Types of VariablesDate: 01/14/99 at 11:24:14 From: Michael Kane Subject: Free variable definition I am currently reading "An Introduction to Mathematical Logic" by Church (I believe it is put out by The Princeton Press), and I am having trouble with the concept of a free variable. As I understand it, a variable is a proper name with multiple denotations. A variable is also a set of numbers in a range. Also it is not proper to reference a variable in terms of its range. Doing so would result in a statement such as "let x be in the set of x," and a statement like that results in a free variable. I am having trouble with this concept. Is it like saying (as an example), "let a number in the set of 1-5 be in the set of 1-5"? If it is, is this improper because a variable in a set is implied to have a single value in the set, and with a free variable, a single value is not implied? Thanks. Michael Kane Date: 01/14/99 at 14:44:01 From: Doctor Schwa Subject: Re: Free variable definition One way to look at it is like this. If you say "it is always true that x = x" then x is not a bound variable - x could still stand for anything. But if you say "there is at least one number that, when squared, is equal to itself. Let's call one of those numbers x," now x is a bound variable, because it has been given a meaning that's not true for all x. "Dummy" variables, or if you are a computer programmer "local" variables, are always bound. For instance, if you say take the sum from k = 1 to 10 of k^2, then the variable k has a special meaning within that sum and referring to it outside the sum doesn't make sense. When you say {x | x is a number between 1 and 5}, that x is a variable being used only inside the { }. Referring to x at all outside the set brackets doesn't make sense. You could say "let y be an element of {x | x blah blah blah}" and then you would know y is a number between 1 and 5, but not which one. y is then free (in the sense that it can be used outside the set brackets) and bound (in the sense that you couldn't later say "let y be an element of {q | q > 10}" or anything like that). If these comments are still unclear to you, please write back again with some more examples for me to discuss! - Doctor Schwa, The Math Forum http://mathforum.org/dr.math/ |
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