Before Abraham Robinson and Kurt Gödel became familiar with Paul Cohen’s Results, both logicians held a naïve Platonic approach to philosophy. In this paper I demonstrate how Cohen’s results influenced both of them. Robinson declared himself a Formalist, while Gödel basically continued to hold onto the old Platonic approach. Why were the reactions of Gödel and Robinson to Cohen’s results so drastically different in spite of the fact that their initial philosophical positions were remarkably similar? I claim that the key to these different responses stems from the meanings that Gödel and Robinson gave to the concept of intuition, as well as to the relationship between epistemology and ontology. I also illustrate that although it might initially appear that Gödel’s and Robinson’s positions after Cohen’s results were quite different, this was not necessarily the case.
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Gödel anticipated that the independence of the continuum hypothesis would eventually be proved. In his 1947 paper 'What is Cantor’s Continuum Problem', he provides examples which demonstrate the inability to prove the continuum hypothesis from the axioms of set theory.
Robinson did not publish any philosophical papers before the famous one from 1964 in which he declared himself to be a formalist. However, from comments he made in several previously published math papers, it is clear that his earlier philosophical outlook had been Platonic. For example, in his 1950 paper on the application of symbolic logic to algebra, Robinson wrote, 'The argument will be developed from the point of view of a fairly robust philosophical realism'. According to Gödel's 1951 paper, “The Platonic view is the only view tenable” (Gödel 1995, pp. 322–323).
As is clearly apparent from his famous 1964 paper, Robinson completely changed his philosophical point of view in response to Cohen's results. Although Gödel somewhat changed his opinions regarding naïve Platonism at this time, it is debatable whether Cohen's results influenced him enough to entirely alter his philosophical stance Wang alleged that Gödel was stimulated by Cohen's results to work on what is called ‘the Gödel program’ (Wang 1987).
Gödel's and Robinson’s position before 1964 seems to have been a fairly straighforward philosophy of Platonic realism. In 1964 Robinson emphasized his rejection of his early Platonist position in the following manner: "The infinite totalities do not exist in any sense of the word. Neverthless, we should continue the business of mathematics "as usual", i.e., we should act as if infinite totalities really existed."(the bold letter are mine) (Robinson 1964, p. 507). In Gödel (1972) said that he and A. Robinson entertained the same sort of objectivism except for Robinson's 'as if' position, which "require a special art of pretending well. But such pretending can never reach the same degree of imagination as one who believes objectivism." (Wang 1996, p. 240).
In one part of the paper, Robinson describes an argument between P—a platonist and R—himself.
Gödel believed that the definition of the concept 'finite' concerning mathemtical proofs had to be changed. He wrote several articles concerning this matter in the early 1950s. Robinson, on the other hand, thought that in order to avoid Hilbert's concept of 'finite', the definitions of what is provable had to be changed (Robinson 1973).
I am not claimimg that these are exact parallels or that Gödel belived that they were. I am simply asserting that in both cases the perception of the existence of more than one world of math became a reality.
Robinson did not explictly say that he was influenced by Kant (instead he agreed with Hilbert and with Gödel on this point); rather, he emphasized the similarity between the investigation of physical objects on one hand and mathematical objects on the other (Robinson 1964, pp. 508–509). Robinson agreed with this assertion as far as it concerned a finite number of elements. Of course, this does not mean that Hilbert, Godel and Robinson agreed about the definition of ‘finite’. Both Hilbert and Godel position themselves as followers of Kant, and Robinson was also familiar with Kant's approach. A seminar on the philosophy of mathematics given by Robinson and Korner at Yale University in 1973 included a discussion about Kant (Dauben 1995, 470). Abramsky, a very close friend of Robinson from the days they spent together as students at the Hebrew University in Jerusalem, spoke about Robinson's wide knowledge of German philosophy in general and of Kant's philosophy in particular (Dauben 1995, p 40, p.110).
This idea originates from phenomenology: Edmund Husserl claims that we can intuite essences, and moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method 'free variation in imagination' or 'ideation'. Tieszen (2005, p. 154) claims—and I agree with him—that the best and clearest examples of this method are to be found in mathematics. If I start squishing a circle, for example, we might ask which properties of the circle change and which remain the same. It is unfortunate that Husserl does not give examples invoving mathematics, but he does describe the methods of ideation in a number of his writings. Tiezsen describes the method of variation in detail, in his article (see Tieszen 2005, pp. 154–156).
The more details we know about a concept, the more truth statements we can apply to it and the clearer it becomes to us.
Since Robinson was concerned with objectivity and therefore in objective concepts, he was very interested in methods for completing formal systems and defining tests for verifying the completeness of formal systems.
Complex and real numbers are examples of order and closed field, and in this sense their theory is complete. Strangely, although one would intuitively imagine that real numbers-R are more complicated than natural numbers N, the theory of R (in terms of first-order logic) is in reality much less complicated than the theory of N. The theory of N contains undecidable sentences, which is very much in contrast to R, in which every sentence is decidable.
http://www.logicmatters.net/2011/11/21/kgfm-19-cohens-interactions-with-godel/On the website 'Logic Matters'.
