[Sigi_BC84]

helene_t, on Jun 23 2006, 08:40 AM, said:

This is contrary to how Sigi phrased it. Sigi says that there are truths that are hidden for us. But this is not hidden for us, it's just hidden for the system.

I deliberately used the phrasing "...there will always be thruths hidden from you while within your chosen system..." with which I meant that while working with a given system of axioms you will not be able to prove certain statements. This is what I meant by "hidden".

To not make a complete fool out of myself I would have to read up quite thoroughly on this stuff before making more statements, but on the above I'm actually quite sure even now :-).

kfgauss: I don't see at the moment where Gödel's Theorem is paradoxical, maybe you could try to explain that.

--Sigi

[kfgauss]

Sigi_BC84, on Jun 24 2006, 02:36 AM, said:

kfgauss: I don't see at the moment where Gödel's Theorem is paradoxical, maybe you could try to explain that.

The theorem isn't paradoxical, but the "this statement is false" idea is used in the proof. I, perhaps somewhat sloppily, referred to this idea as a paradox. Helene expands a bit on how this comes in to the proof above: you figure out how to write the statement "this statement is unprovable" in your formal system (or, perhaps more precisely, "this statement is unprovable in this formal system"). Then there can't be a proof of this in the formal system, but this means it's true (and, of course, unprovable).

Andy

[kfgauss]

P_Marlowe, on Jun 23 2006, 04:12 PM, said:

[snip mention of Russell's Paradox]

You get contradictions, if you allow, that
the set is an element of itself.
And you certainly know, how this gets prevented:
this is forbidden .

This isn't quite correct. Sets are allowed to be elements of themselves.

The resolution to Russell's Paradox involves making sure that sets never get "too big." The "set" of all sets that don't contain themselves is in fact not a set, but what mathematicians call a class (and the sorts of things you can do with classes are more restricted than those for sets).

Andy

[Sigi_BC84]

kfgauss, on Jun 24 2006, 10:02 AM, said:

Sigi_BC84, on Jun 24 2006, 02:36 AM, said:

kfgauss: I don't see at the moment where Gödel's Theorem is paradoxical, maybe you could try to explain that.

The theorem isn't paradoxical, but the "this statement is false" idea is used in the proof. I, perhaps somewhat sloppily, referred to this idea as a paradox. Helene expands a bit on how this comes in to the proof above: you figure out how to write the statement "this statement is unprovable" in your formal system (or, perhaps more precisely, "this statement is unprovable in this formal system"). Then there can't be a proof of this in the formal system, but this means it's true (and, of course, unprovable).

Yes, but doesn't that make it merely an example of "proof by contradiction"?

(NB I understand now what you meant by "at the heart of it it's paradoxical", so no need to argue further on that ;-).

--Sigi

[kfgauss]

Sigi_BC84, on Jun 25 2006, 02:02 AM, said:

kfgauss, on Jun 24 2006, 10:02 AM, said:

Sigi_BC84, on Jun 24 2006, 02:36 AM, said:

kfgauss: I don't see at the moment where Gödel's Theorem is paradoxical, maybe you could try to explain that.

The theorem isn't paradoxical, but the "this statement is false" idea is used in the proof. I, perhaps somewhat sloppily, referred to this idea as a paradox. Helene expands a bit on how this comes in to the proof above: you figure out how to write the statement "this statement is unprovable" in your formal system (or, perhaps more precisely, "this statement is unprovable in this formal system"). Then there can't be a proof of this in the formal system, but this means it's true (and, of course, unprovable).

Yes, but doesn't that make it merely an example of "proof by contradiction"?

(NB I understand now what you meant by "at the heart of it it's paradoxical", so no need to argue further on that ;-).

--Sigi

Yes, certainly.

I intended "paradox" to refer to the "this statement is false" idea, as this is a well-known paradox. Perhaps I sloppily applied the term to the theorem/proof.