Abstract
This article attempts a subjectively based approach, in fact one phenomenologically motivated, toward some key concepts of forcing theory, primarily the concepts of a generic set and its global properties and the absoluteness of certain fundamental relations in the extension to a forcing model M[G]. By virtue of this motivation and referring both to the original and current formulation of forcing I revisit certain set-theoretical notions serving as underpinnings of the theory and try to establish their deeper subjectively founded content and also their influence in reaching relative consistency results by the forcing method. In this perspective, the present approach may be seen as offering an alternative view of the consistency results of K. Gödel and P. Cohen in mathematical foundations reaching a subjective level that may be taken as ultimately conditioning the non-decidability of key infinity statements (such as the Continuum Hypothesis) on the level of formal theory.
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Notes
These two statements are unprovable in ZF since by Gödel’s incompleteness theorem they would imply that ZF is consistent. In the face of it, Cohen considered them as stronger than consistency of the system and most probably ‘true’ on intuitive grounds, in the sense that if we believe in actual sets then it should exist in the ‘real’ world a model M bearing the standard \(\in\)-relation of inclusion and whose objects are such sets (Cohen 2002, p. 1081).
A Baire space is defined to be a topological space with the property that each countable collection of open dense sets has dense intersection. The Baire category theorem can then be basically stated as follows: Every complete metric space and generally every completely metrizable topological space OR every locally compact Hausdorff space are Baire spaces.
There is a host of other formulations of forcing theory mainly in terms of complete Boolean algebras and the theory of topoi, see e.g., Moore (1988) and Lawvere (1971). My approach based on the ‘mainstream’ version of forcing theory, i.e., the one principally involving the notion of generic filters, holds in its general conclusions for all versions.
An excellent presentation of the motives and the stages in reaching the original formulation of forcing theory was given by P. Cohen himself in his later years in a conference at the University of Hawaii (2001) whose proceedings were published in the Rocky Mountain Journal of Mathematics; see Cohen (2002).
As a matter of fact, the proof of the existence of a generic set G relies on the countability of the ground model M by means of the Rasiowa-Sikorski lemma. However in most cases G does not belong to M (it cannot be enumerated within M), therefore its generic properties are rather associated with the actual infinity statements mentioned above.
The notion of absoluteness in set and generally model theories appeared first and was applied in a substantive way in Gödel’s construction of the constructive universe L. Intuitively, it establishes the invariability of certain properties of mathematical objects in extensions of mathematical systems provided with a sound notion of well-foundedness. A general definition of the absoluteness of formulas is this: If \(\varphi\) is a formula of a structure \(X\) with free variables \(x_{1},\ldots .,x_{k}\) and \(X\subseteq Y\), then \(\varphi\) is absolute between \(X\) and \(Y\) if:
$$\begin{aligned} \forall \; x_{1}, x_{2}\ldots .,x_{k}\;(x_{1}\in X\wedge x_{2}\in X\wedge \cdots .\wedge x_{k}\in X\Rightarrow (\varphi ^{X}\Leftrightarrow \varphi ^{Y}) \end{aligned}$$where \(\varphi ^{X}\) and \(\varphi ^{Y}\) are the relativizations of the formula \(\varphi\) to sets or classes \(X\) and \(Y\) respectively.
The meaning of the term immanent may be roughly explained to a non-expert in phenomenology as considering an object or state-of-affairs to be co-substantial to consciousness, this latter meant as a self-constituting temporal flux.
This involves taking the formula P in.
$$\begin{aligned} \exists X\;[\forall n\;(n\in X \leftrightarrow \mathbf{P}(n)] \end{aligned}$$(where P(\(n\)) is a definite property of the members of the set of natural numbers \(N\)) as a formula of full second-order logic in the language of arithmetic.
The two conditions satisfied if \(G\) is a filter in a partially ordered set \(P\) are:
$$\begin{aligned} \mathbf{(i)}\; \forall p,q \in G\;\exists r\in G\;(r\le p\;\wedge r\le q) \end{aligned}$$and
$$\begin{aligned} \mathbf{(ii)}\;\forall p\in G\;\forall q\in P\;(q\le p\rightarrow q\in G) \end{aligned}$$In addition, for \(G\) to be a \(P\)-generic filter one has to perform a second-order quantification over all dense sets of the ground model M. In turn, defining a dense set of the poset \(P\) involves also an existential-universal quantification over an indefinite horizon; see Sect. 1, par. 9, 12.