A minimal model is a sub-model contained within all other models. In the case of the ZF system of axioms, model L (the model built by Gödel), is the minimum model of the system. Cohen and Shepherdson established that it is impossible to prove the existence of a minimal model in ZF. It turns out that it is impossible to find a pure minimal model, since there are non-standard models which define the concept of belonging differently. Thus, any classes of constructible sets are comprised of the same features and are capable of serving as a minimal model (a discovery that has a direct bearing on the meaning of Gödel's model L). This model is no longer necessarily the only constructive model that can serve as a model of ZF; thus, these models do not necessarily have a minimal model.
Gödel stresses that for him, the main importance of Cohen’s result is the fact that he proved the existence of his model L. 'One doesn't see how Cohen's proof work's, but one can see how my proofs work, if they are carried our in light of what we know after Cohen's proof.' (Wang 1996, p. 252).
At the end of his 1964 paper, Godel added a postscript (September 1966) in which he wrote 'Cohen's work, which no doubt is the greatest advance in the foundation of set theory since its axiomatisation, has been used to settle several other important independence questions' (Gödel 1964, p. 269). Godel went on to say that this result is particularly important since 'the axioms of infinity mentioned in footnote 20, to the extent to which they have so far been precisely formulated are not sufficient to answer the question of the truth or falshood of Cantor's continuum hypothesis.
Later in this paper I will show that Husserl influenced Gödel into believing that the correct way to describe a concept (including the concept of the ‘world of sets') is to do so syntactically and not semantically.
By the word 'elements' I am referring to both objects and concepts. I do not intend to give a full picture of Gödel's philosophy regarding the intuition of a concept here (whether abstract or ideal).
Primitive concepts are not abstract and are also not recursive. Gödel does not define or explain what these are; instead, he gives some examples such as the color red and other colors (Van Atten and Kennedy 2003, p. 458).
This path becomes increasingly wider as the process of abstraction continues and new abstract entites become more concrete.
Wang explains this process in detail (as well as the concepts of abstraction and idialisation) (Wang 1996, p. 158, 372).
Tiesen explains in detail the mental process of perception and also the mental process according to Husserl (Tieszen 2002, pp. 368–371).
In Wang, p. 233, Godel provides examples of our ability to perceive concepts clearly and links the definitions of a concept to the axioms that concern it. Wang (p. 216) describes Godel’s opinion of the power of intellectuall intuition to find the right axioms of set theory and generally clarify basic concepts. In p. 332 Wang describes Godel's belief that "we can percive the primitive concepts of metaphysics clearly enough to see the axioms concerning them and, thereby, arrive at a substantive system of metaphysics,….The axioms will be justified because we can see that they are – in the Platonic conceptual world – objectively true".
This means that the concept of a set will only be clear (and therefore objective) when we have a complete set of axioms which describe it. "Set theoretical concepts and theorems describe well-determined reality, in which Cantor's conjuecture must be either true or false. Hence its undecidability from the axioms being assumed today can only mean that these axioms do not contain a complete description of that reality" (Gödel 1964, p. 260 ).
There is a subtle but an important distinction between intuition and clarity. Intuition about an entity should be clear at some level; for example, intuition of elementary arithmetic is fairly strong, while the concept 'elementary arithmetic' is actually infinity, although an understanding of relevant concepts allows us to write formal theorems that are accepted by almost all mathematicians. However, despite the fact that our intuition regarding the arithmetic of higher infinite cardinals and their logic is very weak, most mathematicians would agree that these concepts are clear (even though this is not a basic or a primitive concept). The fact that we can differentiate between different cardinals is a proof that the concept 'cardinal' is clear to us at some level (Gödel 1964, p. 372).
Gödel's main interest was formal language and not models. He spent a considerable amount of time researching the consistency of the GCH adding an axiom of his own, the axiom of constructability, to ZFC. In the seventeis Godel was looking for new, strong infinity axioms in order to complete the set axioms ZFC. Gödel greatly appreciated Cohen's results because they showed that CH really is independent (as he had already suspected). However, he still maintained that models are not appropriate tools for revealing the structure of the world of math.
Gentzen proved the consistency of PA using real numbers in 1936.
Houser, In his article 'Gödel program revisited part 1' (Hauser 2006, pp. 537–539), explains in detail the changes in the character of infinite axioms which Gödel was looking for after the publication of Cohen's results.
Intentionality is a concept used by Husserl to describe the power of the mind to represent or substitute ideas, properties, or states of existence.
I am talking here about infinite proofs, i.e., proofs that are composed of an infinite number of steps (such as Gentzen's proof regarding the consistency of PA, or proofs using infinite forcing techniques).
Although Gödel never published anything further on the subject, three separate drafts appeared postumously in Volume 3 of his collected works. Notably, Etike Ellentuk, Robert M. Solovey and Gaisi Takeuti carried out research that helped to clarify the relationship between different ideas contained in Gödel's manuscrips. The summary of this research appears in Dawson (1997, p. 236).
The meaning of consistency strengh is explained in detailed in (Hauser 2006, p. 538 note 34).
In the sense that the world of math exists indepedently of us and that this world is complete, despite our epistemic limitations. This does not mean that we can grasp the world of math fully using only finite proofs.
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Leven, T. The Role of Intuition in Gödel’s and Robinson’s Points of View. Axiomathes 29, 441–461 (2019). https://doi.org/10.1007/s10516-019-09425-2
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