Abstract
Alternative set theory was created by the Czech mathematician Petr Vopěnka in 1979 as an alternative to Cantor’s set theory. Vopěnka criticised Cantor’s approach for its loss of correspondence with the real world. Alternative set theory can be partially axiomatised and regarded as a nonstandard theory of natural numbers. However, its intention is much wider. It attempts to retain a correspondence between mathematical notions and phenomena of the natural world. Through infinity, Vopěnka grasps the phenomena of vagueness. Infinite sets are defined as sets containing proper semisets, i.e. vague parts of sets limited by the horizon. The new interpretation extends the field of applicability of mathematics and simultaneously indicates its limits. Compared to strict finitism and other attempts at a reduction of the infinite to the finite Vopěnka’s theory reverses the process: he models the finite in the infinite.
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Notes
He was the vice-rector of Charles University in Prague in 1990 and the minister of education in 1990 – 1992.
Zuzana Haniková (2022) deals in detail with the contexts and influences of other theories on AST and its development.
“How to formalise the intuitive notion of feasible numbers? To see what feasible numbers are, let us start by counting: 0,1,2,3, and so on. At this point, Yesenin-Volpin (in his analysis of potential feasibility, 1959) asks: ‘What does this ’and so on’ mean? Up to what extent ’and so on’?’ And he answers: ‘Up to exhaustion!’ Note that by cosmological constraints exhaustion must occur somewhat before, say, 21000.” (Sazonov, 1995, p. 30).
In particular, this means that one cannot freely use the general modus ponens rule E with the corresponding rule I in the system of Natural deduction calculus. Then the implication does not have to be transitive.
Vopěnka ’s principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent. However, before publishing his inconsistency proof, he found a flaw in it.
Azriel Lévy wrote in his review: “It was far enough to convince readers that modern metamathematics can be carried out for TS to the extent that it is carried out for ZF. As a result of the unusual way TS handles set theory and, even more, as a result of the highly formal approach taken in writing this book, the wealth of information in it is almost completely inaccessible to the students of set theory. This is a pity since the book contains many of the results of Vopěnka ’s Czech school of set theory and shows how to obtain the independence proofs of set theory by means of relative interpretations.” (Levy, 1984, p. 1423).
The original book (Vopěnka, 1979) is hardly available. A copy can be found at https://drive.google.com/file/d/17JRj2orUVDw7lrBEmBS1K6OK06RP32Xa/view. Holmes (2017) gives an abbreviated overview of its axioms.
Nonstandard models of Peano arithmetic contains infinite numbers. AST can be formally described as an ω-saturated model of cardinality ℵ1 of Peano arithmetics. Robinson’s Nonstandard Analysis uses ultrafilters (Robinson, 1966) to construct a model of nonstandard real numbers where the differential and integral calculus can be consistently described using infinitely small quantities.
“A model of AST can easily be constructed as follows. Let HF be the set of hereditarily finite sets. Let \((\widetilde V, \widetilde E)\) be the ultrapower of (HF,∈) over some nontrivial ultrafilter on ω. Add to \((\widetilde V, \widetilde E)\) all subsets \(X \subseteq \widetilde V\) such that for no \(x \in \widetilde V\) do we have \(X = \{y; (\widetilde V, \widetilde E) \models y \in x \}\). If we assume the continuum hypothesis, then the resulting model is a model of AST.” (Pudlák and Sochor, 1984, p. 572).
Husserl presents his Principle of All Principles: “Every originary presentive intuition is a legitimising source of cognition, that everything originally (so to speak, in its ‘personal’ actuality) offered to us in ‘intuition’ is to be accepted simply as what it is presented as being, but also only within the limits in which it is presented there” (Husserl, 1982, sec. 24).
This concept of a horizon resembles that of van Peursen (1977). “The horizon is tied to an observer; there is something subjective about it. On the other hand, it appears outside of man; it is eternal; we can never catch it (p. 183). The horizon adds nothing to the world. The horizon does not enrich the world. On the other hand, the world without the horizon is unimaginable, even impossible (p. 184). Man cannot remove it nor reach it. To man, the horizon represents the idea that there is more than what he sees (p. 187).”
Walking distance is still a walking distance if we increment it by one foot (but not 5 miles); a child is still a child 1 hour later (but not 5 years) (Gaifman, 2010, p. 6).
