TREE Function: TREE(5) and TREE(10)

This page delves into the mind-boggling realm of TREE(5) and TREE(10), values of the TREE function that are so enormous they defy comprehension and push the boundaries of mathematical notation.

Important Note: The exact values of TREE(5) and TREE(10) are unknown and cannot be precisely calculated or represented using conventional mathematical notation. The descriptions here are attempts to convey the scale and significance of these numbers.

TREE(5)

TREE(5) represents the maximum length of a sequence of trees where:

Each tree has at most 6 vertices
Vertices are labeled with positive integers not exceeding 5
No tree in the sequence can be homeomorphically embedded into a later tree

TREE(5) is unfathomably larger than TREE(4), which is already beyond human comprehension. To give a sense of scale:

TREE(10)

TREE(10) follows the same principle but allows for trees with up to 11 vertices and labels up to 10. The magnitude of TREE(10) is so vast that it dwarfs TREE(5) to an extent that defies analogy.

Consider the following:

If TREE(5) were represented as a tower of exponents reaching to the moon, TREE(10) would be incomparably larger
The difference between TREE(10) and TREE(5) is greater than the difference between TREE(5) and 1

Comparison and Scale

Function	Relative Scale	Comprehensibility
TREE(3)	"Small" in comparison	Already beyond human intuition
TREE(4)	Unimaginably larger than TREE(3)	Exceeds known mathematical notation
TREE(5)	Incomparably larger than TREE(4)	Beyond any conceivable notation
TREE(10)	Vastly, vastly larger than TREE(5)	Transcends mathematical philosophy

Mathematical Significance

The study of such large numbers as TREE(5) and TREE(10) has profound implications for:

Set theory and the nature of infinity
Proof theory and the limits of formal systems
Computational complexity and the boundaries of computability
Philosophy of mathematics and the concept of "unimaginably large" numbers

These numbers challenge our understanding of mathematical reality and the limits of human cognition in grappling with extreme scales.