Interesting, but I just want to be clear. I have seen several examples of this proof in math books and they use as an example ( to the best of my recollection ) non terminating decimals. Based on this I have assumed that this proof was about irrational numbers.
There is a lot in your answer...
I didn't have an infinite set in mind. I was answering your question.
Conversation went as follows:
Me: Is there some reason that this proof would not work with infinite sets?
You: Why do you think it does not?
Me: I don't know why the proof would not work with infinite sets. With that...
Yes, I agree. These proofs that result in 1=2 usually involve division by zero and you would be quite right to ask "Am I missing something?"
I question it because if there was a list that did absolutely contain all the irrationals then Cantor's proof would provide a number that supposedly is...
Cantor "proved" that if there was a list that purported to include all irrational numbers, then he could find an irrational number that was not on the list.
Please consider two scenarios:
1. The list claims to contain all irrationals but doesn't.
2. The list absolutely contains all...
Reply to Deveno:
First, let me again thank you and HallsofIvy for your replies and input. My purpose in these posts is to get my idea out there and defend it.
Getting back to your quoteQuote:
the digits have to match "all the way down".
It DOES NOT MATTER how "far down we go" in the digits...
I suspect that there are many things in math the are subtle so just to make sure we are on the same page, is this what Cantor was saying: ( My perception of it anyway and this is as simple as I can make it ) ?
"Given a list of all possible irrationals I can show that there are irrational...
No, I'm pretty sure he "generated" the first digit after the decimal place of his "new" number by taking the first digit after the decimal place in the first number on the supposed list of all irrationals and changing it. Same for the second digit. He may have "assumed" that he could continue...
Reply to Deveno:
Sorry, when you said "infinite combinations of a finite number of things" I was assuming something along the lines of combinatorics which I think to mean would give me the power set of the set of digits from 0 to 9. I did not consider that you were indicating that a digit could...
First to clarify: I am not a mathematician, so while I find your explanation interesting I would have to struggle through some of the things that you mentioned. And I will have to look up the "Downward Skolem Lowenheim Theorem" ( First I heard of it! )
However I have been contemplating Cantor's...
It begins with an arbitrary number. I choose pi because it just popped into my head.
( Actually pi-3 )
This odometer ( actually just a metaphor for the process like the Infinite Hotel is used to illustrate some properties of infinite sets ) will work its way through all permutations. It will...
To HallsofIvy,
I assume you have my claim at phar2wild.ca
It goes on to say:
Quote:
Now imagine a very special odometer; different in two ways from the one in your car:
First, instead of operating from right to left, this one operates from left to right
so that the first rotor to complete a...
Deveno,
In reply to your posts: ( In no particular order )
Item 1
Quote: "Apply Cantor's second diagonal method to your list-what is your conclusion?" Unquote.
There is a conflict between what I claim and Cantor's second proof and it is not to be lightly dismissed. If there is a list of all...
Deveno,
Thank you for replying and taking the time; your answer is interesting and I will have to think about it.
Please clarify which Cantor proof you are referring to:
I assume the second is the proof that you can always create another irrational number not on the list
versus the first...
I'm starting with an irrational number and cycling through all digits so that the first ten on the list will each start with a different digit and everything pass the first digit will be identical for the first ten entries on the list. It should not matter which number I start with since every...
Hi,
New member here and have been dabbling with some aspects of George Cantor's work.
I think I have found a way to put the irrationals in one to one correspondence with natural numbers
but thousands of mathematicians over the years might disagree. Is there a subtle error ( or even a
blatant...