Let Con(PA) be the sentence of arithmetic which translates as "Peano Arithmetic is consistent." Then according to Godel's 2nd incompleteness theorem, assuming PA is consistent then PA can neither prove Con(PA) nor its negation. And in fact, if T contains PA and T is (omega-) consistent, then T can neither prove Con(T) nor its negation. In particular, if PA+Con(PA) is consistent then PA+Con(PA) can neither prove Con(PA+Con(PA)) nor its negation.(adsbygoogle = window.adsbygoogle || []).push({});

But consider the following piece of reasoning: if PA is consistent, then Con(PA) is true, so PA+Con(PA) is consistent, so Con(PA+Con(PA)) is true. My question is, why can't this reasoning by formalized in PA, so that within PA you can prove Con(PA) implies Con(PA+Con(PA))? If you could prove that, then since you can obviously prove Con(PA) within PA+Con(PA), you would be able to prove Con(PA+Con(PA)) within PA+Con(PA), which is a contradiction. So where am I going wrong?

We can even talk about this in terms of model theory. There are nonstandard models of PA in which Con(PA) does not hold: basically, you have infinitely large natural numbers, and you have infinitely long proofs of a contradiction in PA. This does not mean that PA is inconsistent, because there are no proofs of finite length of this contradiction. So are there also nonstandard models of PA+Con(PA) in which Con(PA+Con(PA)) does not hold?

Any help would be greatly appreciated.

Thank You in Advance.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Question about Godel's 2nd Theorem

Loading...

Similar Threads for Question Godel's Theorem |
---|

B What question should you ask? |

I Question about simplifying Sigma notation |

I Shopping List Game: Probability Question |

I A simple question about probability theory |

B Correlation question |

**Physics Forums | Science Articles, Homework Help, Discussion**