Take standard first-order semantics. Consider a model $M$ of a certain theory $T$. Now suppose that we wish to include a subset a $\Psi \subset M$ to represent first-order formulas $\phi$. We denote the element corresponding to a formula $\phi$ by $\bar{\phi} \in \Psi$. Suppose we wanted all subsets of $M$ to also have internal representation via elements set $\Omega \subset M$. Given $X \subset M$ let the representative be denoted by $\bar{X} \in \Omega$. Then we wish to have a predicate $m(x,y)$ such that
\[ M \Vdash m(\bar{X},\overline{\phi(x)}) \text{ iff } \{z \in M: z \in \mathcal{I}\phi(x)\} = X \]
where $\mathcal{I}$ is the Tarskian interpretation in $M$. We will investigate later if this can lead to a paradox or limitation result.
Now suppose that in FOL we had an operator $[\,]$ which transforms a formula $\phi$ into a constant $[\phi]$: the free variables in $\phi$ this time (unlike in Bealer's system) are not free in $[\phi]$, it is a constant (or closed term). Suppose we had a predicate $Sub(a,b)$ which holds precisely when $a = [\psi(x)]$ for some $\psi(x)$ with single free variable $x$ and $b = [\psi([\psi(x)])]$. This expresses the reflection-into-self of $a$.
Now suppose we had a truth-like-predicate $T(a)$ which holds precisely when $a = [\phi]$ for some sentence $\phi$ which is, for instance, provable in a deductive system $D$ or true in a given class of models $M$, etc.
We assume we have the basic properties: i) $T([\phi]) \rightarrow \phi$ and ii) $T([\neg \phi])$ iff $ \neg T([\phi])$.
Consider now the formula $G(x) \equiv \neg\exists y. Sub(x,y) \,\&\, T(y)$ and let $Sub([G(x)], g)$.
Suppose that $T(g)$. Then by hypothesis $Sub([G(x)], g)$ and so by definition of $G(x)$, $\neg G([G(x)])$ and hence by i) and ii) we get $\neg T([G([G(x)])])$, that is, $\neg T(g)$, a contradiction.
Note how $g$ involves a further reflection-into-self mediated by negation. The result is thus a limitation on predicates satisfying i) and ii). It seems very likely that the above argument could be transposed to Bealer's system.
In Bealer's system if we postulate $y\Delta [\phi(x)]_x \equiv \phi(y)$ then we can obtain a Russell-type paradox (without relying on practically any other axiom).
A very interesting kind of 'limit theorem' involves how formal systems (such as $HA^\omega$) cannot represent the totality of total recursive functions.
Also the curious fact that we can define the set of indefinable real numbers.
This reflection-into-self, can it be given a geometric embodiment (if we wish to consider the connection between geometry and logic as in topos theory) ?
The downward Löwenheim-Skolem theorem is not the same kind of limit result. We consider that it is formalized within ZF(C) itself. If we do not accept this (i.e. the self-reflectivity of ZF(C)) then we are done. Otherwise we see that ZF(C) reflected-into-itself is aware that it is not really set theory, but only a countable reflection of set theory. This is a positive result. A limited system is itself 'aware' of its own limitation.
Self-reflection in logic and set theory can be conceptualised as the ability of a logical system, such as ZF (Zermelo-Fraenkel set theory with the Axiom of Choice, or ZFC), to contain statements that describe or analyse its own structure and consistency. This is analogous to a mirror reflecting itself within a larger arrangement of mirrors.
ReplyDeleteGeometrically, we can imagine a space where every point or object represents a statement or a set, and the relations between these points mirror logical relations such as membership, subset, and so forth. In this space, envision a specific subset that acts as a 'mirror' - this could take the form of a shape or a region (such as a circle, sphere, or any closed boundary) which 'reflects' or 'contains' a smaller version of the entire space. This smaller version represents a countable model of the larger space, illustrating the idea of the Löwenheim-Skolem theorem, which states that any infinite theory with a model also has a countable model.
In topos theory, a topos can be viewed as a category that behaves like the category of sets and includes a notion of logic and relations. Topoi can be visualised geometrically as spaces where objects (sets in classical set theory) are replaced by more general structures. The reflection within a topos could then be interpreted as a sub-topos or a site that reflects the properties or the logic of the entire topos in a simplified or reduced form.
The awareness of limitations within a system can be represented by this 'mirror' or reflective subset having properties that specifically denote or recognise that they are merely a reflection, not the entirety. This could be visualised as a reflective boundary within the geometric representation that contains annotations or markers that denote its status as a reflection, not the original.
To be continued (because a comment is too long).