For this reason, Vopěnka claimed that “all sets are finite in Cantor’s sense.” Sochor (1984, p. 172) even asserts “There is no infinite set”. The reason is that he uses slightly different terminology and considers all sets to be phenomenologically “finite”. He calls Vopěnka’s infinite sets “inaccessible”.
Let φ(x) be a set formula that defines the class X = {x; φ(x)} and \(X \subseteq a\). We wish to prove there is a set b such that b = X. The set formula (∃b)(∀x)(x ∈ b ⇔ (x ∈ a ∧ φ(x)) is valid for ∅ and if it is valid for a then it is valid for its successor a ∪{c}. According to the Induction axiom it is valid for all a ∈ V.
Every subclass has the smallest element.
It is easy to construct an isomorphism. We assign the smallest elements of both classes to each other. The remained subclasses have the smallest elements again, we assign them again to each other, and so on.
It is not a set because the induction entails that a linearly ordered set-theoretically definable set has a least and the greatest element.
This agrees with Pascal’s concept of two infinities: the infinitely large and the infinitely small. While they are infinitely distinct, they correspond to one another: from the knowledge of one follows the knowledge of the other. “La principale comprend les deux infinites qui se rencontrent dans toutes: l’une de grandeur, l’autre de petitesse. … Ces deux infinis, quoique infiniment différents, sont néanmoins relatifs l’un à l’autre, de telle sorte que la connaissance de l’un mène nécessairement à la connaissance de l’autre (Pascal, 1866, pp. 288, 295).”
The term monad was originally borrowed by Robinson from Leibniz in his Nonstandard Analysis. Vopěnka took it from Robinson, and used it in the same meaning.
If \(C = \{c_{1}, c_{2}, c_{3}, \dots \}\) then \(x \in C \Leftrightarrow (\exists n)(n \in \mathbb {F}\mathbb {N} \wedge x \in C_{n})\).
\(x \in \mathbb Q\) is infinitely small iff \((\forall n)(n \in \mathbb {F}\mathbb {N} \Rightarrow |x|< \frac {1}{n})\); great iff \((\forall n)(n \in \mathbb {F}\mathbb {N} \Rightarrow |x|> n)\).
If \(x,y \in \mathbb {B}\mathbb {Q}\) then \(x \doteq y \Leftrightarrow (\forall n)(n \in \mathbb {F}\mathbb {N} \Rightarrow |x - y| < \frac {1}{n})\).
This continuum concept can serve as a response to Zeno. Zeno’s paradoxes are designed to refute both Aristotle’s and Democritus’ views (Fletcher, 2007, p. 567). Among other things, he challenged the notion of the continuum as a plurality of things. He argued that if there are many things, then they need not have any size at all; otherwise, there would be unlimited objects. If things have no size, then they do not exist at all.
Monads do not have observable size. But they are something: They have a body. Joining or removing a monad is indistinguishable for an observer. However, the composition of infinitely many monads forms an observable part of the continuum.
Unit distance means that the horizon of “depth” corresponds to the horizon of “distance”.
“The past is that which has been present, the future that which will be present. So there cannot be either a past or a future unless there is, independently of past or future, such a thing as how things are now.” (Dummett, 2000, p. 501)).
This definition corresponds to that of a uniformly continuous function in non-standard analysis. (Albeverio , p. 27)
Levy gives an example: An account of gene flow in a population that assumes an infinite population size is idealised, in that; obviously, no real-world population is infinite.
See for instance Boolos et al. (2002, pp. 302 - 312).
Let \(x \doteq y\) and \(y \doteq z\). Then for any \(n \in \mathbb {F}\mathbb {N}\) it is true that \(|x - y| < {1 \over n}\). \(\mathbb {F}\mathbb {N}\) is closed under arithmetic operations, so also \(|x - y| < {1 \over {2n}}\). The same holds true for \(y \doteq z\). Consequently, \(|x - z| \leq |x -y| + |y - z| < {1 \over {2n}} + {1 \over {2n}} = {1 \over n}\). Thus \(x \doteq z\).
