[FOM] 777: Goedel's Second: Proofs/4

Harvey Friedman hmflogic at gmail.com
Thu Dec 28 11:52:45 EST 2017

```RADICALLY (?) SIMPLIFIED(?) PROOF OF GOEDEL'S SECOND INCOMPLETENESS THEOREM

This is a sequel to https://cs.nyu.edu/pipermail/fom/2017-December/020729.html

1. {e: T proves (e not in W_e)}
2. r.e. agreement theorems
3. proving GSIT
4. For recursion theorists

I feel more comfortable with GSIT as the abbreviation for Goedel's
Second Incompleteness Theorem.

1. {e: T proves (e not in W_e)}

Here we simply copy the proof given in
https://cs.nyu.edu/pipermail/fom/2017-December/020729.html

GSIT. Let T be a consistent r.e., axiomatized theory containing a
weak fragment K of arithmetic. Then T does not prove Con(T).

Proof: Let n* be natural so that W_n* = {e: T proves (e not in W_e)}. Then

1) n* in W_n* if and only if T proves (n* not in W_n*).
2) n* in W_n* implies T proves (n* in W_n*).
3) n* not in W_n* and T does not prove (n* not in W_n*).

Note that 3) follows from 1),2), and that T is consistent. The above
argument has been carried out external to T.

Note also that the above argument can be carried out in T + Con(T). In
particular, the first conjunct of 3) gets modified to: T + Con(T)
proves n* not in W_n*. So if T proves Con(T) then T proves n* not in
W_n*, contradicting the unmodified second conjunct of 3). Therefore T
does not prove Con(T). QED

2. r.e. agreement theorems

Two sets A,B agree at x if and only if (x in A iff x in B).

THEOREM 2.1. There is an r.e. set which agrees somewhere with every r.e. set.

Proof: {e: e in W_e} agrees with any W_n at n, because

n in W_n if and only if n in {e: e in W_e}.

QED

We don't need the following Theorem 2.3 for GSIT.

LEMMA 2.2. Suppose A agrees somewhere with every r.e. set. Let f:N
into N be a recursive bijection. Then f[A] agrees somewhere with every
r.e. set. I.e., the property "agreeing somewhere with every r.e. set"
is preserved under recursive isomorphism.

Proof: Let A,f be as given. Let B be an r.e. set. Then A and f^-1[B]
agree somewhere, say at n. Hence f[A] and B agree at f(n). QED

THEOREM 2.3. Every complete r.e. set agrees somewhere with every r.e. set.

Proof: Let A be complete r.e. By classical recursion theory, all
complete r.e. sets are recursively isomorphic, and {e: e in W_e} is
complete r.e.. Hence A and {e: e in W_e} are recursively isomorphic.By
Theorem 2.1, {e: e in W_e} has "somewhere agreement with every r.e.
set", and so by Lemma 2.2, A has "somewhere agreement with every r.e.
set". QED

We do need the following very explicit form of Theorem 2.1 for GSIT.
This is obvious from the proof of Theorem 2.1.

THEOREM 2.4. There is an r.e. set A and a very weak system W of
arithmetic such that the following holds. For any r.e. set B, there
exists n such that W proves A,B agree at n.

NOTE: We can contemplate Theorem 2.4 nicely without having to look at
the blatantly "technical" construction {e: T proves (e not in W_e)}.

3. PROVING GSIT

GSIT. Let T be a consistent r.e., axiomatized theory containing a
weak fragment K of arithmetic. Then T does not prove Con(T).

Proof: By Theorem 2.4, let A be an r.e. set such that the following
holds. For any r.e. set B, there exists n such that K proves A,B agree
at n.

Let B be the r.e. set {e: T proves (e not in A)}. Fix n* such that

K proves (n* in A if and only if T proves (n not in A)).

Then

1) n* in A if and only if T proves (n* not in A).
2) n* in A implies T proves (n* in A).
3) n* not in A and T does not prove (n* not in A).

Note that 3) follows from 1),2), and that T is consistent. The above
argument has been carried out external to T.

Note also that the above argument can be carried out in T + Con(T). In
particular, the first conjunct of 3) gets modified to: T + Con(T)
proves n* not in A. So if T proves Con(T) then T proves n* not in
A, contradicting the unmodified second conjunct of 3). Therefore T
does not prove Con(T). QED

4. FOR RECURSION THEORISTS

Analyze the property of an r.e. set A that: A agrees somewhere with
every r.e. set.

Also consider various effective forms of the property "A agrees
somewhere with every r.e. set".

