Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Variations on a Theme by Friedman

Ali Enayat, Göteborgs Universitet

September 5, 2013

Honorary Doctorate Harvey Friedman, Universiteit Ghent

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Friedman’s Theme

• Friedman. Every countable nonstandard model of ZF orPA is isomorphic to a proper initial segment of itself..

•

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Friedman’s Theme

• Friedman. Every countable nonstandard model of ZF orPA is isomorphic to a proper initial segment of itself..

•

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Friedman’s Theme

• Friedman. Every countable nonstandard model of ZF orPA is isomorphic to a proper initial segment of itself..

•

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Popular TV meets Logic

•

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Popular TV meets Logic

•

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Jim Schmerl’s Account

• Harvey was on the Flip Wilson show. It must have been in1971 (perhaps plus/minus 1) since I was at Yale at thetime and Joram Hirschfeld was just finishing his thesisthen.

• He heard Harvey talk about embedding models of PA asinitial segments and that gave him an idea that ended upin his thesis.

• Hirschfeld showed that every countable model of PA canbe embedded in a nontrivial homomorphic image of thesemiring R of recursive functions.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Jim Schmerl’s Account

• Harvey was on the Flip Wilson show. It must have been in1971 (perhaps plus/minus 1) since I was at Yale at thetime and Joram Hirschfeld was just finishing his thesisthen.

• He heard Harvey talk about embedding models of PA asinitial segments and that gave him an idea that ended upin his thesis.

• Hirschfeld showed that every countable model of PA canbe embedded in a nontrivial homomorphic image of thesemiring R of recursive functions.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Jim Schmerl’s Account

• Harvey was on the Flip Wilson show. It must have been in1971 (perhaps plus/minus 1) since I was at Yale at thetime and Joram Hirschfeld was just finishing his thesisthen.

• He heard Harvey talk about embedding models of PA asinitial segments and that gave him an idea that ended upin his thesis.

• Hirschfeld showed that every countable model of PA canbe embedded in a nontrivial homomorphic image of thesemiring R of recursive functions.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Jim Schmerl’s Account

• Harvey was on the Flip Wilson show. It must have been in1971 (perhaps plus/minus 1) since I was at Yale at thetime and Joram Hirschfeld was just finishing his thesisthen.

• He heard Harvey talk about embedding models of PA asinitial segments and that gave him an idea that ended upin his thesis.

• Hirschfeld showed that every countable model of PA canbe embedded in a nontrivial homomorphic image of thesemiring R of recursive functions.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.• §1. Preliminaries; pp. 544-551.• §2. Standard Systems of nonstandard admissible sets; pp.

552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.• §6. Categoricity relative to ordinals; pp.570-572.• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.• §1. Preliminaries; pp. 544-551.• §2. Standard Systems of nonstandard admissible sets; pp.

552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.• §6. Categoricity relative to ordinals; pp.570-572.• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.

• §1. Preliminaries; pp. 544-551.• §2. Standard Systems of nonstandard admissible sets; pp.

552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.• §6. Categoricity relative to ordinals; pp.570-572.• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.• §1. Preliminaries; pp. 544-551.

• §2. Standard Systems of nonstandard admissible sets; pp.552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.• §6. Categoricity relative to ordinals; pp.570-572.• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.• §1. Preliminaries; pp. 544-551.• §2. Standard Systems of nonstandard admissible sets; pp.

552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.• §6. Categoricity relative to ordinals; pp.570-572.• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.• §1. Preliminaries; pp. 544-551.• §2. Standard Systems of nonstandard admissible sets; pp.

552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.• §6. Categoricity relative to ordinals; pp.570-572.• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.• §1. Preliminaries; pp. 544-551.• §2. Standard Systems of nonstandard admissible sets; pp.

552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.• §6. Categoricity relative to ordinals; pp.570-572.• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.• §1. Preliminaries; pp. 544-551.• §2. Standard Systems of nonstandard admissible sets; pp.

552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.

• §6. Categoricity relative to ordinals; pp.570-572.• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.• §1. Preliminaries; pp. 544-551.• §2. Standard Systems of nonstandard admissible sets; pp.

552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.• §6. Categoricity relative to ordinals; pp.570-572.

• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

A Landmark Paper

• H. Friedman, Countable models of set theories, inCambridge Summer School in Mathematical Logic(Cambridge, 1971), pp. 539-573. Lecture Notes in Math.,Vol. 337. Springer, Berlin, 1973.

• Introduction; pp. 539-543.• §1. Preliminaries; pp. 544-551.• §2. Standard Systems of nonstandard admissible sets; pp.

552-556.

• §3. The ordinals in nonstandard admissible sets; pp.557-562.

• §4. Initial segments of nonstandard power admissible sets;pp. 563-565.

• §5. Submodels of Σ1∞-CA; pp. 566-569.• §6. Categoricity relative to ordinals; pp.570-572.• References and Errata; p.573.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (1)

• 1962. In answer to a question of Dana Scott, RobertVaught showed that there is a model of true arithmeticthat is isomorphic to a proper initial segment of itself.This result is later included in a joint paper of Vaught andMorley.

• 1973. Friedman’s self-embedding theorem.• 1977. Alex Wilkie showed the existence ofcontinuum-many initial segments of every countablenonstandard model of M of PA that are isomorphic to M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (1)

• 1962. In answer to a question of Dana Scott, RobertVaught showed that there is a model of true arithmeticthat is isomorphic to a proper initial segment of itself.This result is later included in a joint paper of Vaught andMorley.

• 1973. Friedman’s self-embedding theorem.• 1977. Alex Wilkie showed the existence ofcontinuum-many initial segments of every countablenonstandard model of M of PA that are isomorphic to M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (1)

• 1962. In answer to a question of Dana Scott, RobertVaught showed that there is a model of true arithmeticthat is isomorphic to a proper initial segment of itself.This result is later included in a joint paper of Vaught andMorley.

• 1973. Friedman’s self-embedding theorem.

• 1977. Alex Wilkie showed the existence ofcontinuum-many initial segments of every countablenonstandard model of M of PA that are isomorphic to M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (1)

• 1962. In answer to a question of Dana Scott, RobertVaught showed that there is a model of true arithmeticthat is isomorphic to a proper initial segment of itself.This result is later included in a joint paper of Vaught andMorley.

• 1973. Friedman’s self-embedding theorem.• 1977. Alex Wilkie showed the existence ofcontinuum-many initial segments of every countablenonstandard model of M of PA that are isomorphic to M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (2)

• 1978. Hamid Lessan showed that a countable model Mof ΠPA2 is isomorphic to a proper initial segment of itself iffM is 1-tall and 1-extendible.

• Here 1-tall means that the set of Σ1-definable elements ofM is not cofinal in M, and 1-extendible means that thereis an end extension M∗ of M that satisfies I∆0 andThΣ1(M) = ThΣ1(M∗).

• 1978. Craig Smorynski’s influential lectures andexpositions systematized and extended Friedman-styleembedding theorems around the key concept of (partial)recursive saturation.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (2)

• 1978. Hamid Lessan showed that a countable model Mof ΠPA2 is isomorphic to a proper initial segment of itself iffM is 1-tall and 1-extendible.

• Here 1-tall means that the set of Σ1-definable elements ofM is not cofinal in M, and 1-extendible means that thereis an end extension M∗ of M that satisfies I∆0 andThΣ1(M) = ThΣ1(M∗).

• 1978. Craig Smorynski’s influential lectures andexpositions systematized and extended Friedman-styleembedding theorems around the key concept of (partial)recursive saturation.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (2)

• 1978. Hamid Lessan showed that a countable model Mof ΠPA2 is isomorphic to a proper initial segment of itself iffM is 1-tall and 1-extendible.

• Here 1-tall means that the set of Σ1-definable elements ofM is not cofinal in M, and 1-extendible means that thereis an end extension M∗ of M that satisfies I∆0 andThΣ1(M) = ThΣ1(M∗).

• 1978. Craig Smorynski’s influential lectures andexpositions systematized and extended Friedman-styleembedding theorems around the key concept of (partial)recursive saturation.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (2)

• 1978. Hamid Lessan showed that a countable model Mof ΠPA2 is isomorphic to a proper initial segment of itself iffM is 1-tall and 1-extendible.

