Logic Proof-Sketches in Rhymed Verse!
This come about after trying to memorize some bits of Dr. O’Dunlaing’s 371 notes for our end of year exam. The notes can currently be found on his website,
but may not always be, and so the following might lose their followability pretty quickly (though his proofs are similar to those in Mendelson’s book on logic).
Diagonalization lemma:
Diagonalization here does state
For any formula numerical,
There is a sentence which one can equate
With its insertion theorem P-Arithmetical.
Firstly diagonalize for all x2,
To imply the initial formulae.
Now, diagonalize this to form a sentence new,
And now the proof must go both way:
Hypothesize, insert, ponens and close,
Retract, ponens, generalize, deduce – post twice suppose.
Gödel’s First Incompleteness Theorem:
In Gödel one, consistency implies
The theorem of big G is not a truth,
And if it’s consistent omega-wise
neither can you there its negation deduce.
If it can be proved there is a proof,
Expressible within the system’s word
Then there exists a proof which, in a puff,
Implies the language to be quite absurd.
In omega for each you cannot prove,
Thus there exists no proof, as we were wont to shew.
Gödel-Rosser Theorem:
G.R. says this can’t be proved either way:
For all the second the proof of first implies,
There is a prior proof of it’s nega-
tion. Take this sentence and diagonalize.
Positively, proof and second part
You can at most substitute to exist.
Insert negative there and depart
To early lemma that shows disproof exist;
If not: write out, make clear, and find a proof of this.