Gödel’s Proof and Intuitionism
Page last updated 25 Sept 2020
Some people assert that when Gödel stated in his incompleteness proof that his Proposition V was ‘intuitionistically unobjectionable’, that what he meant by that was that it was in accordance with what is called ‘intuitionistic logic’. Intuitionistic logic arose from the philosophy of intuitionism, but there is more to intuitionism than intuitionistic logic alone. In intuitionism all mathematical entities are ‘constructed’ from primitive fundamental entities that arise from intuitive concepts of the mind.
It is reasonable to infer that Gödel meant that his Proposition V was in accordance with the principles of intuitionism rather than what is now called ‘intuitionistic logic’, since
- Gödel made the assertion in relation to the claim that his proof was constructive, and in relation to any assertions of existence. See Gödel’s Proposition VI, particularly his footnote 45a. Intuitionistic logic, of itself, cannot ‘construct’ anything. In order for that to happen, there has to be one or more mathematical entities to which that logic can be applied. In intuitionism, these are constructed from primitive fundamental entities that arise from intuitive concepts of the mind.
- At the time of Gödel’s proof, intuitionism was in a state of flux, and a fully defined system of intuitionistic logic (which was originally based on the notions of intuitionism as it was then) was only published in 1930 by Heyting, the same year in which Gödel wrote his paper. Gödel himself notes that a proof of his Proposition V would be rather involved. It is therefore difficult to fathom why Gödel would state that it is ‘easily seen’ that his Proposition V was intuitionistically unobjectionable, if what he meant that it was easily seen that every single step of this unwieldy proof would, if written out in full detail, be formulated completely in accordance with intuitionistic logic, when a full description of such logic had only just been published and would have been unfamiliar to most mathematicians of the time.
- Gödel made no reference to intuitionistic logic. Although today, intuitionistic logic is accepted as a subject that is independent of the philosophy of intuitionism, that was not the case at the time of Gödel’s paper, so if he had wanted to make that distinction, one would have expected that he would have actually done so.
As such, it is reasonable to infer that Gödel was making the distinction that, rather than his proof only asserting that a certain unprovable proposition must exist without actually constructing one, his proof provided a proposition which could be seen to be constructible according to the principles of intuitionism – and that this applied to the entities referred to in his Proposition V.
And if the entities referred to in a proof of Gödel’s Proposition V were constructible according to the principles of intuitionism, they had to be ‘easily seen’ to be ‘constructible’ according to the principles of intuitionism – from the fundamental mathematical concepts that are given by the intuition of the mind – so plainly apparent as to require no detailed consideration.
However, this is where there are fundamental difficulties that Gödel failed to address, leaving them to the vagaries of what might be meant in the philosophy of intuitionism by the fundamental mathematical concepts that are given by the intuition of the mind. Intuitionism, according to Brouwer, involved the notion that intuitionistic mathematics is an activity of the mind that is over and above any considerations of language – that language is used to communicate mathematical ideas, rather than creating them. That raises several questions in relation to Gödel’s proof.
For example, in Gödel’s paper, the symbols of the formal system are simply assumed to exist; they are not constructed from other more primitive entities. One might debate whether from the intuitionist stance, the symbols of the formal system that Gödel refers to are themselves entities given by the intuition of the mind, or whether the symbols of the formal system are actually abstract intuitive mind concepts to which we later attach symbols of language. Gödel does not state his position on this, but it does not affect the following paragraphs.
For the formal system, the symbols and combinations of symbols of the formal system have no meaning until the axioms and the rules of inference are given. And since the symbols of the formal system are more primitive concepts than those axioms and rules of inference, it follows that if we attach a meaning to the symbol combinations of the formal system, that must occur subsequent to the initial act of intuiting the symbols of the formal system.
So that raises the question, for example, why is there a symbol of the formal system that Gödel refers to as 0, and which denotes zero? Why would the primitive intuitive mind concept of a formal system symbol, by mere coincidence, happen to be referred to by the very same symbol as that used for the concept of zero? Why is it the case that Gödel’s proof relies on this rather fortuitous ‘coincidence’ (among others)?
It follows that when Gödel asserts that his Proposition V is ‘intuitionistically unobjectionable’, he makes assumptions about the intuitive aspect of intuitionism, which far from making the intuitionistic aspect of any putative proof ‘easily seen’, raises more questions than it answers. In making that assertion, Gödel was himself making intuitive assumptions about the intuitive aspects of intuitionism, so that not only was he assuming that any proof of his Proposition V had to be correct, but also that it would also be in accordance with the principles of intuitionism, and in accordance with the principles of intuition on which intuitionism was based.
The irony is that when Gödel appealed to the principles of intuitionism in support of his Proposition V without any detailed consideration of what that might involve, he thereby sidestepped the considerations of language that give rise to the innate error of his paper.
Finally, it should be noted that we cannot rely on Gödel’s assertions made several years subsequent to the publication of his proof. Although intuitionistic logic became more and more acceptable, the notion that intuition and intuitionism might be a valid basis for a philosophy of mathematics fell out of favour. Gödel, like anyone else, was quite capable of changing his position depending on circumstances.
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