There is a broad discussion opened among philosophers, logicians and set-theorists as to the nature of mathematical-logical objects and to the accompanying philosophical doctrines. The position of the author clearly does not side with the proponents of platonic rationalism in mathematics; it rather converges toward R. Tieszen’s position of constituted platonism in Tieszen (2011) which can be roughly described as an attempt to accommodate platonic rationalism with transcendental phenomenology. A lengthier reference to the philosophical issue at hand will be made in the last section.
A set \(Y\subseteq X\) is first-order definable over \(<X,\in >\) if there is a first-order formula \(\varphi (\nu _{0}, \nu _{1},\ldots ,\nu _{n})\) with \(a_{1},\ldots ,a_{n}\) all in \(X\) such that \(Y=\{z\in X;\;\varphi ^{X}(z, a_{1},\ldots ,a_{n})\}\).
The \(\alpha\)-sequence \(f_{\alpha }(n)\) in M[G] is, in fact, an \(\alpha\)-sequence of generic functions from \(\omega\) into 2; this way, taking \(\kappa =\omega _{2}\) for \(\alpha < \kappa\), \(\omega _{2}\) new reals are introduced in M[G].
In the same approach one can interpret the absoluteness of the operation of union mentioned above by virtue of its definition as the union of a family \(\mathcal {A}\) of sets, i.e.,
$$\begin{aligned} \bigcup \mathcal {A}=\{x:\exists Y\in \mathcal {A}\;(x\in Y)\}. \end{aligned}$$A partial order \((P, \;\le )\) has the Countable Chain Condition (c.c.c.) iff every antichain (any family of pairwise incompatible elements) of the poset \(P\) is countable. Letting \(P\ne \emptyset\), the elements \(p, q\in P\) are defined as compatible if,
$$\begin{aligned} (\hbox {for}\; p,q\in P)\; \exists r\in P\;(r\preceq p \wedge r\preceq q) \end{aligned}$$that is, \(r\) extends both \(p\) and \(q\) in the usual intuition of extension. For example, if \(p\), \(q\) are finite partial functions from \(\omega\) to 2 and \(p\preceq q\) iff \(q\subset p\), then \(p\) and \(q\) are compatible iff they agree on \(\hbox {dom}(p)\cap \hbox {dom}(q)\), in which case \(p\cup q\) is a common extension of \(p\) and \(q\).
This term with a dash after meta is meant by the author as pointing to some subjectively rooted origin of the corresponding concept.
It should be also noted that proving \(2^{\omega }\) to be exactly equal to \(\omega _{2}\) in M[\(G\)], on the supposition that the ground model M is a model of GCH and AC, requires not only the application of the c.c.c. condition for the poset \(P\) but also Zorn’s lemma which is considered an actual infinity assumption (Kunen 1982, pp. 208, 209).
I have referred to S. Feferman’s view of the matter in Sect. 3, par. 1.
The recourse to actual infinity assumptions proves to be an inescapable necessity in the construction of foundational theories such as Gödel’s constructible universe L and Cohen’s introduction of the forcing model M[G]. One may mention, for instance, the introduction of impredicatively specified sets up to limit ordinals and also the application of the Axiom of Choice in Gödel’s proof of the consistency of CH with the other ZF axioms; while in forcing theory one may note the implicit application of the higher-infinity statement \(\hbox {MA}(\kappa )\), (\(\kappa > \omega\)), as this latter ensures the global properties of the generic set G out of the countable ‘environment’ of ground model M, as well as the application of second-order quantifications over generic sets.
Quine stated in (1986, p. 400), in response to the views of Ch. Parsons, that non-denumerable infinities are acceptable to him insofar as they are the only known systematizations of more welcome matters whereas anything beyond them, (e.g. inaccessible cardinals), should be considered as a mathematical recreation with no ontological rights.
For a subjectively founded interpretation of universal sentences of Predicate Calculus by means of an infinite regression of mental-constitutional states in the process of constitution expect author’s The Metaphysical Source of Logic by way of Phenomenology.
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Stathis Livadas—Non-affiliated research scholar.
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Livadas, S. The Subjective Roots of Forcing Theory and Their Influence in Independence Results. Axiomathes 25, 433–455 (2015). https://doi.org/10.1007/s10516-015-9269-8
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DOI: https://doi.org/10.1007/s10516-015-9269-8