• Here 1-tall means that the set of Σ1-definable elements ofM is not cofinal in M, and 1-extendible means that thereis an end extension M∗ of M that satisfies I∆0 andThΣ1(M) = ThΣ1(M∗).

• 1978. Craig Smorynski’s influential lectures andexpositions systematized and extended Friedman-styleembedding theorems around the key concept of (partial)recursive saturation.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (3)

• 1979. Leonard Lipshitz showed that a countablenonstandard model of PA is Diophantine correct iff it canbe embedded into arbitrarily low nonstandard initialsegments of itself (the result was suggested by StanleyTennenbaum).

• 1981. Jeff Paris noted that an unpublished constructionof Robert Solovay shows that every countable recursivelysaturated model of I∆0 + BΣ1 is isomorphic to a properinitial segment of itself.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (3)

• 1979. Leonard Lipshitz showed that a countablenonstandard model of PA is Diophantine correct iff it canbe embedded into arbitrarily low nonstandard initialsegments of itself (the result was suggested by StanleyTennenbaum).

• 1981. Jeff Paris noted that an unpublished constructionof Robert Solovay shows that every countable recursivelysaturated model of I∆0 + BΣ1 is isomorphic to a properinitial segment of itself.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (3)

• 1979. Leonard Lipshitz showed that a countablenonstandard model of PA is Diophantine correct iff it canbe embedded into arbitrarily low nonstandard initialsegments of itself (the result was suggested by StanleyTennenbaum).

• 1981. Jeff Paris noted that an unpublished constructionof Robert Solovay shows that every countable recursivelysaturated model of I∆0 + BΣ1 is isomorphic to a properinitial segment of itself.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (4)

• 1985. Costas Dimitracopoulos showed that everycountable nonstandard model of I∆0 + BΣ2 is isomorphicto a proper initial segment of itself.

• 1987. Jean-Pierre Ressayre proved an optimal result: forevery countable nonstandard model M of IΣ1 and forevery a ∈M there is an embedding j of M onto a properinitial segment of itself such that j(x) = x for all x ≤ a;moreover, this property characterizes countable models ofIΣ1 among countable models of I∆0.

• 1988. Independently of Ressayre, Dimitracopoulos andParis showed that every countable nonstandard model ofIΣ1 is isomorphic to a proper initial segment of itself.

• Dimitracopoulos and Paris also generalized Lessan’saforementioned result by weakening ΠPA2 toI∆0 + exp +BΣ1.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (4)

• 1985. Costas Dimitracopoulos showed that everycountable nonstandard model of I∆0 + BΣ2 is isomorphicto a proper initial segment of itself.

• 1987. Jean-Pierre Ressayre proved an optimal result: forevery countable nonstandard model M of IΣ1 and forevery a ∈M there is an embedding j of M onto a properinitial segment of itself such that j(x) = x for all x ≤ a;moreover, this property characterizes countable models ofIΣ1 among countable models of I∆0.

• 1988. Independently of Ressayre, Dimitracopoulos andParis showed that every countable nonstandard model ofIΣ1 is isomorphic to a proper initial segment of itself.

• Dimitracopoulos and Paris also generalized Lessan’saforementioned result by weakening ΠPA2 toI∆0 + exp +BΣ1.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (4)

• 1985. Costas Dimitracopoulos showed that everycountable nonstandard model of I∆0 + BΣ2 is isomorphicto a proper initial segment of itself.

• 1987. Jean-Pierre Ressayre proved an optimal result: forevery countable nonstandard model M of IΣ1 and forevery a ∈M there is an embedding j of M onto a properinitial segment of itself such that j(x) = x for all x ≤ a;moreover, this property characterizes countable models ofIΣ1 among countable models of I∆0.

• 1988. Independently of Ressayre, Dimitracopoulos andParis showed that every countable nonstandard model ofIΣ1 is isomorphic to a proper initial segment of itself.

• Dimitracopoulos and Paris also generalized Lessan’saforementioned result by weakening ΠPA2 toI∆0 + exp +BΣ1.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (4)

• 1985. Costas Dimitracopoulos showed that everycountable nonstandard model of I∆0 + BΣ2 is isomorphicto a proper initial segment of itself.

