First off, thank you for the video. It's uncanny, you know, how you seem to be able to find good quality videos on the www and by quality I'm not referring to the video resolution. You've made what is essentially chance into an art. It must take both intelligence and loads of luck to boot to turn what is essentially a roll of a die into a skill. Kudos! Thanks again.
Last I checked, Godel's incompleteness employs a variation of the liar sentence which, as you know, is "this sentence is false." According to the video, Godel's version of it is, K (for Kürt) = "the sentence with Godel number g is unprovable", the sentence with Godel number g being K itself. Thus, if K's provable, then it's unprovable [inconsistent because of the contradiction] and if K's unprovable then some mathematical truths are unprovable [incomplete].
As you might've already guessed, at the heart of Godel's therems lies the liar paradox. Before I go any further I need to draw your attention to the rather odd fact that Godel and anyone else who uses different versions of the liar sentence for whatever purposes is, all said and done, resorting to a L-I-A-R. Would you or anyone put to service a
liar to
prove something, anything? Perhaps I'm being too dramatic and perhaps I'm barking up the wrong tree; after all, the word "liar" may have been used just to grab our attention - only for effect, nothing else.
That out of the way, let's revisit K = the sentence with Godel number g is unprovable and the argument presented in the video which hopefully is a variation, salva veritate, of Godel's own.
Argument A [Adele, Godel's wife]
1. K is provable [assume for reductio ad absurdum]
2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable]
3. K is unprovable [1, 2 Modus Ponens]
4. K is provable and K is unprovable [contradiction] [..Math is inconsistent]
Ergo,
5. K is unprovable [1 - 4 reductio ad absurdum][..Math is incomplete]
A few points that seem worth mentioning.
a) Look at N (Nimbursky, middle name of Godel's wife) = premise 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable]. The assumption that has to be made for argument A to do its job of breaking math as it were is that N
makes sense, in logical terms,
makes sense implies that a truth value can be assigned to it.
The first clue that something's off is that N is a derivative of the liar sentence and we know that the liar sentence
doesn't make sense. One could say that the liar sentence is a poisoned well so to speak and every bucket of water, N being one, drawn from it will be lethal or, in this case, highly dubious. Common sense! No?
b) Consider now the fact that argument A is a reductio ad absurdum which, as you know, derives a conclusion and uses that to reject/negate one or
more of the assumptions made in the preceding lines of an argument. If you're not familiar, a reductio argument looks like this:
1. p
2. q & ~q [inferred from p]
Ergo,
3. ~p
Now in the argument A, the following assumptions/premises occur
1. K is provable
2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable]
The two assumptions lead to the contradiction below,
4. K is provable and K is unprovable
We are now justified in rejecting "one" of the premises but it doesn't necessarily have to be the one Godel has rejected which is 1. K is provable. After all, a reductio absurdum doesn't actually identify
which premise is false. A reductio ad absurdum is like a detective in faer earlier stages of a murder investigation - fae knows only that
someone is the murderer but doesn't know
who the murderer is. Thus, I could reject N = 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable and if I do that Godel's argument falls apart.
Given premise 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable] has highly ignoble origins (the liar sentence), shouldn't we reject it rather than reject 1. K is provable, a perfectly reasonable proposition?
c) There's another issue with statements like N = 2. If K is provable then K is unprovable [K = the sentence with Godel number g (K itself) is unprovable].
Given a proposition P,
1. If ~P then P
2. ~~P or P [from 1 implication]
3. P or P [2 double negation]
4. P [3 tautology]
The statement, If ~P then P can be thought of as P itself, it can be reduced to P. In other words, the conditional if ~P then P is an
illusion of sorts because it actually means P
Let's look at the version of the liar sentence that Godel uses which is, if K is provable then K is unprovable.
1. If K is provable then K is unprovable
2. ~K is provable or K is unprovable [from 1 implication]
3. K is unprovable or K is unprovable [from 2, ~K is provable = K is unprovable]
4. K is unprovable [3 tautology]
In essence, 1. K is provable then K is unprovable is logically equivalent to (I've used only equivalence rules of natural deduction), is nothing but, the statement 4. K is unprovable wearing heavy disguise.
What this means is that Godel's argument as presented in the video becomes,
1. K is provable [assume for reductio ad absurdum]
2. K is unprovable [If K is provable then K is unprovable = K is unprovable]
3. K is provable and K is unprovable [1, 2 Conjunction]
Ergo,
4. K is unprovable [1 - 3 reductio ad absurdum]
Did you notice what went wrong? The conclusion, 4. K is unprovable is also a premise 2. K is unprovable. A petitio principii.