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Assume v. Presume

'Ey up, I is back, suitably grilled and catching up on this thread, although I may be suffering the effects of too much 'milk of amnesia' (ouzo).

I read a passage in a paperback which used the word 'koan' and may sum up the discussion on the previous pages.

"A koan is a kind of riddle or paradox without a solution, used by Zen masters to demonstrate the ultimate futility of logic and to provide (with some pupils) instant enlightenment."
I thought this phrase was rather relevant to the topics discussed and to some of the participants.
 
RobH1 said:
'Ey up, I is back, suitably grilled and catching up on this thread, although I may be suffering the effects of too much 'milk of amnesia' (ouzo).

I read a passage in a paperback which used the word 'koan' and may sum up the discussion on the previous pages.

"A koan is a kind of riddle or paradox without a solution, used by Zen masters to demonstrate the ultimate futility of logic and to provide (with some pupils) instant enlightenment."
I thought this phrase was rather relevant to the topics discussed and to some of the participants.
Click to expand...

Do not assume that logic is futile. I have a proof that it is not futile, but the proof is not instantly enlightening. Any source of instant enlightenment is (a) wrong, (b) right but inaccurate, or (c) pretends grammatically to mean something but is in fact vacuously meaningless, ie, is neither right nor wrong, because the concept cannot apply to it. Not all syntactically correct statements have truth values. Hungry red statements, like this one, are particularly false when true. This statement is false. This statement is false. Only one of the previous two statements is true, the other is false. The preceding three statements, and this one, are all false. All the statements in this paragraph are true.

Everything in the preceding paragraph is false, as is everything in this one.
 
Logic is only applicable within the axiomitic world from which it derives.

One of my favourites paradoxes concerns the set of all sets which do not contain themselves as a member. Does this set contain itself as a member or not?

The logician's answer to escape this paradox is to say that it is not a set at all but a member of an encompassing collection of objects called classes. All sets are classes but not all classes are sets.
 
Mike0001 said:
Any source of instant enlightenment ....(c) pretends grammatically to mean something but is in fact vacuously meaningless, ie, is neither right nor wrong.
Click to expand...
This is the option that most politicians seem to adopt most of the time
 
gomezz said:
Logic is only applicable within the axiomitic world from which it derives.

One of my favourites paradoxes concerns the set of all sets which do not contain themselves as a member. Does this set contain itself as a member or not?

The logician's answer to escape this paradox is to say that it is not a set at all but a member of an encompassing collection of objects called classes. All sets are classes but not all classes are sets.
Click to expand...


The mathematician's answer is to say that you cannot arbitrarily define sets that way. You must carefully define the axioms that sets satisfy, including only those that you need. The "set of all sets not members of themselves" then never actually appears unless you postulate its existence, and that then gives an inconsistent set of axioms.
 
gomezz said:
In this case not inconsistent but incomplete.
Click to expand...

Inconsistent, surely. Leading to a contradiction.

When you add the "set of all sets that are not elements of themselves" to mathematics, you immediately get contradictions, ie, inconsistencies. (If this set is a member of itself then it isn't, and vice versa.) No amount of extra axioms will solve that inconsistency.

[Note how I used "amount" there to irritate Black Hole!]
 
Not really. The contradiction arises from a insufficiently precise definition of a set.

Being able to prove that it does belong and to prove that it does not belong would demonstrate an axiomatic inconsistency.
 
gomezz said:
Not really. The contradiction arises from a insufficiently precise definition of a set.

Being able to prove that it does belong and to prove that it does not belong would demonstrate an axiomatic inconsistency.
Click to expand...

Unrestricted extensionality and the substitution of ¬(x∈x) for the membership predicate of a set is all that is needed for a contradiction.

[Bet we have lost Black Hole and perhaps everyone else now.]
 
All I need is to post for a contradiction.

(BTW - I employed group theory to devise a general method to solve the Rubik's Cube when it came out)
 
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