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:

## Monday, November 02, 2009

### Gödel Without Tears -- 4

Posted by Peter Smith at 1:20 PM

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## 2 comments:

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".

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

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

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

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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 ...".

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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".)

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

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