u/Inevitable_Garage706

And is the contract necessary to reach the conclusion that 0.999...<1?

To clarify, an axiom is a foundational assumption that is built off of via logical deductions.

As an example, we can set the following axioms to be true:

Axiom 1: a>0

Axiom 2: b>0

From these axioms, we can use a series of logical deductions to conclude that a+b>0. Here's how:

Step 1: a>0 (by Axiom 1)

Step 2: a+b>b (adding b to both sides preserves the inequality)

Step 3: a+b>b>0 (by Axiom 2)

Step 3: a+b>0 (by the Transitive Property of Inequality)

This is how axioms work.

Now what axioms do you set to be true in your system, and what logical deductions are made to conclude that 0.999...<1?

It's okay if you don't have all of your axioms ready to go. All I'm asking for is axioms and logical deductions that are sufficient to prove that 0.999...<1 within your system. We can worry about additional axioms later.

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