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You make a specific point that the last bead out is the axiom, I have asked you why you have chosen this specific point.
If you had said the first bead in then you would have a point to make.
The axiom is the starting point, which the other problems require knowledge of to be solved. An axiom by it's very definition is the root not the branch of the tree.
Axiom comes first, first the axiom is proved then you can move on to other problems.
Okay, I understand what're you're asking. I will show you what I did by example.
Let's say we have the proposition P which is some popsicles are red.
This implies three things that come before it.
S: There is a quantity called some.
R: There is a color called red.
P: There is an object called popsicle.
To go back even further.
A: Quantity is an abstract concept that represents the amount of something in a group.
E: Red represents the frequency on the electromagnetic spectrum.
O: Objects are possible to exist and do exist.
What I am saying is a proposition n, assumption or statement may have a previous set of propositions(n-1) that lead up to proposition n. Why can't one work backwards or reverse engineer from a proposition , assumption, theorm, or statement to get to the set of Axiom(s).
Furthermore why are the jars inside each other, are you implying that the string is interconnected to the beads between jars?
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The relationship between axioms is not necessarily a linear relationship.
Yes, you're right it isn't linear. I was keeping it very simple and I didn't want to bog us down with so much complexity. One can have more than one axiom for a given proposition. I kept it concise as much as possible. In addition, I had other things pressing so I didn't have time to go into the more complexities of this. I have an 16 year old ODD brother in law who I have to deal with. In addition, I'm trying to start a business.
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It's obvious the beads and jars are symbols, very pretty ones at that, but used in this way they are not very accurate either.
It's because in the figurate language example I did proceed from the axiom I worked backwards and proceeded to the axiom.
The best example I can give for what I did is watch a movie but backwards.