Bayesianism handles AI's hallucinations and Godel's undecidability

AI's hallucinations and Godel's undecidability are expected consequences of a frequentist approach to resolution of uncertainty. Bayesianism replaces the unknown with an (eventually arbitrary) prior.

AI's hallucinations and Godel's undecidability within frequentism

In Bayesianism, the probability prior replaces the unknown, so it is necessarily subjective. Frequentist probabilities try to be objective as much as possible, so they cannot replace the unknown. Any theory which is useful in real-world problems must address the unknown, explicitly or implicitly. AI's hallucinations and Godel's undecidability are examples of implicit ways of addressing the unknown. Since the unknown cannot be avoided for all practical purposes, it makes sense to address it as soon as possible, ideally in the mathematical logic itself.

The advantage of probabilities over true/false is that it allows updates. If a statement is said to be true, it is then hard to update it to false. Of course, we can leave a statement as undetermined or unknown, but then we have to define what this means in some logic. For instance, general recursive functions provide a model of computation which is Turing complete (thus it must deal with the unknown due to the halting problem being undecidable), these are partial functions, meaning that it is not required that every element of the first set is associated to an element of the second set, often used when the exact domain of definition of a function is difficult to specify. Working with partial functions is much harder than with total functions, note that unbounded operators in an Hilbert space are also partial functions and are much harder to work with than working with bounded operators.

In reference kopperman67, it is shown that standard probability spaces (through a parametrization involving a Hilbert space) can be defined in a particular logic, with remarkable properties that are hard to match, even if we are not interested in the unknown. One can easily argue that these properties are much better than set theory given by the ZFC axioms in first-order logic or another logic, for most practical purposes.

Ferminonic Fock space as a state of knowledge

Using a Ferminonic Fock space as a state of knowledge allows the definition of probabilities over a continuous set of statements, this is crucial to deal with uncertain statements about higher-order objects. A set of null-measure in a Fock-space is converted into a vacuum state, which represents complete ignorance (a prior probability) and not zero probability. This allows us to define probabilities for a subset of statements from abstraction, higher-order and ∞-categories logics.

Fock space vs. ZFC axioms, Higher-order logic and ∞-categories

There will always be undecidable problems, such as the halting problem. However, we can go very far with decidable mathematics, since the theory of real Hilbert spaces is by itself complete and decidable kopperman67, even for an infinite-dimensional Hilbert space. This implies that we can (and one could argue that "we must") deal with the incompleteness of mathematics using probabilities for what it is unknown.

Moreover, for Physics we do not need Peano's theory (algebra of integer numbers). A direct sum over Hilbert spaces corresponding to integer numbers modulo N, suffices. Also, the tensor product of two Fock spaces produces another Fock-space, so we do not need an infinite-dimensional tensor-product.

Once we consider the Hilbert space complete logic theory kopperman67 (and the probabilities associated) as the logic, then we can use Fock-spaces to deal with higher-order objects, in a way that is compatible with probability theory. This renders abstraction, higher-order and ∞-categories logics redundant, since Fock-spaces within first-order logic are enough to deal with higher-order objects.

Note that only first-order logic and abstraction logic are complete, not higher-order logic and ∞-categories. First-order logic is not expressive enough (which defeats the purpose of being complete) and it is not clear how to include probabilities in abstraction, higher-order and ∞-categories logics @Beauquier02 (which also defeats the purpose of being complete, since then we can't deal with incomplete mathematics). Note that we can also include probability theory in higher-order logic, as long as the statements involving probabilities are about the probabilities of events that involve first-order logic only @Rashkovich92, but this is still not expressive enough.

