I have been away due to overtime.

Euclid was the gold standard until the early 1800's. At that time geometers found that they could build consistent geometries by starting with contradictory forms of the parallel postulate. This brought into question the platonic conception that mathematics is a description of reality. (There are certainly not contradictory truths in reality.) The development of calculus also brought up some issues. After a century of work, the mathematical community settled on the idea of formal systems. Apparently now Physicist Max Tegmark has formed the "mathematical universe hypothesis" which heads back the other way. I was not aware this prior to this discussion.

With regards to your system, in your mind, you are assigning meanings to the various words. Your proofs rely on picturing things in your mind. Once the mental images are removed, you will find that nothing is obvious or has been proven at all.

An A has at most two Bs. (note I have not restricted my A to being C or D or anything)

Theorem: An A with zero Bs contains at least one E or extends to F.

Theorem: An A with one B has at least one E.

Theorem: An A with two Bs is G.

If your system was a system then anything for which the axioms held, the theorems would also hold. So let A = boy and B = eye. Then your postulate is that "a boy has at most two eyes". None of your theorems follow. You might find specific choices for the remaining letters that are true, but as theorems they do not follow.

 

Do you have a specific question about Godel? I will try to answer, but Godel is more of a graduate school topic.

 

Hello again, Darrough, with all due respect, you are a difficult person to hold a discussion with since you only ever seem to address part of a post.

So if post#81 was addressing my post#, what about the missing part (that of the supporting definitions and common notions)?
And if your post#82 was addressing my post#80, I was interested in your take on the relevence of Godel to axiomatic systems in both mathematics and physics, without prejudgement or challenge, bearing in mind the subject of this thread.

Talking of the subject, I have realised that computer folks have been very busy re-defining words used by other more mature disciplines and I forsee much more mayhem as a result.

I recently saw a computer person answer the question "Whis is div(V) in quite a different fashion that has nothing to do with 3D vectors.

 

"Definitions" and "common notions" both reflect the idea that mathematics is a description of reality. For this reason, they were left behind. Definitions still appear as an axiom with an equivalence while common notions could be either axioms or inference rules.

Godel established some limits on formal systems. One cannot prove the consistency of a system from within the system. Also, no system is complete. Actually, if anything, it was Godel that dealt the final blow to mathematical platonism of Euclid.