Struggling with Gödel’s incompleteness theorem

Phillip asked:

Is it an implication of Godel’s Incompleteness theorems that all statements/ claims are equally valid/true because all statements/claims are based on unprovable assumptions?

Answer by Craig Skinner

Ah, Godel – the man without whom mathematics would be complete!

No, there is no such implication as the one in your question.

His theorems are technical proofs about formal axiomatic systems, and have implications only for such systems – for maths & logic, for classical (algorithmic) computing, and for cognitive science insofar as cognition includes formal deduction.

Plane geometry was axiomatized by Euclid, non-Euclidean geometries (using variant parallel lines axioms) followed in the 18th/19th century, and arithmetic was axiomatized in the 19th century.

It was rather assumed that all true theorems and no false ones could ultimately be derived from the axioms, so that mathematics was complete and consistent. Hilbert and followers struggled to prove this. But hopes were permanently dashed by Godel, Turing and Church who all proved in their different ways that the goal is impossible.

Godel proved that any axiomatic system contains some true statements not provable in the system (incompleteness), and that no axiomatic system can ever be proved to be consistent. His genius was to use the notation of a system (arithmetic) not just to derive theorems within the system, but to construct a theorem ABOUT the system, which theorem was obviously true but couldn’t be proved in the system.

His theorems are hard going if you have Uni-level maths, near impossible otherwise. A standard ordinary-language illustration is as follows:

Consider the statement: This sentence is unprovable.

If it is true, what it says is correct, so it IS unprovable, and we have a true statement which cant be proved (system is incomplete).

If it is false, what it says is incorrect, so it ISN’T unprovable, it’s provable, and we have a false statement provable within the system (system is inconsistent).
It follows that in any consistent axiomatic system there exist true but unprovable statements.

As regards maths, we happily work with incompleteness, and although consistency cant be proved, we reasonably assume it since no inconsistency has ever shown up.

Godel’s arithmeticized statement, and plain-language equivalents, are self-referential. They are not just statements in a language, but about it. And this can seem paradoxical.

A simple example is that these (contradictory) sentences both come out true:

‘This sentence contanes exactly two erors’
‘This sentence contanes exactly three erors’

The first sentence has two spelling mistakes, so two errors.
The second sentence has the same two errors, plus a mistaken claim about there being three errors, which constitutes the third error.

To deal with this, rules have been drawn up governing first- and second-level expressions (languages and metalanguages) which I wont go into.

Human consciousness is self-referential – I talk about ‘my self’, ‘knowing my own mind’, and right now am aware of writing this sentence, and am aware that I am so aware, and of that awareness, and so on. So Godel is sometimes cited in explanations as to how life and consciousness can arise from dumb matter, notably Hofstadter’s brilliant book ‘Godel. Escher, Bach: an Eternal Golden Braid’ which makes extensive use of self-referential theorems, art and music

A classical computer (algorithmic software analogous to a formal axiomatic system) cant grasp the truth of a Godel-type sentence, it cant step outside the system. Humans can. Some people (notably Penrose) think this is an essential difference between humans and any possible computer. This view may provide some spiritual comfort for us, but I see no good reason to hold it, and advances in computing look set to produce systems that wonder whether humans can grasp the truth of Godel sentences.

Finally, many new big ideas in science or maths get misapplied. A Newtonian ‘science’ of human affection was attempted with people being ‘attracted’ to each other, ‘gravitating’ to the more attractive others, affection diminishing inversely as the square of the distance between lovers, and other nonsense. Darwin was swiftly misapplied to justify ruthless laissez-faire capitalism (‘Social Darwinism’). Then Einstein (‘everything’s relative’), Heisenberg (‘nothing’s certain’), quantum mechanics (‘there is no mind-independent reality’ or ‘expand your mind by resonance with quantum field vibrations for only 69.99’), Godel, catastrophe theory, chaos theory and more.


Answer by Helier Robinson

No. First of all, not all antecedents of arguments are unprovable assumptions. Descartes’s cogito is an example. Secondly, Godel’s theorems apply to formal systems such as Whitehead and Russell’s Principia, and not all philosophic arguments are such. More basically, Godel’s incompletness theorem says that a formal system large enough to include arithmetic necessarily cannot prove itself to be complete; and this does not mean that all statements are equally valid.


Answer by Shaun Williamson

No it isn’t and that is all can say because I don’t know where you got this idea from. Godel’s proof applies to our system of mathematics. It doesn’t apply to our system of logic. You should also keep in mind that we have proofs in mathematics and in logic. The idea of proof does not apply to everything.

Suppose you look out of the window and say ‘It is raining’. Someone else says ‘Prove it’. That can only be interpreted as a joke because we don’t have a system of proofs for remarks about the weather nor do we know what such a proof would look like.


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