4 ms·

I've been wondering about the Russell Paradox... S = the set of all sets not members of themselves x = 1 / 0 Isn't the problem for x the same as for S? No

by ced 17y ago

I've been wondering about the Russell Paradox... S = the set of all sets not members of themselves x = 1 / 0 Isn't the problem for x the same as for S? Not all mathematical expressions are well-defined, and likewise for all "set expressions". 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?

ubernostrum 17y ago
The problem is that there is no X for which "X / 0" is defined, at least in the sort of traditional mathematics you're talking about. 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.
- ced 17y ago
  Mwell, I'm not really satisfied by your answer. Both "1" and "0" are well-defined expressions; division is also well-defined, but, in order to avoid logical contradictions, we cannot meaningfully define the division of 1 and 0. To me, that sounds exactly like what you described above, and I thought that the axiom of choice was the analog of the "you can't divide by 0" rule.
  - ubernostrum 17y ago
    Maybe look at it this way: 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?
kragen 17y ago
Yes, the problem is exactly analogous. And whenever you manipulate equations of rational expressions, or manipulate any equations by using division, you have to be careful that you don't accidentally invalidate your logic by inadvertently dividing by zero. A classic example is http://www.math.utoronto.ca/mathnet/falseProofs/first1eq2.html 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 conclude y = z Corrected procedure: given: xy = xz conclude: y = z ∨ x = 0 In many circumstances you can then immediately eliminate the x=0 case, e.g. when x is a numeric constant other than 0, or when x=0 → y=z by some other chain of reasoning. In other circumstances it turns out to complicate your proof enormously, and in cases like the fallacious proof I mentioned above, it actually makes the proof impossible. 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.

by ced 17y ago

ubernostrum 17y ago

The problem is that there is no X for which "X / 0" is defined, at least in the sort of traditional mathematics you're talking about. 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.

ced 17y ago
Mwell, I'm not really satisfied by your answer. Both "1" and "0" are well-defined expressions; division is also well-defined, but, in order to avoid logical contradictions, we cannot meaningfully define the division of 1 and 0. To me, that sounds exactly like what you described above, and I thought that the axiom of choice was the analog of the "you can't divide by 0" rule.
- ubernostrum 17y ago
  Maybe look at it this way: 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?

kragen 17y ago

Yes, the problem is exactly analogous. And whenever you manipulate equations of rational expressions, or manipulate any equations by using division, you have to be careful that you don't accidentally invalidate your logic by inadvertently dividing by zero. A classic example is http://www.math.utoronto.ca/mathnet/falseProofs/first1eq2.html 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 conclude y = z Corrected procedure: given: xy = xz conclude: y = z ∨ x = 0 In many circumstances you can then immediately eliminate the x=0 case, e.g. when x is a numeric constant other than 0, or when x=0 → y=z by some other chain of reasoning. In other circumstances it turns out to complicate your proof enormously, and in cases like the fallacious proof I mentioned above, it actually makes the proof impossible. 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.