my forty years on his shoulders - Department of Mathematics

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18THEOREM 4.1. (Self reference lemma). Let A be a formula of RT. There exists aclosed term t of RT such that T proves t = #(A[x 1 /t]).Proof: Let s = #(A[x 1 /SSUB(x 1 )]). Write s = #B, where B = A[x 1 /SSUB(x 1 )].Note thatB[x 1 /#(B)] = B[x 1 /s] = A[x 1 /SSUB(s)].We now apply condition 9 to B. We have thatSSUB(#(B)) = #(B[x 1 /#(B)])is provable in T. HenceSSUB(s) = #(A[x 1 /SSUB(s)])is provable in T. Thus the closed term SSUB(s) is as required. QEDLEMMA 4.2. (“I am not provable” Lemma). There exists a closed term t such thatT proves t = #(¬PROV[x 1 /t]).Proof: By Theorem 4.1, setting A = ¬PROV. QEDWe fix a closed term t provided by Lemma 4.2.LEMMA 4.3. Suppose T proves ¬PROV[x 1 /t]. Then T is inconsistent.Proof: Assume T proves ¬PROV[x 1 /t]. By condition 12,PROV[x 1 /#(¬PROV[x 1 /t])]is provable in T. By Lemma 4.2, T proves PROV[x 1 /t]. Hence T is inconsistent.QEDLEMMA 4.4. T proves PROV[x 1 /t] → PROV[x 1 /PR(t)]. T proves PROV[x 1 /t] →PROV[x 1 /NEG(PR(t))].Proof: Let A = ¬PROV[x 1 /t]. By condition 11,
19PROV[x 1 /#(A)] → PROV[x 1 /PR(#(A))]is provable in T. By Lemma 4.2,PROV[x 1 /t] → PROV[x 1 /PR(t)]is provable in T. By condition 10, T provesPR(t) = #(PROV[x 1 /t]).By condition 8, T provesNEG(#(PROV[x 1 /t])) = #(¬PROV[x 1 /t]) .By Lemma 4.2, T provesNEG(PR(t)) = t.The second claim follows immediately. QEDWe let CON be the sentence(∀x 1 )(¬(PROV ∧ PROV[x 1 /NEG(x 1 )])).THEOREM 4.5. (Abstract second incompleteness). Let T obey conditions 1-12.Suppose T proves CON. Then T is inconsistent.Proof: Suppose T is as given. By Lemma 4.4, T provesPROV[x 1 /t] → PROV[x 1 /PR(t)] ∧ PROV[x 1 /NEG(PR(t))].Since T proves CON, T proves¬(PROV[x 1 /PR(t)] ∧ PROV[x 1 /NEG(PR(t))]).Hence T proves ¬PROV[x 1 /t]. By Lemma 4.3, T is inconsistent. QEDInformal statements of Gödel's Second Incompleteness Theorem aresimple and dramatic. However, current versions of the Formal SecondIncompleteness are complicated and awkward. Even the abstract form of second
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Page 45 and 46: 45THEOREM 9.5. (Friedman 2005). The
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Page 57 and 58: 57Monographs, 24, Oxford: Clarendon
Page 59 and 60: 59Vol. 41, No. 3, September pp. 209
Page 61 and 62: 61Friedman, H., Robertson, N., and
Page 63 and 64: 63Society, 8:437-479, 1902.Hilbert,
Page 65 and 66: 65Mostowski, A. 1952. Sentences und
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