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 thoughts on “Gödel Without Tears — 4”

  1. 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".)

Leave a Comment

Your email address will not be published.

Scroll to Top