Evaluating the Infinite (2509.19389v1)

Abstract: I present a novel mathematical technique for dealing with the infinities arising from divergent sums and integrals. It assigns them fine-grained infinite values from the set of hyperreal numbers in a manner that refines the standard theories of summation and integration. This has implications in statistics (helping us work with distributions whose mean or variance is infinite), decision theory (allowing comparison of options with infinite expected values), economics (allowing evaluation of infinitely long streams of utility without discounting), and ethics (allowing evaluation of infinite worlds). There are even implications for finite cases, as the ability to handle these infinities undermines a common argument for bounded utility and the discounting of future utility.

• The paper introduces a hyperreal framework that refines divergent cases by assigning distinct infinite values using ultrafilters.
• It generalizes summation and integration, preserving linearity and the Fundamental Theorem of Calculus while addressing infinite expectations and utility streams.
• The approach highlights challenges such as ultrafilter dependence and grid limitations, paving the way for further research in infinite analysis.

Hyperreal Summation and Integration: Fine-Grained Evaluation of Divergent Sums and Integrals

Introduction

This paper introduces a novel mathematical framework for evaluating divergent sums and integrals by assigning them fine-grained infinite values from the set of hyperreal numbers. The approach refines standard summation and integration theories, which typically collapse all divergent cases to a single undifferentiated infinity, and instead provides a rich structure for distinguishing between different infinite quantities. The implications are significant for fields such as statistics, decision theory, economics, and ethics, where infinite expectations, utility streams, and populations pose persistent technical and philosophical challenges.

Construction of Hyperreal Numbers

The hyperreal numbers, denoted $^*\mathbb{R}$, extend the real numbers to include both infinitesimal and infinite elements. The construction is based on equivalence classes of sequences of real numbers, with the identification of sequences determined by a free ultrafilter on $\mathbb{N}$. This ultrafilter selects "large" sets of indices, allowing the definition of hyperreals such as $\omega = [(1,2,3,\ldots)]$ and infinitesimals like $[(1,1/2,1/3,\ldots)]$. The transfer principle ensures that all first-order properties of the reals are inherited by the hyperreals, providing a robust foundation for extending arithmetic and analysis.

Hyperreal Summation and Integration

The central innovation is the generalization of finite sums and definite integrals to hyperreal bounds. For a function $f$, the hyperreal sum and integral are defined as:

• Hyperreal sum: $$\sum_{i=a}^{b} f(i) = [( \sum_{i=a_1}^{b_1} f(i), \sum_{i=a_2}^{b_2} f(i), \ldots )]$$
• Hyperreal integral: $$\int_{a}^{b} f(x) dx = [( \int_{a_1}^{b_1} f(x) dx, \int_{a_2}^{b_2} f(x) dx, \ldots )]$$

Infinite sums and improper integrals are evaluated by taking the bound to be a hyperreal infinite, such as $\omega$. This method assigns distinct infinite values to divergent series, e.g., $\sum_{i=1}^{\omega} 1 = \omega$, $\sum_{i=1}^{\omega} 2 = 2\omega$, and $\sum_{i=1}^{\omega} i = \omega(\omega+1)/2$. The approach preserves linearity, additivity, and the Fundamental Theorem of Calculus, even in the infinite case.

Ultrafilter Dependence and Indeterminacy

A key technical issue is the dependence of some hyperreal sums and integrals on the choice of ultrafilter. For oscillatory or poorly behaved series, the value may be indeterminate, varying across ultrafilters. The paper advocates a middle path: classifying statements as determinately true, false, or indeterminate depending on ultrafilter invariance. In some cases, averaging across ultrafilters yields values consistent with classical summation methods (e.g., Abel, Euler, Borel summation).

Numerosity and Infinite Set Size

Hyperreal summation provides a numerosity measure for infinite sets, capturing intuitive differences in size that cardinality ignores. For example, the numerosity of the positive integers is $\omega$, while the set of even numbers has numerosity $\omega/2$. This approach reflects the internal structure and density of sets, offering a complementary perspective to Cantorian cardinality.

Applications in Probability, Decision Theory, and Economics

Infinite Expectations

The framework enables the assignment of precise infinite expectations to probability distributions with divergent means, such as the St Petersburg gamble. The expected value is no longer a monolithic $+\infty$ but a specific hyperreal, e.g., $[\log_2(\omega)]$, which is a "lesser infinity"—infinite but smaller than $\omega$. This allows meaningful comparison between prospects with infinite expected values, resolving violations of dominance and continuity axioms in standard expected utility theory.

Infinite Utility Streams

In economics, infinite utility streams (e.g., $(1,1,1,\ldots)$ vs. $(2,2,2,\ldots)$) are distinguished by their hyperreal sums ($\omega$ vs. $2\omega$), avoiding the need for discounting or bounded utility. The method satisfies strong Pareto and finite anonymity, and can handle exponentially growing streams and sparse sequences, assigning them appropriate lesser infinite values.

Infinite Populations in Ethics

For ethical evaluation of infinite worlds, the hyperreal integral over spatial utility density yields meaningful infinite values, such as $8p\omega^3$ for constant density. This enables comparison of infinite populations and distributions, though the approach requires a privileged origin and further development for physically realistic universes.

Limitations and Open Problems

The paper identifies several challenges:

• Ultrafilter dependence: Some sums/integrals are indeterminate or arbitrary.
• Grid problem: Only countable sequences of points are considered, potentially missing divergent behavior between them.
• Choice of $\omega$: No canonical reason for selecting a particular infinite hyperreal as the unit.
• Spillover and closure: Summing/integrating over hyperreal domains can include unintended contributions; the system lacks closure under further summation/integration of non-real outputs.
• Fanaticism and sure-thing principle: The approach is "fanatical" in that infinitesimal probabilities of infinite outcomes dominate all finite options, and it violates certain versions of the sure-thing principle.

Connections to Physics and Infinitesimal Probability

The method has potential applications in quantum field theory, where divergent sums and integrals arise. Hyperreal summation may provide a rigorous foundation for regularization and renormalization techniques. Additionally, the framework supports the development of infinitesimal probability theory, assigning nonzero infinitesimal probabilities to events in infinite processes.

Implications and Future Directions

The hyperreal approach offers a mathematically elegant and conceptually intuitive refinement of infinite summation and integration, with broad applicability across mathematics, statistics, decision theory, economics, and ethics. It challenges the use of extended reals and cardinal numbers for evaluating infinite options, advocating for a fine-grained, Leibnizian conception of infinite value. Future work includes characterizing ultrafilter-independent functions, developing averaging methods, and exploring alternative infinite number systems such as the surreals.

Conclusion

This paper presents a compelling case for the use of hyperreal numbers in evaluating divergent sums and integrals, providing a fine-grained structure for infinite values that preserves desirable mathematical and philosophical properties. The approach resolves longstanding technical problems in multiple disciplines and opens new avenues for rigorous analysis of infinite expectations, utility streams, and populations. While challenges remain, the framework represents a significant advance in the mathematical treatment of infinity, with promising implications for both theory and practice.