From a point process to a random field

(Bochner space)

The number operator in a Fock-space defines a (classical statistical) point process. But in a Fock-space we can change basis, from a continuous base space to a discrete base space, including non-local (with respect to the base space) correlations. In such a discrete base space we can define a projection operator corresponding to the non-null eigenvalues of the number operator (that is, the degree of freedom corresponding to the number operator is cancelled, only averages over such degree of freedom are considered). Note that in the 2D base-space required to define an infinite tensor-product, there may be more than one wave-function corresponding to one point of the x-axis: then we sum (using the inner-product) over all wave-functions with index greater or equal to the minimal index. Since this can be done for an arbitrary discretization, then the Fock-space for a 2D base-space represents a random field and the "points" in the point process refer to where there is available information.

The only way to define values of the field is in a basis such that the x-axis of the base space is discrete. When the x-axis is continuous, we only have the number operator at our disposal which is unbounded, no projection operator for a single x from a continuous x-axis. Note that using gauge-symmetries we can still define what happens at a single x from a continuous x-axis, however the normalization of the number operator is non-local an thus it cannot be defined for a single x, thus we cannot define a projection for a single x from a continuous x-axis. This has widespread implications in Physics and Mathematics, since random fields are usually assumed to be well-defined for a continuous index in the literature.

A follow-up question is then what are the alternatives to a point process to define a random field, or even just to define an infinite tensor-product. Since the existence of an infinite-tensor product is equivalent to the axiom of choice, we know that there is no alternative other than assuming that there is an infinite-tensor product and then studying its properties. But since the Fock-space verifies these properties and it exists, then assuming that such object exists doesn't add much (we already know it). Note that in the Fock-space the infinite tensor product appears as a completion of a countable basis, thus Cauchy sequences can be used and we never have to deal with infinite tensor-products directly, unlike in the axiomatic approach where relating finite and infinite tensor-products requires assumptions such as the "continuum limit" in Lattice QCD, for instance.

Relating the Fock-space with a uniform measure in an infinite-dimensional sphere

In the limit that the dimensions of an hypersphere become infinite, the uniform measure over its surface becomes the gaussian measure and the Gegenbauer polynomials (which define the hyperspherical harmonics) become the Hermite polynomials. Since the Gegenbauer polynomials are orthogonal with the uniform measure, then the uniform measure is the prior and it means that we have no information, while any other orthogonal state means we do have information in the form of a final wave-function (the result of the integral of the wave-function over the hyperspherical surface times the wave-function corresponding to the point in the surface).

The normalization of the Gegenbauer polynomials, for integers $n,\alpha\geq 0$ is:

$$N^{(\alpha)}_n=\Big(\frac{\alpha}{2}\Big)^{\frac{1}{2}-n}\int_{-1}^1\left[C_n^{(\frac{\alpha}{2})}(x)\right]^2\sqrt{1-x^2}^{\alpha-1}dx=\Big(\frac{\alpha}{2}\Big)^{\frac{1}{2}-n}\frac{\pi2^{1-\alpha}\Gamma(n+\alpha)}{n!(n+\frac{\alpha}{2})[\Gamma(\frac{\alpha}{2})]^2}$$

$$\lim_{\alpha\to\infty}N^{(\alpha)}_n=\frac{\sqrt{\pi}2^n}{n!}=\int\left[\frac{H_n(x)}{n!}\right]^2e^{-x^2}dx$$

According to reference @Lopez99:

$$\lim_{\alpha\to\infty}\Big(\frac{\alpha}{2}\Big)^{-n/2}C_n^{\Big(\frac{\alpha}{2}\Big)}\Big(\sqrt{\frac{2}{\alpha}}x\Big) =\frac{1}{n!}H_n(x)$$

Note that

$$\lim_{\alpha\to\infty} \sqrt{1-\frac{2x^2}{\alpha}}^{\alpha-1}=e^{-x^2}$$

$$d\sigma(k)=\prod_{j=0}^{k-1}\sqrt{1-x_j^2}^{j-1}\mathrm{d}x_j$$

Above is the area element of a k-sphere, with the integral in $dx_0=\sin\varphi_0\mathrm{d}\varphi_0$ different from the others, since the integration over $\varphi_0$ is in the interval $[0,2\pi]$, while for all other angles it is in the interval $[0,\pi]$.