The gulf between the Russell of Principia and Godel, Tarski, Turing etc. turned upon new ideas regarding the 'effective axiomitization' or effective generation of formal systems whose theorems were recursively enumerable. On the one hand there would be the set of all true statements in a field which would be consistent and complete. Sadly, this 'true Arithmetic' would not be 'first order'. There would be a 'Godel sentence' which was self-referential in that formal language and thus independent of it (because of the problem of self-referential negation or 'liar's paradox'). No standard natural number would have the property to make the Godel sentence true. Of course, there could be a non-standard model but in that case we would be mixing up apples and oranges.
For Russell, I suppose, Godel's result looks like a semantic paradox. From Godel's point of view, Russell simply hadn't developed his theory well enough. Perhaps if Russell had continued to work in Mathematical logic he and Godel could have had a very productive dialogue. Not so Wittgenstein. His notorious paragraph on Godel's theorem runs as follows
I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says ‘P is not provable in Russell’s system’. Must I not say that this proposition on the one hand is true, and on the other hand is unprovable?
No. You say Godel's proposition is true and has been proved in Russell's or Peano's system or any other first order system consistent with either. There are propositions about things which can be constructed but which can't be proven to be true or false in a particular system. In other words, there is a way to show 'within' a first-order system that it has its limits. But showing there is a limit does not necessarily involve crossing that limit.
Of course, a crazy person may 'interpret' any proposition to mean 'cats are dogs'. An interpretation- i.e. the assignment of meanings to symbols in a formal language- may be utterly arbitrary. If it assigns the value 'True' to a sentence or a theory, it is a model of that sentence or theory. But only models which serve some useful purpose are worthy of attention.
For suppose it were false; then it is true that it is provable.
Supposing a thing to be false does not make it 'provable'. Sherlock Holmes never said 'It is false to claim there is any such person as Professor Moriarty. This proves he and he alone is guilty of this murder.'
And that surely cannot be! And if it is proved, then it is proved that is not provable. Thus it can only be true, but unprovable.”
It is true that Professor Moriarty exists. But this does not mean it is unprovable that he killed so and so.
Just as we ask, “‘Provable’ in what system?”, so we must also ask, “‘true’ in what system?” ‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘false in Russell’s system’ means: the opposite has been proved in Russell’s system.
Godel's claim is proved in PM. Though it refers to what is unprovable it is not itself unprovable. Similarly, though I may talk about my cat, I am not myself a cat.
– Now what does your “suppose it is false” mean? In the Russell sense it means ‘suppose the opposite is proved in Russell’s system’; if that is your assumption you will now presumably give up the interpretation that it is unprovable.
No. We are keeping the 'interpretation' by which propositions in PM are proved and extending it to a proposition about what is unprovable in PM. But that proposition is not itself unprovable for the same reason that people who talk about their cats are not themselves cats.
And by ‘this interpretation’ I understand the translation into this English sentence. – If you assume that the proposition is provable in Russell’s system, that means it is true in the Russell sense, and the interpretation “P is not provable” again has to be given up.[…]
No. Godel's claim is provable. What is unprovable is something Godel constructed for that purpose. If I prove a particular truth about my cat only a crazy person would interpret my words or actions as a confession that I myself am a cat.
Back in 2000, Floyd and Putnam claimed that Wittgenstein’s argumentation is based on Gödel’s assumption of ω-consistency- which obtains if a collection of sentences is syntactically consistent (that is, does not prove a contradiction) and avoids proving certain infinite combinations of sentences that are intuitively contradictory.
This assumption is irrelevant. Wittgenstein simply confused a number constructed by Godel with a proof which refers to it as unprovable in PM.
Timm Lampert writes
'According to Floyd and Putnam this claim is grounded in Wittgenstein’s acceptance of Gödel’s mathematical proof showing that PM must be ω-inconsistent, if ¬P is provable. From this it follows that the predicates ‘NaturalNo(x)’ and ‘Proof(x,t)’ occurring in P cannot be interpreted as ‘x is a natural number’ and ‘x is the number of a proof of the formula with the number t’ because one has to allow for non-standard interpretations of the variable’s values being other than numbers.'
