S = the set of all sets not members of themselves
x = 1 / 0
Aside: There was a wonderful quote in Jaynes' Logic of Science, decrying the kind of airy mathematics that Chaitin is doing...
Should one design a bridge using theory involving infinite sets or the axiom of choice? Might not the bridge collapse?
But "the set of all sets which meet criterion X" is well-defined and is an extremely common thing to encounter. Similarly, "X is a member of itself" is well-defined and is fairly common.
So you end up with this definition -- the set of all sets which are not members of themselves -- which just puts together these two common, well-defined (at least, we hope they're well-defined) concepts and ends up at a contradiction.
There is no X for which "X / 0" is defined.
There are many, many, many X (in fact, potentially all X such that X is a set) for which "X is a member of itself" is defined.
In other words, division by zero universally leads to contradiction no matter what you put in for X. But self-membership is perfectly well-behaved for lots and lots of values, and only starts to get troublesome on certain particular cases. This is curious: why should something which works well in many, many cases end up at a contradiction based not on the logical form of the statement, but rather on the particular values we substitute in place of its variables?
A classic example is http://www.math.utoronto.ca/mathnet/falseProofs/first1eq2.ht... — you get from a correct equation to 2 = 1 by simply canceling an innocuous factor (a² - ab) from both sides, the problem being that the innocuous factor happens to be equal to 0.
You can avoid this problem by changing your reasoning procedure. Old incorrect procedure; given:
xy = xz
y = z
y = z ∨ x = 0
So the surprising discovery that this basic set-creation operation — the set of all objects meeting some criterion — contained a pitfall analogous to that of division occasioned some consternation. What was the analogous correction to make to reasoning with set theory? And once that was settled, all of the proofs made under the old system had to be checked wherever they used this fundamental operation.
I don't understand this. This seems to assume that the meaning of a phrase is contained within the phrase itself. But if you instead assume that the phrase is a pointer to meaning that is contained somewhere else then the paradox goes away.
And then Godel blew everything up anyway :)
Think of it this way:
If a stranger were to approach you on the street, say, "I am a liar," then walk away, you'd likely think that the fellow is a little depressed and perhaps a bit remorseful about a habit of lying whenever it suited him. If the same fellow, in the same circumstances, had instead said "I am lying to you right now" before wandering off, you're left with two choices -- seriously psychotic or smartass.
People are reluctant to think of systems such as language and mathematics as seriously psychotic, so a cottage industry has sprung up trying to prove that these systems are merely smartasses out to infect our internal source code repositories with Brainfuck or some other Turing tarpit. It doesn't matter how many times one invokes the demons of deconstructionism, these paradoxes are and will remain very real.
This sentence contains ten words and is a true statement.
If you haven't read Godel, Escher, Bach, now is the time.
c) file not found
If you're interested, though, let me know and I'll see what I can dig up for you from back when I was studying this stuff.
If we assume for every positive number there is only one negative number then this must be true: +∞/-∞=1
More or less:
Never mind, just trying to understand Cantor's brilliant mind.