Monday, November 02, 2009

Gödel Without Tears -- 4

Here now is the fourth episode [slightly corrected] which tells you -- for those who don't know -- what first-order Peano Arithmetic is (and also what Sigma_1/Pi_1 wffs are). A thrill a minute, really. Done in a bit of a rush to get it out to students in time, so apologies if the proof-reading is bad!

Here are the previous episodes:

  1. Episode 1, Incompleteness -- the very idea (version of Oct. 16)
  2. Episode 2. Incompleteness and undecidability (version of Oct. 26)
  3. Episode 3. Two weak arithmetics (version of Nov. 1)


a.c. said...

Sentence near the top of p 2:

Q, then, is a very weak arithmetic. Still, it will turn out to be ‘modest amount of arithmetic’ needed to get Theorem 2 to fly, and also containing Q gives us a ‘sufficiently strong arithmetic’ in the sense of Theorem 6.

Should the "it will turn out to be" be "it will turn out that the"?

Also should be a comma after "and also containing Q".

- - - - -

Minor detail at the end of section 12 on p 6:

"The answer will emerge over shortly enough" reads oddly to me.

- - - - -

Last sentence before 13.1: missing space between "and" and PI_1.

- - - - -

2nd sentence of 1st para of 13.1:

"We can now express such claims in formal arithmetics like Q and PA wffs of the shape ..."

Should be "... using wffs of the shape ...".

- - - - -

Proof of Theorem 17 on p 8:

Two (separate) "then"s in the 2nd sentence when only one ought to be needed.

The proof concludes:

Contraposing, if T is consistent, it proves any Π1 sentence it proves is true.

Is that what was intended? Or should it be only that if T is consistent, any Π1 sentence it proves is true. (Taking out the "it proves" before "any".)

Peter Smith said...

Many, many thanks a.c. :-) Corrected.