# Easton's theorem for the tree property below aleph_omega

## TL;DR

This paper proves that starting from many supercompact cardinals, the tree property can be made consistent at all leph_n for n<, with an arbitrary continuum function below leph_, showing the tree property does not restrict the continuum function.

## Contribution

It demonstrates the consistency of the tree property at all leph_n for n< with arbitrary continuum functions below leph_, extending Easton's theorem.

## Key findings

- Tree property holds at all leph_n for n< under certain conditions.
- The continuum function can be arbitrarily specified below leph_, with the only restriction being at leph_{n+1}.
- The result relies on starting with infinitely many supercompact cardinals.

## Abstract

Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal $\aleph_n$, $1 < n <\omega$, is consistent with an arbitrary continuum function below $\aleph_\omega$ which satisfies $2^{\aleph_n} > \aleph_{n+1}$, $n<\omega$. Thus the tree property has no provable effect on the continuum function below $\aleph_\omega$ except for the restriction that the tree property at $\kappa^{++}$ implies $2^\kappa>\kappa^+$ for every infinite $\kappa$.