(This is not a homework question, but I still hope for some help so that I can understand these math concepts better for future math courses.)
I can't make sense of Cantor's Diagonal Argument as presented by some notes I found from Virginia Common Wealth University. I am far from a mathematician, but I have enough university calculus and linear algebra to understand what the notes are talking about.
Here are the notes that I will refer to, http://www.people.vcu.edu/~rhammack/BookOfProof/Cardinality.pdf from VCU.
On page 218, they use a function that shows a bijection between |N| and |Z|. However, there are probably some functions that would not work such as f(n) = n; this would leave out all of the negative numbers of course. I take this to mean that a function failing to show a bijection does not imply that a bijection does not exist.
Then on page 219, they use some function that fills the right column with what seems to be random real numbers, okay. They also state the rule that leaves out a real number called b.
Before I get into my main issue, why does there need to be this extravagant rule to get b; why not just make the requirement that f can't equal some number? And can't we just make the exact same kind of rule for matching |N| to |Z|? And it would still exhaust all n leaving out an element of |Z|.
Anyways, my main issue is that they only show one function that does not match every n to every f(n). To address this, on page 220 it says "since this argument applies to any function f: N -> R ..." where are they getting that from? I don't understand how they know this is true for any function.
Either I am missing something - and I feel that I must - or Cantor's Diagonal Argument is not a very convincing argument at all.
I can't make sense of Cantor's Diagonal Argument as presented by some notes I found from Virginia Common Wealth University. I am far from a mathematician, but I have enough university calculus and linear algebra to understand what the notes are talking about.
Here are the notes that I will refer to, http://www.people.vcu.edu/~rhammack/BookOfProof/Cardinality.pdf from VCU.
On page 218, they use a function that shows a bijection between |N| and |Z|. However, there are probably some functions that would not work such as f(n) = n; this would leave out all of the negative numbers of course. I take this to mean that a function failing to show a bijection does not imply that a bijection does not exist.
Then on page 219, they use some function that fills the right column with what seems to be random real numbers, okay. They also state the rule that leaves out a real number called b.
Before I get into my main issue, why does there need to be this extravagant rule to get b; why not just make the requirement that f can't equal some number? And can't we just make the exact same kind of rule for matching |N| to |Z|? And it would still exhaust all n leaving out an element of |Z|.
Anyways, my main issue is that they only show one function that does not match every n to every f(n). To address this, on page 220 it says "since this argument applies to any function f: N -> R ..." where are they getting that from? I don't understand how they know this is true for any function.
Either I am missing something - and I feel that I must - or Cantor's Diagonal Argument is not a very convincing argument at all.