The construction of real numbers from non-standard rational numbers is described in Albeverio et al. (1986, p.14). Let Q∗ denotes the class of nonstandard rational numbers, it is a dense, linearly ordered, non-Archimedean field. Let \(\mathbb Q_{b}\) denote the set of bounded rational numbers, \(\mathbb Q_{i}\) the set of infinitely small rational numbers, \(\mathbb Q_{i} \subseteq \mathbb Q_{b} \subseteq \mathbb Q^{*}.\) \(\mathbb Q_{i}\) form the maximal ideal in a ring \(\mathbb Q_{b}\). The result of the factorisation of \(\mathbb Q_{b}\) modulo \(\mathbb Q_{i}\) is the same as a factorisation by the infinite nearness \(\doteq \). We obtain the field isomorphic to real numbers.
$$ \mathbb Q_{b} /\mathbb Q_{i} \quad = \quad \mathbb Q_{b}/\doteq \quad \cong \quad \mathbb R. $$The class Q∗ corresponds to \(\mathbb Q\) of AST, \(\mathbb Q_{b}\) to \(\mathbb {B}\mathbb {Q}\), infinite nearness has the same definition.
Discussion of a similar systems can be found in Dean (2018, pp. 309 - 313). Dean inquires models of a theory Sτ formulated over a language extending that of first-order arithmetic with a new predicate F(x) such that F(0) ∧ (F(x) ⇒ F(S(x)) ∧ (F(x) ⇒ (∀y)(y < x ⇒ F(y)) but ¬F(τ) for a sufficiently great term τ. An interpretation of F(x) is x is feasible. It can express any soritical predicate. If we interpret it as x is finite then Sτ and \(\mathbb {F}\mathbb {N}\) have the same models. Dean suggested the neo-feasibilist theory of vagueness as a possible solution. This theory employs a nonstandard model of natural numbers. The term τ, which represents a non-feasible number, is realised by an infinite number, and the soritical predicate is interpreted as a proper cut on natural numbers). Dean cited Vopěnka as the only person to use nonstandard methods in connection with vagueness (Dean, 2018, p. 296).
“I look at something which is moving, but moving too slowly for me to be able to see that it is moving. After one second, it still looks to me as though it was in the same position; similarly, after three seconds. After four seconds, however, I can recognise that it has moved from where it was at the start, i.e. four seconds ago.” (Dummett, 1975, 315).
The most famous Zeno’s paradoxes are based on the tension between the real continuum and the ideal mathematical continuum. According to Dichotomy paradox “there is no motion because that which is moving must reach the midpoint before the end.” (McKirahan, 2010, p. 181). Since the argument can be repeated again and again, one must go through infinitely many places before arriving at the goal. No finite distance can ever be travelled: all motion is impossible. Indeed, an interval of ideal real numbers that has the length one can be halved again and again, and still, it is an interval of real numbers. The distance from the end will subsequently be \(\frac {1}{2},\frac {1}{2^{2}}, \frac {1}{2^{3}}, \dots , \frac {1}{2^{n}}, \frac {1}{2^{n + 1}}, \dots \) but never 0. It is impossible to reach the goal. However, in a concrete situation, there is a finite number n such that the distance \( \frac {1}{2^{n}}\) is indiscernible from the endpoint, \(\frac {1}{2^{n}} \approx 0\). We are at the goal, at the same monad, in n steps The Achilles and the tortoise paradox is based on the same principle. Their distance becomes indiscernible after finitely many steps. They are within the same monad. If the race continues, Achilles and the tortoise go on from the same position.
“We have seen that when we consider the two main alternatives which apparently allow one to make sense of a sort of ’modelling’ of countable infinity in the finite, namely nonstandard methods and feasibility, we face a dilemma. If we take into account proofs of arbitrary finite length, we might have consistency proofs, …, but we do not obtain any reduction of the infinite to the finite. On the other hand, if we consider proofs of length at most k, with k a standard integer, we have only proofs (possibly in relatively weak theories) of ’almost consistency’, and we do not obtain real consistency proofs.” (Bellotti, 2008, p. 23).
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Acknowledgements
I am grateful to Alena Vencovská for reading the text and hel** me with the mathematical questions. I have also used her prepared English translation of Vopěka’s latest book New Infinitary Mathematics. For valuable comments I thank Miroslav Holeček and Pavel Zlatoš. I would also like to thank three anonymous reviewers for all their comments and for their patience. They considerably helped to improve the text.
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Trlifajová, K. Infinity and continuum in the alternative set theory. Euro Jnl Phil Sci 12, 3 (2022). https://doi.org/10.1007/s13194-021-00429-7
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DOI: https://doi.org/10.1007/s13194-021-00429-7