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)

2 comments:

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.