HyperbolicMan
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What EXACTLY is a variable?
This probably sounds like a really stupid question, buts been giving me a terrible headache . . .
I've always had the intuitive understanding I learned in high school algebra that a variable can represent known or unknown quantity. I recently received an introduction to set theory and I began to rethink my understanding of a variable. It seems to me that a better way to describe a variable would be to say that it is a symbol that can refer to either a specific or an arbitrary element of a set. For example, we could say that "x=5" or that "x is a real number."
My question is: What does it really mean to refer to an arbitrary element of a set? If we accept that variables can represent specific elements of a set, then is there a way to define what we mean by a variable referring to an arbitrary member of a set? Vice-versa? I cannot seem to get around this without falling back on intuition.
I think the answer to this question is very important, because in almost all the proofs I've ever seen, proving that a proposition is true for an arbitrary element of a set implies that the proposition is true for every element of the set.
Thanks
This probably sounds like a really stupid question, buts been giving me a terrible headache . . .
I've always had the intuitive understanding I learned in high school algebra that a variable can represent known or unknown quantity. I recently received an introduction to set theory and I began to rethink my understanding of a variable. It seems to me that a better way to describe a variable would be to say that it is a symbol that can refer to either a specific or an arbitrary element of a set. For example, we could say that "x=5" or that "x is a real number."
My question is: What does it really mean to refer to an arbitrary element of a set? If we accept that variables can represent specific elements of a set, then is there a way to define what we mean by a variable referring to an arbitrary member of a set? Vice-versa? I cannot seem to get around this without falling back on intuition.
I think the answer to this question is very important, because in almost all the proofs I've ever seen, proving that a proposition is true for an arbitrary element of a set implies that the proposition is true for every element of the set.
Thanks