Large Cardinals and Their Impact on Number Theory and Theorems

Number theory, a branch of mathematics that deals with the properties and relationships of numbers, has been vastly influenced by the concept of large cardinals. These abstract objects play a significant role in the consistency of foundational mathematical theories such as Zermelo-Fraenkel set theory with the axiom of choice (ZFC). In this article, we explore the connection between large cardinals and number-theoretic statements, emphasizing the work of notable mathematicians such as Harvey Friedman.

Introduction to Large Cardinals

Large cardinals are extensions of the usual notion of cardinal numbers in set theory. They are essentially cardinal numbers that are so large that their existence cannot be proven within standard set theories like ZFC. Instead, their existence is taken as an assumption to develop stronger and more powerful mathematical theories. By assuming the existence of large cardinals, mathematicians can prove the consistency of certain mathematical theories, such as ZFC.

The Role of Large Cardinals in Number Theory

One of the fascinating aspects of large cardinals is their ability to influence number-theoretic statements. For instance, the consistency of ZFC, denoted by Con(ZFC), can be expressed as a number-theoretic statement. Although this statement is very large and not particularly interesting in its form, it highlights the profound connection between set theory and number theory.

Harvey Friedman, a renowned mathematician, has conducted extensive research into finding combinatorial statements that cannot be proven within ZFC alone. These statements, often referred to as Friedman's Finite Incompleteness Theorem, are order-theoretic in nature but not explicitly number-theoretic. One of the notable examples is his work on order-invariant graphs and the finite incompleteness theorem, which demonstrates the limitations of ZFC in proving certain statements.

Friedman's Example: Order Invariant Graphs and Finite Incompleteness

One of the key insights from Friedman's research is the example of order-invariant graphs. These graphs involve an order structure and are not straightforwardly number-theoretic. An example of such a statement is the one found in the paper "ORDER INVARIANT GRAPHS AND FINITE INCOMPLETENESS," which can only be proven using a large cardinal hypothesis.

This statement is checkable by a Turing machine, making it both computationally and conceptually interesting. Friedman's work has further led to the development of Turing machine-independent theorems, which are not restricted by the specific hardware used to prove them. This highlights the importance of abstract mathematical structures in understanding the limits of formal systems.

Specific Examples and Proofs

Consider the Proposition 2.4 mentioned in a paper by Friedman, which states:

For all R contained in N^2k, some R intersect A^2 maps some infinite subcube of A to some cube that is mapped to some subcube of A-1 delta R[A].

This proposition is provable in a large cardinal theory known as SMAH (Subtle Mahlo Axiom). The theorem is further enriching by showing that it is equiconsistent with the consistency of SMAH, meaning that if SMAH is consistent, then so is the proposition.

Moreover, the proposition is provably equivalent over ACA (Arithmetical Comprehension Axiom) to the statement Con(SMAH). This equivalence highlights the deep connection between number theory, foundational set theory, and large cardinal hypotheses.

Concrete Functions and Factorials

Many of these large cardinal implications can be expressed in a more concrete form using specific functions and factorials. For instance, explicit Pi01 forms are obtained through the use of concrete functions and factorials. Any specific subset of natural numbers that grows at least exponentially can be used in these constructions, further demonstrating the flexibility of these theorems.

Concluding Remarks

While not all statements about tuples of natural numbers are number-theoretic, many theorems in number theory can be expressed in the language of second-order arithmetic. This connection underscores the foundational importance of set theory and large cardinals in understanding the consistency and completeness of mathematical theories.

Friedman's work and the use of large cardinals in number theory not only expand the boundaries of mathematical knowledge but also challenge our understanding of the limits of formal systems. These findings are not only of theoretical interest but also contribute to the broader landscape of mathematical logic and foundational research.

Keywords: Large cardinals, number theory, consistency, Friedman