Peano`s Arithmetic
... In 1884 Peano became a professor at the university. In the five years that followed, Peano produced many significant mathematical results. For example, he proved that if a function f(x, y) is continuous, then the first order differential equation dx/dy = f(x, y) has a solution [4]. But Peano’s most ...
... In 1884 Peano became a professor at the university. In the five years that followed, Peano produced many significant mathematical results. For example, he proved that if a function f(x, y) is continuous, then the first order differential equation dx/dy = f(x, y) has a solution [4]. But Peano’s most ...
Supplemental Reading (Kunen)
... makes sense from a strictly finitistic point of view: the axioms of ZFC do not say anything, but are merely certain finite sequences of symbols. The assertion ZFC 1 4 means that there is a certain kind of finite sequence of finite sequences of symbols-namely, a formal proof of 4. Even though ZFC con ...
... makes sense from a strictly finitistic point of view: the axioms of ZFC do not say anything, but are merely certain finite sequences of symbols. The assertion ZFC 1 4 means that there is a certain kind of finite sequence of finite sequences of symbols-namely, a formal proof of 4. Even though ZFC con ...
The Ring of Integers
... That was a lot of work to show that the function [x] is actually defined, particularly since this fact was probably obvious to you form the start! When you learn about “ordinary” math like calculus, or number theory, you usually “start in the middle”: A lot of things (that are hopefully plausible) ...
... That was a lot of work to show that the function [x] is actually defined, particularly since this fact was probably obvious to you form the start! When you learn about “ordinary” math like calculus, or number theory, you usually “start in the middle”: A lot of things (that are hopefully plausible) ...
Computational foundations of basic recursive function theory
... T treated as equal if they yield the same result. But it is not necessary to construe bar types as computations of elements, as will be seen in the semantics section below. It is significant that the bar types are defined after the basic types. We first understand the ordinary mathematical objects, ...
... T treated as equal if they yield the same result. But it is not necessary to construe bar types as computations of elements, as will be seen in the semantics section below. It is significant that the bar types are defined after the basic types. We first understand the ordinary mathematical objects, ...
Lecture 8 - McGill University
... • An integer N can be expressed as: N = PF X N’, where PF is a prime factor and N’ is another number which is N/PF – i.e. we can reduce the size of N by taking out its factor PF • We can keep doing this until eventually N = 1. Special case if N is actually a prime number. • This is O(π(√N)) = O(√ ...
... • An integer N can be expressed as: N = PF X N’, where PF is a prime factor and N’ is another number which is N/PF – i.e. we can reduce the size of N by taking out its factor PF • We can keep doing this until eventually N = 1. Special case if N is actually a prime number. • This is O(π(√N)) = O(√ ...
Introduction to HyperReals
... Since b is finite there are real numbers s and t with s < b < t. Let A = { x | x is real and x < b }. A is non-empty since it contains s and is bounded above by t. Thus there is a real number r which is the least upper bound of A. We claim r b. Suppose not. Thus r b and Hence r-b is positive or ...
... Since b is finite there are real numbers s and t with s < b < t. Let A = { x | x is real and x < b }. A is non-empty since it contains s and is bounded above by t. Thus there is a real number r which is the least upper bound of A. We claim r b. Suppose not. Thus r b and Hence r-b is positive or ...
Frege`s Other Program
... not equinumerous. Such a possibility is precluded if numbers are conceived of as objects, and in fact our counterexample crucially depends on assumptions as to what concepts have value ranges. The importance of this sort of counterexample is that it goes straight to the heart of the matter as regard ...
... not equinumerous. Such a possibility is precluded if numbers are conceived of as objects, and in fact our counterexample crucially depends on assumptions as to what concepts have value ranges. The importance of this sort of counterexample is that it goes straight to the heart of the matter as regard ...
Scoring Rubric for Assignment 1
... unclear. Theory is not relevant or only relevant for some aspects; theory is not clearly articulated and/or has incorrect or incomplete components. Relationship between theory and research is unclear or inaccurate, major errors in the logic are present. 0 – 4 pts Conclusion may not be clear and the ...
... unclear. Theory is not relevant or only relevant for some aspects; theory is not clearly articulated and/or has incorrect or incomplete components. Relationship between theory and research is unclear or inaccurate, major errors in the logic are present. 0 – 4 pts Conclusion may not be clear and the ...
071 Embeddings
... embedded in another larger lattice, and of these axioms defining both an ideal and a filter in that larger structure. Every Boolean lattice is capable of being embedded in a yet larger lattice, and the collection all Boolean lattices forms an unbounded collection – in other words, a proper class. On ...
... embedded in another larger lattice, and of these axioms defining both an ideal and a filter in that larger structure. Every Boolean lattice is capable of being embedded in a yet larger lattice, and the collection all Boolean lattices forms an unbounded collection – in other words, a proper class. On ...