************************************************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 777th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-699 can be found at

700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
700: Large Cardinals and Continuations/14  8/1/16  11:01AM
701: Extending Functions/1  8/10/16  10:02AM
702: Large Cardinals and Continuations/15  8/22/16  9:22PM
703: Large Cardinals and Continuations/16  8/26/16  12:03AM
704: Large Cardinals and Continuations/17  8/31/16  12:55AM
705: Large Cardinals and Continuations/18  8/31/16  11:47PM
706: Second Incompleteness/1  7/5/16  2:03AM
707: Second Incompleteness/2  9/8/16  3:37PM
708: Second Incompleteness/3  9/11/16  10:33PM
709: Large Cardinals and Continuations/19  9/13/16 4:17AM
710: Large Cardinals and Continuations/20  9/14/16  1:27AM
711: Large Cardinals and Continuations/21  9/18/16 10:42AM
712: PA Incompleteness/1  9/23/16  1:20AM
713: Foundations of Geometry/1  9/24/16  2:09PM
714: Foundations of Geometry/2  9/25/16  10:26PM
715: Foundations of Geometry/3  9/27/16  1:08AM
716: Foundations of Geometry/4  9/27/16  10:25PM
717: Foundations of Geometry/5  9/30/16  12:16AM
718: Foundations of Geometry/6  101/16  12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7  10/2/16  1:59PM
721: Large Cardinals and Emulations//23  10/4/16  2:35AM
722: Large Cardinals and Emulations/24  10/616  1:59AM
723: Philosophical Geometry/8  10/816  1:47AM
724: Philosophical Geometry/9  10/10/16  9:36AM
725: Philosophical Geometry/10  10/14/16  10:16PM
726: Philosophical Geometry/11  Oct 17 16:04:26 EDT 2016
727: Large Cardinals and Emulations/25  10/20/16  1:37PM
728: Philosophical Geometry/12  10/24/16  3:35PM
729: Consistency of Mathematics/1  10/25/16  1:25PM
730: Consistency of Mathematics/2  11/17/16  9:50PM
731: Large Cardinals and Emulations/26  11/21/16  5:40PM
732: Large Cardinals and Emulations/27  11/28/16  1:31AM
733: Large Cardinals and Emulations/28  12/6/16  1AM
734: Large Cardinals and Emulations/29  12/8/16  2:53PM
735: Philosophical Geometry/13  12/19/16  4:24PM
736: Philosophical Geometry/14  12/20/16  12:43PM
737: Philosophical Geometry/15  12/22/16  3:24PM
738: Philosophical Geometry/16  12/27/16  6:54PM
739: Philosophical Geometry/17  1/2/17  11:50PM
740: Philosophy of Incompleteness/2  1/7/16  8:33AM
741: Philosophy of Incompleteness/3  1/7/16  1:18PM
742: Philosophy of Incompleteness/4  1/8/16 3:45AM
743: Philosophy of Incompleteness/5  1/9/16  2:32PM
744: Philosophy of Incompleteness/6  1/10/16  1/10/16  12:15AM
745: Philosophy of Incompleteness/7  1/11/16  12:40AM
746: Philosophy of Incompleteness/8  1/12/17  3:54PM
747: PA Incompleteness/2  2/3/17 12:07PM
748: Large Cardinals and Emulations/30  2/15/17  2:19AM
749: Large Cardinals and Emulations/31  2/15/17  2:19AM
750: Large Cardinals and Emulations/32  2/15/17  2:20AM
751: Large Cardinals and Emulations/33  2/17/17 12:52AM
752: Emulation Theory for Pure Math/1  3/14/17  12:57AM
753: Emulation Theory for Math Logic  3/10/17  2:17AM
754: Large Cardinals and Emulations/34  3/12/17  12:34AM
755: Large Cardinals and Emulations/35  3/12/17  12:33AM
756: Large Cardinals and Emulations/36  3/24/17  8:03AM
757: Large Cardinals and Emulations/37  3/27/17  2:39AM
758: Large Cardinals and Emulations/38  4/10/17  1:11AM
759: Large Cardinals and Emulations/39  4/10/17  1:11AM
760: Large Cardinals and Emulations/40  4/13/17  11:53PM
761: Large Cardinals and Emulations/41  4/15/17  4:54PM
762: Baby Emulation Theory/Expositional  4/17/17  1:23AM
763: Large Cardinals and Emulations/42  5/817  2:18AM
764: Large Cardinals and Emulations/43  5/11/17  12:26AM
765: Large Cardinals and Emulations/44  5/14/17  6:03PM
766: Large Cardinals and Emulations/45  7/2/17  1:22PM
767: Impossible Counting 1  9/2/17  8:28AM
768: Theory Completions  9/4/17  9:13PM
769: Complexity of Integers 1  9/7/17  12:30AM
770: Algorithmic Unsolvability 1  10/13/17  1:55PM
771: Algorithmic Unsolvability 2  10/15/17  10:14PM
772: Algorithmic Unsolvability 3  10/19/17  2:41AM
773: Goedel's Second: Proofs/1  12/18/17  8:31PM
774: Goedel's Second: Proofs/2  12/18/17  8:36PM
775: Goedel's Second: Proofs/3  12/19/17  12:48AM
776: Logically Natural Examples 1  12/21/17  8:45AM

Harvey Friedman
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