Why? Either 'recursive enumeration' is allowed in PM or it isn't. If it is then whatever 'interpretation' of PM yields the proofs it itself gives us can be extended to Godel's claim. It is certainly true that any lunatic can interpret anything at all as a command to stab bystanders. But does this 'non-standard interpretation mean that Doctors can't prove, on the basis of medical evidence, the fellow should be put in a padded cell?
Floyd/Putnam write
'That the Gödel theorem shows that (1) there is a well-defined notion of "mathematical truth" applicable to every formula of PM; and (2) that, if PM is consistent, then some "mathematical truths" in that sense are undecidable in PM, is not a mathematical result but a metaphysical claim.'
No. It is either a piece of mathematics not reliant on any interpretation or, ad captum vulgi, a claim about the canonical interpretation actually used for PM not some crazy interpretation whereby every formula in it is an instruction from God to start stabbing people.
But that if P is provable in PM then PM is inconsistent
because adding an axiom which would make a formal system for arithmetic complete would make it inconsistent.
and if ¬P is provable in PM then PM is ω-inconsistent is precisely the mathematical claim that Gödel proved.
What Wittgenstein is criticizing is the philosophical naivete involved in confusing the two, or thinking that the former follows from the latter.
The first result is about recursive enumerability. For a finitary first order language, it exists precisely because of incompleteness. After all, what is finite is likely to have limits and there may be 'internal' ways to show those limits exist even if it is impossible to cross them. We know we won't live forever even though nobody has returned from the dead to explain the exact nature of a limit we will all meet with soon enough.
But not because Wittgenstein wants simply to deny the metaphysical claim; rather, he wants us to see how little sense we have succeeded in giving it.
This interpretation is joined by the claim that Wittgenstein’s “aim is not to refute the Gödel theorem”, “for nothing in that proof turns on any such translation into ordinary language”. According to Floyd and Putnam Wittgenstein himself is just stating what Gödel holds if the latter insists that his proof is independent of any interpretation and rests on consistency assumptions.
Why bother giving a proof if the manner in which it will be interpreted is 'independent' rather than in conformity with what went before? Russell's own interpretation of PM is the one Godel is using (though one can equally say he is just doing math without any interpretation). His cleverness was in constructing a number and showing that its truth or falsity is unprovable in PM. But his proposition which was about that number, wasn't itself that number under any interpretation of PM consistent with the proofs it itself provides.
The claim is simply this: if one assumes (and, a fortiori if one actually finds out) that ¬P is provable in Russell's system one should (or, as Wittgenstein actually writes, one "will now presumably") give up the "translation" of P by the English sentence 'P is not provable'.
One could do so with a non-standard model or at the price of rejecting consistency in favor of completeness.
To see that Wittgenstein is on to something here, let us imagine that a proof of ¬P has actually been discovered.
Proofs can be wrong or else can show that there is some flaw in the axiom system because an absurdity has been proven.
Assume, for the time being, that Russell's system (henceforth PM) has not actually turned out to be inconsistent, however. Then, by the first Incompleteness Theorem, we know that PM is ω-inconsistent. But what does ω-incon consistency show? ω-inconsistency shows that a system has no model in which the predicate we have been interpreting as 'x is a natural number' possesses an extension that is isomorphic to the natural numbers
So, one could have a non-standard model. But then 'interpretation' has changed. We are on a slippery slope to just arbitrarily reading anything into anything else. I suppose, a certain sort of fatalist might come to the conclusion that 'Language speaks us' and that free will, or consciousness, is just an illusion. What is objective is actually subjective and what is normative is actually alethic and vice versa. I suppose this sort of paranoid ideation appeals to eggheads who feel the world is doomed to go in the wrong direction because people are listening to false prophets. Anyway, Godel believed in God. Thus, he must be one of the bad guys.
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