• 1987. Jean-Pierre Ressayre proved an optimal result: forevery countable nonstandard model M of IΣ1 and forevery a ∈M there is an embedding j of M onto a properinitial segment of itself such that j(x) = x for all x ≤ a;moreover, this property characterizes countable models ofIΣ1 among countable models of I∆0.

• 1988. Independently of Ressayre, Dimitracopoulos andParis showed that every countable nonstandard model ofIΣ1 is isomorphic to a proper initial segment of itself.

• Dimitracopoulos and Paris also generalized Lessan’saforementioned result by weakening ΠPA2 toI∆0 + exp +BΣ1.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (4)

• 1985. Costas Dimitracopoulos showed that everycountable nonstandard model of I∆0 + BΣ2 is isomorphicto a proper initial segment of itself.

• 1987. Jean-Pierre Ressayre proved an optimal result: forevery countable nonstandard model M of IΣ1 and forevery a ∈M there is an embedding j of M onto a properinitial segment of itself such that j(x) = x for all x ≤ a;moreover, this property characterizes countable models ofIΣ1 among countable models of I∆0.

• 1988. Independently of Ressayre, Dimitracopoulos andParis showed that every countable nonstandard model ofIΣ1 is isomorphic to a proper initial segment of itself.

• Dimitracopoulos and Paris also generalized Lessan’saforementioned result by weakening ΠPA2 toI∆0 + exp +BΣ1.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (5)

• 1991. Richard Kaye’s text gave a systematic presentationof various Friedman-style embedding theorems.

• Theorem. (Fine-tuned Friedman Theorem) Suppose M isa countable nonstandard model of IΣ1 and {a, b} ⊆ Nwith a < b. The following statements are equivalent:

• (1) For every Σn-formula σ(x , y) we have:

M |= ∃y σ(a, y) −→ ∃y < b σ(a, y).

• (2) There is a Σn-elementary-initial embeddingj :M→M with j(a) = a and a < j(M) < b

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (5)

• 1991. Richard Kaye’s text gave a systematic presentationof various Friedman-style embedding theorems.

• Theorem. (Fine-tuned Friedman Theorem) Suppose M isa countable nonstandard model of IΣ1 and {a, b} ⊆ Nwith a < b. The following statements are equivalent:

• (1) For every Σn-formula σ(x , y) we have:

M |= ∃y σ(a, y) −→ ∃y < b σ(a, y).

• (2) There is a Σn-elementary-initial embeddingj :M→M with j(a) = a and a < j(M) < b

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (5)

• 1991. Richard Kaye’s text gave a systematic presentationof various Friedman-style embedding theorems.

• Theorem. (Fine-tuned Friedman Theorem) Suppose M isa countable nonstandard model of IΣ1 and {a, b} ⊆ Nwith a < b. The following statements are equivalent:

• (1) For every Σn-formula σ(x , y) we have:

M |= ∃y σ(a, y) −→ ∃y < b σ(a, y).

• (2) There is a Σn-elementary-initial embeddingj :M→M with j(a) = a and a < j(M) < b

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (5)

• 1991. Richard Kaye’s text gave a systematic presentationof various Friedman-style embedding theorems.

• Theorem. (Fine-tuned Friedman Theorem) Suppose M isa countable nonstandard model of IΣ1 and {a, b} ⊆ Nwith a < b. The following statements are equivalent:

• (1) For every Σn-formula σ(x , y) we have:

M |= ∃y σ(a, y) −→ ∃y < b σ(a, y).

• (2) There is a Σn-elementary-initial embeddingj :M→M with j(a) = a and a < j(M) < b

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (5)

• 1991. Richard Kaye’s text gave a systematic presentationof various Friedman-style embedding theorems.

• Theorem. (Fine-tuned Friedman Theorem) Suppose M isa countable nonstandard model of IΣ1 and {a, b} ⊆ Nwith a < b. The following statements are equivalent:

• (1) For every Σn-formula σ(x , y) we have:

M |= ∃y σ(a, y) −→ ∃y < b σ(a, y).

