)), this is because if Q is a wqo, then Pf[ (Q) is also a wqo (the U(Pf[ (Q)) case also follows from Theorem 1.3). The U(P ] (Q)) case follows from Theorem 1.3 because for every A ∈ P(Q) there is a B ∈ Pf (Q) that is equivalent to A in the sense that A ≤]Q B and B ≤]Q A. Notic ...

... A(Pf[ (Q)) and U(Pf[ (Q)), this is because if Q is a wqo, then Pf[ (Q) is also a wqo (the U(Pf[ (Q)) case also follows from Theorem 1.3). The U(P ] (Q)) case follows from Theorem 1.3 because for every A ∈ P(Q) there is a B ∈ Pf (Q) that is equivalent to A in the sense that A ≤]Q B and B ≤]Q A. Notic ...

Incompleteness in the finite domain

... tures without any formal supporting evidence. There are, essentially, two reasons for stating some sentences as conjectures. First, we believe that some basic theorems of proof theory should also hold true with suitable bounds on the lengths of proofs. The prime example is the Second Incompleteness ...

... tures without any formal supporting evidence. There are, essentially, two reasons for stating some sentences as conjectures. First, we believe that some basic theorems of proof theory should also hold true with suitable bounds on the lengths of proofs. The prime example is the Second Incompleteness ...

Incompleteness in the finite domain

... and bounded arithmetic seem to follow a general pattern. For example, as we noted above, polynomial time computations are associated with the theory S21 by a witnessing theorem. If we take S22 , which we believe is a stronger theory, then the corresponding function class is PNP ,2 which we believe i ...

... and bounded arithmetic seem to follow a general pattern. For example, as we noted above, polynomial time computations are associated with the theory S21 by a witnessing theorem. If we take S22 , which we believe is a stronger theory, then the corresponding function class is PNP ,2 which we believe i ...

Gödel`s Theorems

... by the familiar rules. But what is the real content of the thought that the truthvalues of all basic arithmetic propositions are thereby ‘fixed’ ? Here’s one initially attractive way of giving non-metaphorical content to that thought. The idea is that we can specify a bundle of fundamental assumptio ...

... by the familiar rules. But what is the real content of the thought that the truthvalues of all basic arithmetic propositions are thereby ‘fixed’ ? Here’s one initially attractive way of giving non-metaphorical content to that thought. The idea is that we can specify a bundle of fundamental assumptio ...

Foundations of Computation - Department of Mathematics and

... When these operators are used in expressions, in the absence of parentheses to indicate order of evaluation, we use the following precedence rules: The exclusive or operator, ⊕, has the same precedence as ∨. The conditional operator, →, has lower precedence than ∧, ∨, ¬, and ⊕, and is therefore eval ...

... When these operators are used in expressions, in the absence of parentheses to indicate order of evaluation, we use the following precedence rules: The exclusive or operator, ⊕, has the same precedence as ∨. The conditional operator, →, has lower precedence than ∧, ∨, ¬, and ⊕, and is therefore eval ...

lecture notes in logic - UCLA Department of Mathematics

... “singleton operation”, i.e., for every x, x ∈ z =⇒ {x} ∈ z. (7) Choice: for every set x whose members are all non-empty and pairwise disjoint, there exists a set z which intersects each member of x in exactly one point, i.e., if y ∈ x, then there exists exactly one u such that u ∈ y and also u ∈ z. ...

... “singleton operation”, i.e., for every x, x ∈ z =⇒ {x} ∈ z. (7) Choice: for every set x whose members are all non-empty and pairwise disjoint, there exists a set z which intersects each member of x in exactly one point, i.e., if y ∈ x, then there exists exactly one u such that u ∈ y and also u ∈ z. ...

Gödel Without (Too Many) Tears

... But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effective decidability. So let’s say: Defn. 1 A property P (defined over some domain of objects D) is effectively decidable iff there’s an algorithm (a finite set of instructions for a det ...

... But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effective decidability. So let’s say: Defn. 1 A property P (defined over some domain of objects D) is effectively decidable iff there’s an algorithm (a finite set of instructions for a det ...

you can this version here

... But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effective decidability. So let’s say: Defn. 1. A property P (defined over some domain of objects D) is effectively decidable iff there’s an algorithm (a finite set of instructions for a de ...

... But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effective decidability. So let’s say: Defn. 1. A property P (defined over some domain of objects D) is effectively decidable iff there’s an algorithm (a finite set of instructions for a de ...

MARTIN`S CONJECTURE, ARITHMETIC EQUIVALENCE, AND

... restricted to the class of uniformly Turing invariant functions. Theorem 1.2 (Slaman and Steel [25]). Part I of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem 1.3 (Steel [26]). Part II of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem ...

... restricted to the class of uniformly Turing invariant functions. Theorem 1.2 (Slaman and Steel [25]). Part I of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem 1.3 (Steel [26]). Part II of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem ...

Document

... that specifies at least one member of the set, and a recursive part that specifies how additional members of the set can be generated in terms of given members. • The prototypical example is ℕ, the set of natural numbers. It can be defined as follows: – Basis statement: 0 ℕ – Recursive part: if n ...

... that specifies at least one member of the set, and a recursive part that specifies how additional members of the set can be generated in terms of given members. • The prototypical example is ℕ, the set of natural numbers. It can be defined as follows: – Basis statement: 0 ℕ – Recursive part: if n ...

... that specifies at least one member of the set, and a recursive part that specifies how additional members of the set can be generated in terms of given members. • The prototypical example is ℕ, the set of natural numbers. It can be defined as follows: – Basis statement: 0 ℕ – Recursive part: if n ...