I'm reading it on a Kindle so the formulas are a bit small for my eyes. It's a tough read but the book was highly recommended and it was only £7-ish.
I've only had it a few days.
Bob
]]>My link did work but now doesn't. ??? I have found the article again and corrected the link. At the moment it works.
I didn't say he proves everything from scratch. I said
He starts with the real number axioms. Then he proves everything from scratch;
I believe this is much the same as what you are saying.
According to Wiki his two theorems apply to theories other than set theory too. It's a long time since I did this at University, but that was how it was introduced then too.
Bob
]]>First Theorem - Undecidablity
If axiomatic set theory is consistent, there exists theorems which can neither be proved or disproved.
Second Therorem - Incompleteness
There is no constructive procedure which will prove axiomatic set theory to be consistent.
[edit] In simple words, 1. There are truths which cannot be proved or disproved.
2. You cannot prove things from scratch; you'll have to make axioms
So in the context of my discussion with Agnishom, he was pointing out that some theorems cannot be proved.
When I said 'everything', I meant everything that follows in his book, rather than everything that ever was/is/shall be.
Sorry, that was not what I am telling. I was reffering to the Second Theorem. He cannot even prove everything in his book from scratch because it is impossible. There has to be axioms somewhere.
The second theorem was the main reason for the cancellation of the Hilbert's program.
P.S: @bob bundy: your link does not work
]]>what is godel's theorem?
He expressed it in mathematical logic symbols but you can roughly sum it up as in a mathematical theory there is a statement, A, that states that "A cannot be proved".
Proof.
Is the statement true or false?
Let's suppose it is false.
Then A can be proved, which contradicts the statement. Therefore A is true.
So in the context of my discussion with Agnishom, he was pointing out that some theorems cannot be proved.
Have a look at
http://en.wikipedia.org/wiki/G%C3%B6del … s_theorems
Bob
]]>It would be silly to attempt that as you'd have to prove 'A' and also 'the converse of A'. That would suggest inconsistency and he goes to a lot of trouble to avoid that. (For example: he starts by proving that zero is a unique identity for addition.)
Bob
]]>Then he proves everything from scratch;
That violates Godel's Theorem
]]>Could you Please post the proof and the book(if possible)?
The book is as stated above.
It's not going to be easy to condense the proof.
He starts with the real number axioms. Then he proves everything from scratch; each result then being used to help prove the next bits. So there's a lot of work leading up to 1.77 and it would be hard to work out if anything could be left out.
I haven't studied that chapter in detail; I got the book for the chapters on complex numbers. I'll look and see if there's an easy way I can show what he does. I suspect it depends on the way he defines decimals, so you may not like it even when you've got the proof in front of you.
Give me a few days and I'll try to come up with something.
Bob
]]>vo' teH Sov batleth 'er'In
Welcome to the forum.
I don't know how that happened either .... not me .... I cannot rename posts. I have checked your account and you have two posts recorded but only one (number 8 here) is showing. Mystery.
Now we are all keen to know what went missing.
............ 0.999... does not exist?
I am studying 'Elementary real and complex analysis' by Shilov.
In section 1.77 he proves, as a theorem from the axioms for the real number system, that it is impossible to have an infinite sequence of 9s in the decimal form of a real number. That would make the question meaningless. !!!
Bob
]]>What did you post?
]]>You can repost it as I have no clue what happened. Please edit post #8 and put it in there. Then I will delete this one. I will try to rename the thread. Renamed it as best as I could, I do not remember the exact name.
]]>