• (2) There is a Σn-elementary-initial embeddingj :M→M with j(a) = a and a < j(M) < b

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (6)

• 1997. Kazuyuki Tanaka extended Ressayre’saforementioned result by showing that every countablenonstandard model of WKL0 has a nontrivialself-embedding in the following sense:Given (M,A) |= WKL0 there is a proper initial segment Iof M such that:

(M,A) ∼= (I ,A � I ),

where

A � I := {A ∩ I : A ∈ A}.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Synoptic History (6)

• 1997. Kazuyuki Tanaka extended Ressayre’saforementioned result by showing that every countablenonstandard model of WKL0 has a nontrivialself-embedding in the following sense:Given (M,A) |= WKL0 there is a proper initial segment Iof M such that:

(M,A) ∼= (I ,A � I ),

where

A � I := {A ∩ I : A ∈ A}.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (1)

• Definition. A (total) function f from M to M is a totalM-recursive function if the graph of f is definable in Mby a parameter-free Σ1-formula.

• Theorem. (V. Shavrukov, 2013) Suppose M is acountable nonstandard model of IΣ1, and {a, b} ⊆ M witha < b. The following statements are equivalent:

• (1) There is an initial embedding j :M→M witha < j(M) < b.

• (2) f (a) < b for all M-total recursive functions f..

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (1)

• Definition. A (total) function f from M to M is a totalM-recursive function if the graph of f is definable in Mby a parameter-free Σ1-formula.

• Theorem. (V. Shavrukov, 2013) Suppose M is acountable nonstandard model of IΣ1, and {a, b} ⊆ M witha < b. The following statements are equivalent:

• (1) There is an initial embedding j :M→M witha < j(M) < b.

• (2) f (a) < b for all M-total recursive functions f..

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (1)

• Definition. A (total) function f from M to M is a totalM-recursive function if the graph of f is definable in Mby a parameter-free Σ1-formula.

• Theorem. (V. Shavrukov, 2013) Suppose M is acountable nonstandard model of IΣ1, and {a, b} ⊆ M witha < b. The following statements are equivalent:

• (1) There is an initial embedding j :M→M witha < j(M) < b.

• (2) f (a) < b for all M-total recursive functions f..

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (1)

• Definition. A (total) function f from M to M is a totalM-recursive function if the graph of f is definable in Mby a parameter-free Σ1-formula.

• Theorem. (V. Shavrukov, 2013) Suppose M is acountable nonstandard model of IΣ1, and {a, b} ⊆ M witha < b. The following statements are equivalent:

• (1) There is an initial embedding j :M→M witha < j(M) < b.

• (2) f (a) < b for all M-total recursive functions f..

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (1)

• Definition. A (total) function f from M to M is a totalM-recursive function if the graph of f is definable in Mby a parameter-free Σ1-formula.

• Theorem. (V. Shavrukov, 2013) Suppose M is acountable nonstandard model of IΣ1, and {a, b} ⊆ M witha < b. The following statements are equivalent:

• (1) There is an initial embedding j :M→M witha < j(M) < b.

• (2) f (a) < b for all M-total recursive functions f..

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (2)

• Theorem. (T. Wong, to appear). Suppose (M,A) is acountable nonstandard recursively saturated model ofRCA∗0. The following are equivalent: .

• (1) There is a self-embedding of (M,A) onto a properinitial segment of itself.

• (2) (M,A) |= WKL∗0..

• RCA∗0 is formulated in the language of second-orderarithmetic, and consists of basic recursive axioms foraddition, multiplication, and exponentiation; augmentedwith ∆10-Comprehension and ∆

00-Induction.

•WKL∗0

I∆0 + Exp + BΣ1=

WKL0IΣ1

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (2)

• Theorem. (T. Wong, to appear). Suppose (M,A) is acountable nonstandard recursively saturated model ofRCA∗0. The following are equivalent: .

• (1) There is a self-embedding of (M,A) onto a properinitial segment of itself.

• (2) (M,A) |= WKL∗0..

• RCA∗0 is formulated in the language of second-orderarithmetic, and consists of basic recursive axioms foraddition, multiplication, and exponentiation; augmentedwith ∆10-Comprehension and ∆

00-Induction.

•WKL∗0

I∆0 + Exp + BΣ1=

WKL0IΣ1

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (2)

• Theorem. (T. Wong, to appear). Suppose (M,A) is acountable nonstandard recursively saturated model ofRCA∗0. The following are equivalent: .

