display all the ideas for this combination of texts
8 ideas
9135 | We now see that generalizations use variables rather than abstract entities [Sorensen] |
Full Idea: As philosophers gradually freed themselves from the assumption that all words are names, ..they realised that generalizations really use variables rather than names of abstract entities. | |
From: Roy Sorensen (Vagueness and Contradiction [2001], 8.4) | |
A reaction: This looks like a key thought in trying to understand abstraction - though I don't think you can shake it off that easily. (For all x)(x-is-a-bird then x-has-wings) seems to require a generalised concept of a bird to give a value to the variable. |
10075 | A 'partial function' maps only some elements to another set [Smith,P] |
Full Idea: A 'partial function' is one which maps only some elements of a domain to elements in another set. For example, the reciprocal function 1/x is not defined for x=0. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1 n1) |
10612 | An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P] |
Full Idea: If a function f maps the argument a back to a itself, so that f(a) = a, then a is said to be a 'fixed point' for f. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 20.5) |
10074 | A 'total function' maps every element to one element in another set [Smith,P] |
Full Idea: A 'total function' is one which maps every element of a domain to exactly one corresponding value in another set. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) |
10076 | The 'range' of a function is the set of elements in the output set created by the function [Smith,P] |
Full Idea: The 'range' of a function is the set of elements in the output set that are values of the function for elements in the original set. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 02.1) | |
A reaction: In other words, the range is the set of values that were created by the function. |
10605 | Two functions are the same if they have the same extension [Smith,P] |
Full Idea: We count two functions as being the same if they have the same extension, i.e. if they pair up arguments with values in the same way. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 11.3) | |
A reaction: So there's only one way to skin a cat in mathematical logic. |
10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
Full Idea: The so-called Comprehension Schema ∃X∀x(Xx ↔ φ(x)) says that there is a property which is had by just those things which satisfy the condition φ. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 22.3) |
10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
Full Idea: 'Theorem': given a derivation of the sentence φ from the axioms of the theory T using the background logical proof system, we will say that φ is a 'theorem' of the theory. Standard abbreviation is T |- φ. | |
From: Peter Smith (Intro to Gödel's Theorems [2007], 03.4) |