Types of Variables

Date: 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/