From _Godel's Proof_, Nagel and Newman, ISBN 0-7100-7078-0, p.98:
"[Godel's conclusions]...show that the prospect of finding for every deductive system (and, in particular, for a system in which the whole of arithmetic can be expressed) an absolute proof of consistency that satisfies the finitistic requirements of Hilbert's proposal, though not logically impossible, is most unlikely. They also show that there is an endless number of true arithmetical statements which cannot be formally deduced from any given set of axioms by a closed set of rules of inference."
Consider the following statements:
"Somewhere, over the rainbow, bluebirds sing..."
"The photon simply had to go through one slit or the other..."
"There is a greatest number that can stated in the English language in 19 syllables..."
"There are propositions within arithmetic which are true but cannot be proven true within the rules of arithmetic..."
"There exists a successor to zero..."
Go ahead, spend the day thinking about these statements. There is at least one problem here. "Which of these can be represented by logical symbols within an axiomatic system?" The questions of "Truth" come long after the answer to the axiomatization question and may not "legally" come up at all.
"Godel showed (i) how to construct an arithmetical formula G that represents the meta-mathematical statement: "The formula G is not demonstrable". This formula G says of itself that it is not demonstrable...But (ii) Godel showed that G is demonstrable if, and only if, its formal negation ~ G is demonstrable..."
G'sP, p. 85.
This is all heavy slogging through a blizzard but it "demonstrates" that the proof is of Axiomatic Systems. Whatever else might apply to poetry is problematic but not excluded from thought. Focus first, however, on Godel's Domain in terms of axiomatic systems.
To your last question: When I taught HS Geometry, our school used the UCSMP "system". This text opened up with a coupla' sections on "What is a point?". The kids would go nuts. "This is STOOPIT!" No. It's not. For some, though, everything had to be argumentative - without the "Logic" and "Rhetoric".
It may take time but allow yourself some freedom to accept that, "There might be more than one line through a given point parallel to a given line". But maybe not, as well. "...And what is a point, anyway?"
You've asked the right questions.
Look at Feynman graphs. "Two sideways "Vs" connected by a squiggly line! Simple!" Yes, and very deep.
Hope this helps.
Charles