Continuation:
ReplyDeleteThis concept of self-reflection can also be extended to consider the consequences of self-reference in the context of the diagonal lemma (or Cantor's diagonal argument). The diagonal lemma is a method of constructing a statement in formal languages that asserts its own unprovability, thereby creating a self-referential paradox. This mirrors the self-reflective nature of logical systems like ZFC, where statements can reflect and indeed question the completeness and consistency of the system itself. In a geometric representation, this might be visualised as a point or region within the 'mirror' that uniquely corresponds to itself in a manner that highlights paradoxes or limits of the system.
In quantum mechanics, the concept of a point or region within a 'mirror' that uniquely corresponds to itself, highlighting paradoxes or limits of the system, finds a compelling analogy in the behaviour of quantum states and the principle of superposition. Quantum systems are fundamentally probabilistic, and their states can be superpositions of different classical states, similar to how a point in the geometric mirror might represent a combination of multiple logical statements or conditions.
This analogy extends further in the context of quantum entanglement, where particles become interconnected in ways that the state of one (no matter the distance from the other) instantly correlates with the state of another. This could be visualised as a point in the mirror reflecting not just itself but also forming a bridge to another point, thus creating a complex and interconnected reflective structure that defies classical intuition.
Furthermore, the concept of self-referential paradoxes in the geometric mirror parallels the famous Schrödinger’s cat thought experiment in quantum mechanics, where the cat is both alive and dead until observed. This experiment highlights the inherent paradoxes and limits of quantum mechanics, akin to a point in the mirror that both exists in multiple states (reflects multiple truths) and yet is constrained by the boundaries of quantum theory.
Thus, the visual metaphor of a point in a geometric 'mirror' aligns well with quantum mechanics by illustrating the non-intuitive, paradoxical, and interlinked nature of quantum states.
I also consider if incorporating monoidal and quotient structures into the discussion of self-reflection in logical and geometric frameworks enhances our understanding of the interplay between abstract mathematical structures and logic. These structures offer a rich vocabulary for discussing concatenation and identity, as well as equivalence and simplification within such systems.
When these concepts are applied to the notion of self-reflection, monoidal and quotient structures provide mechanisms for the logical system to introspect and reconfigure itself. In a geometric metaphor, monoidal operations could be visualised as dynamically creating new regions or shapes by combining existing ones, thus expanding the reflective capacity of the system. Quotient structures could be represented as smoothing or merging areas within the mirror that reflect similar or equivalent aspects, thereby simplifying the overall structure and making it easier to comprehend or analyze.
We did not focus on diagonalization but focused instead on formal self-reflectivity seen as the transformation a structure undergoes when being inputed a representation of itself. The diagonalization perspective is also interesting and is given a very general form in Lawvere's well-known paper "Diagonal Arguments and Cartesian Closed Categories". It would be interesting the investigate further the subtle relationship between these two perspectives.
ReplyDeleteYes, a topos (and even a more general category) does contain a miniature version of itself within itself that determines its internal logic. This is given by the structure of the poset of subjobjects Sub(A) of an object A. Also for a topos the subobject classifier Omega can be seen as a miniature version of the topos itself. For a topos of sheaves it yields the structure of the underlying topological space (or site).
The comments on quantum theory, monoidal and quotient structures are most interesting, as is the intriguing analogy monoidal bifunctor = multiplication, quotient construction = division. This is quite in line with our investigations into a universal logical-categorical interpretation of Hegelian logic. We have started to investigate linear logic fragments which have a close connection to monoidal categories. Hegel himself in his Greater Logic in the section on Quantity gives a very abstract interpretation of basic arithmetical operations which might be given an abstract category theoretic interpretation. For these and many other topics see:
ReplyDeletehttps://chryssipus.blogspot.com/2024/04/universal-duality.html
https://chryssipus.blogspot.com/2024/04/proclus-elements-of-theology-as.html
https://chryssipus.blogspot.com/2024/03/on-hegel.html
More about multiplication and division. An interesting kind of 'multiplication' is given by the convolution of two functions which expresses a kind of synthesis of all possible interactions between the values of the two functions at two different points. Under certain conditions the fourier transform of the convolution of two functions is the ordinary product of the fourier transforms of the two functions. One could then define a kind of generalize 'quotient' of two functions f and g as the inverse fourier transform of the quotient F(f)/ F(g) of their two fourier transforms provided of course that, for instance, F(g) is not zero almost everywhere.
ReplyDeleteThat is really interesting about boundaries as mirrors. My paper https://www.researchgate.net/publication/351575493_Modern_incarnations_of_the_Aristotelian_concepts_of_Continuum_and_Topos focuses on the significance of a 'boundary'. But your comment, with a focus on quantum theory, suggests a more sophisticated Hegelian interpretation (I am thinking of the last stages of Quality in the Greater Logic).
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