Conjecture of Interconnectedness

Formal Statement: In any mathematical system ( S ), for every pair of theorems ( T_a ) and ( T_b ), there exists a finite sequence of theorems ( T₁, T₂, ..., T_n ) within ( S ) such that each theorem in this sequence forms a logical connection from ( T_a ) to ( T_b ).

Mathematical Framework:

1. A 'logical connection' is defined as a relationship where theorem ( T_{i+1} ) can be logically inferred from or is directly related to theorem ( T_i ).
2. The sequence ( T₁, T₂, ..., T_n ) represents a pathway in the network of theorems, where each step is a logical advancement from the previous one.
3. The 'finiteness' of this sequence ensures that the pathway connecting any two theorems is not infinitely long, making the connection attainable within a finite number of steps.

Discussion: The Conjecture of Interconnectedness posits that the mathematical universe is a tightly-knit fabric of theorems and concepts, where every theorem is interlinked with others through a series of logical steps or connections. This conjecture mirrors the 'small world' phenomenon in network theory, where nodes (theorems) are much more interconnected than they may initially appear.

This conjecture implies that even highly abstract theorems in areas like set theory can be connected through a series of logical steps to more concrete theorems in applied mathematics, and vice versa. It highlights the fundamental unity and coherence in mathematical knowledge, suggesting that all mathematical areas, despite their differences, are interconnected parts of a larger, cohesive structure..

In relation to Godels incompleteness theorems:

A key point to consider is the nature of the mathematical theories we are dealing with. This conjecture seems to inherently assume we are working within a maximally consistent and coherent theory. However, Gödel's Incompleteness Theorems suggest that in such a system, we inevitably encounter limitations, specifically in the form of incompleteness.

In light of this, it might be more prudent to think of the network of theorems and their connections as finite, rather than infinite. This perspective aligns well with the practical reality of our mathematical work. At any given time, the theories we operate within are finite, much like how we handle the number pi. When we work with pi in computations and theoretical explorations, we use a finite representation of it, despite its infinite nature.

Similarly, while the theoretical network of theorems in mathematics could be infinitely expansive, any practical exploration or mapping of this network will be finite. This doesn't diminish the value or the insight the Conjecture of Interconnectedness offers. Instead, it provides a more grounded and realistic framework for understanding the interconnected nature of mathematical theories.

Just as we accept pi in its finite form for practical purposes, acknowledging the finite nature of our exploration and understanding of the network of mathematical theorems seems both logical and in harmony with the nature of mathematical exploration and discovery. It's an acknowledgment of the vast, potentially infinite complexity of mathematics, while also recognizing the limits of our current understanding and the finite nature of our theoretical frameworks.