A discussion with Chat GPT-4 on Rayo’s Number

(Q1) I don’t think Rayo’s Number is well-defined. What do you think?

Rayo’s number is an example of a very large number defined in a highly abstract and complex way. It was created by Agustín Rayo in 2007 as part of a “big number duel” against Adam Elga. The formal definition of Rayo’s number involves set theory and first-order logic, and it is intentionally designed to surpass any previously defined specific large number.

The definition of Rayo’s number is as follows:

“Rayo’s number is the smallest number bigger than any finite number named by an expression in the language of first-order set theory with less than a googol ((10^{100})) symbols.”

The key aspects to consider regarding its well-definedness are:

1. Language of First-Order Set Theory: This language includes variables, quantifiers, logical connectives, and the binary relation of set membership. It is a well-defined formal language used in mathematical logic.
2. Descriptive Power: The definition essentially says that Rayo’s number is larger than any number that can be uniquely described using a finite expression in this language, with the expression being less than a googol symbols in length.
3. Ambiguity and Subjectivity: While the language of first-order set theory is well-defined, the concept of “naming a number” using such a language can be seen as subjective and potentially ambiguous. This is because what constitutes a description or naming of a number can vary and may depend on the intricacies of set theory and logic.
4. Comparison with Other Large Numbers: Unlike more conventional large numbers, even those in the realm of large cardinal numbers or those used in combinatorics or number theory, Rayo’s number is not defined by a clear mathematical process or function. Instead, it relies on the expressive power of a particular language and set theory, making it more abstract and less tangible.
5. Practical Computability: Rayo’s number, by its very definition, is not computable. It is not just extraordinarily large; it is so large that no physical system in the known universe could describe or utilize it in a computational sense.

In conclusion, whether Rayo’s number is “well-defined” can depend on your perspective. From a formal logic and set theory viewpoint, it has a definition that makes sense within the context of these fields. However, from a practical or computational perspective, it may seem less well-defined or even irrelevant due to its extreme abstraction and the impossibility of computation or application.

(Q2) But what about the unadressed question: Is there an intended model of set theory?

The question of whether there is an “intended model” of set theory is a deep and philosophically charged one in the realm of mathematical logic and the philosophy of mathematics. It touches upon how mathematicians and philosophers interpret the nature of mathematical objects and the foundations of mathematics.

1. Models of Set Theory: In mathematical logic, a model of a set of axioms is a structure in which all the axioms are true. For set theory, particularly Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), models are used to understand and explore the implications of the axioms.
2. The Concept of an “Intended Model”: An “intended model” would be a model that perfectly captures the essence or the true nature of the sets as conceived by the creators of the theory. However, in the context of set theory, the notion of an “intended model” is ambiguous and controversial. This is because set theory attempts to formalize the intuitive notion of a “set,” but there are various interpretations of what sets “truly are.”
3. Platonism vs. Formalism: The existence of an “intended model” of set theory often depends on one’s philosophical stance:

• Platonism in mathematics suggests that there is an abstract, ideal realm where mathematical objects (like sets) exist, and that our mathematical theories attempt to describe this realm. Under this view, one might argue for the existence of an intended model of set theory.
• Formalist views, on the other hand, treat mathematical statements and theories as purely symbolic, devoid of any intrinsic meaning, focusing on the manipulation of symbols according to specified rules. From this perspective, the concept of an “intended model” is less meaningful.

1. Independence Results: In set theory, there are many propositions whose truth or falsity cannot be determined from the ZFC axioms (e.g., the Continuum Hypothesis). This suggests that there are multiple models of ZFC in which different truths hold. These results complicate the notion of an “intended model,” as different models can satisfy the same axioms yet differ in significant ways.
2. The Role of Intuition: The development of set theory has been guided by both formal axiomatic methods and mathematical intuition. Some mathematicians might argue that this intuition points towards an intended model, but such arguments are more philosophical than mathematical.

In conclusion, the question of whether there is an intended model of set theory is more of a philosophical question than a mathematical one. It depends on one’s views about the nature of mathematical objects and the goals of mathematical practice. In the strict mathematical sense, set theory (like ZFC) is understood through its axioms, and models are used to explore the consequences of these axioms, without necessarily claiming to be the definitive representation of some platonic ideal of set theory.

(Q3) Could Rayo’s Number vary, based on the model of set theory?

Rayo’s number, by its definition, is intricately linked to the formal language of first-order set theory. Since it is the smallest number greater than any finite number definable in first-order set theory with less than a googol symbols, its value is, in principle, tied to the expressive power of that language. However, the question of whether Rayo’s number could vary based on the model of set theory used is an interesting one and delves into deeper aspects of mathematical logic and set theory.

1. Dependence on the Language and Axioms: Rayo’s number is defined in terms of the language of first-order set theory and implicitly relies on the axioms that govern this theory (such as Zermelo-Fraenkel set theory with the Axiom of Choice, ZFC). The expressive power of this language, and hence the specific numbers that can be defined within a googol symbols, can depend on the axioms assumed.
2. Independence Phenomenon in Set Theory: In set theory, especially in ZFC, there are many statements that are independent of the axioms. This means that these statements can be true in some models of ZFC and false in others. If the definition of Rayo’s number involves concepts or constructions that are independent of ZFC, then its value could, in theory, vary between different models of ZFC.
3. The Role of Specific Models: The specifics of what can be defined within the constraints of Rayo’s definition (a googol symbols in the language of set theory) could potentially differ in different models, especially if these models have different properties regarding very large sets or the structure of the set-theoretic universe.
4. Philosophical Considerations: From a philosophical standpoint, if one believes that there is a single “true” model of set theory (a contentious and philosophical stance), then Rayo’s number would have a specific value in that model. However, if one adopts a more pluralistic or formalistic view of mathematical models, then the concept of Rayo’s number could be seen as varying across different models.

In practical terms, Rayo’s number is so large and abstract that these considerations are more of theoretical interest than practical consequence. The number is beyond the realm of physical realizability or direct mathematical application. However, these discussions do highlight important themes in the philosophy of mathematics and the nature of mathematical models and definitions.