• (1) There is a self-embedding of (M,A) onto a properinitial segment of itself.

• (2) (M,A) |= WKL∗0..

• RCA∗0 is formulated in the language of second-orderarithmetic, and consists of basic recursive axioms foraddition, multiplication, and exponentiation; augmentedwith ∆10-Comprehension and ∆

00-Induction.

•WKL∗0

I∆0 + Exp + BΣ1=

WKL0IΣ1

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (2)

• Theorem. (T. Wong, to appear). Suppose (M,A) is acountable nonstandard recursively saturated model ofRCA∗0. The following are equivalent: .

• (1) There is a self-embedding of (M,A) onto a properinitial segment of itself.

• (2) (M,A) |= WKL∗0..

• RCA∗0 is formulated in the language of second-orderarithmetic, and consists of basic recursive axioms foraddition, multiplication, and exponentiation; augmentedwith ∆10-Comprehension and ∆

00-Induction.

•WKL∗0

I∆0 + Exp + BΣ1=

WKL0IΣ1

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (2)

• Theorem. (T. Wong, to appear). Suppose (M,A) is acountable nonstandard recursively saturated model ofRCA∗0. The following are equivalent: .

• (1) There is a self-embedding of (M,A) onto a properinitial segment of itself.

• (2) (M,A) |= WKL∗0..

• RCA∗0 is formulated in the language of second-orderarithmetic, and consists of basic recursive axioms foraddition, multiplication, and exponentiation; augmentedwith ∆10-Comprehension and ∆

00-Induction.

•WKL∗0

I∆0 + Exp + BΣ1=

WKL0IΣ1

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (2)

• Theorem. (T. Wong, to appear). Suppose (M,A) is acountable nonstandard recursively saturated model ofRCA∗0. The following are equivalent: .

• (1) There is a self-embedding of (M,A) onto a properinitial segment of itself.

• (2) (M,A) |= WKL∗0..

• RCA∗0 is formulated in the language of second-orderarithmetic, and consists of basic recursive axioms foraddition, multiplication, and exponentiation; augmentedwith ∆10-Comprehension and ∆

00-Induction.

•WKL∗0

I∆0 + Exp + BΣ1=

WKL0IΣ1

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (3)

• Theorem. (K. Yokoyama, to appear). Suppose (M,A) isa countable nonstandard model of RCA0. The followingare equivalent:

.

• (1) (M,A) |= Π11-CA..

• (2) There is a self-embedding of (M,A) onto a properinitial segment of itself such that j(M) is a “Ramsey cut”in M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (3)

• Theorem. (K. Yokoyama, to appear). Suppose (M,A) isa countable nonstandard model of RCA0. The followingare equivalent:

.

• (1) (M,A) |= Π11-CA..

• (2) There is a self-embedding of (M,A) onto a properinitial segment of itself such that j(M) is a “Ramsey cut”in M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (3)

• Theorem. (K. Yokoyama, to appear). Suppose (M,A) isa countable nonstandard model of RCA0. The followingare equivalent:

.

• (1) (M,A) |= Π11-CA..

• (2) There is a self-embedding of (M,A) onto a properinitial segment of itself such that j(M) is a “Ramsey cut”in M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (3)

• Theorem. (K. Yokoyama, to appear). Suppose (M,A) isa countable nonstandard model of RCA0. The followingare equivalent:

.

• (1) (M,A) |= Π11-CA..

• (2) There is a self-embedding of (M,A) onto a properinitial segment of itself such that j(M) is a “Ramsey cut”in M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (4)

• Theorem. (A.E., to appear) The following conditions areequivalent for a countable nonstandard model of PA:

.

• (1) M has a self-embedding onto a proper initial segmentof itself such that Fix(j) = K 1(M).

.

• (2) N is a strong cut of M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (4)

• Theorem. (A.E., to appear) The following conditions areequivalent for a countable nonstandard model of PA:

.

• (1) M has a self-embedding onto a proper initial segmentof itself such that Fix(j) = K 1(M).

.

• (2) N is a strong cut of M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (4)

• Theorem. (A.E., to appear) The following conditions areequivalent for a countable nonstandard model of PA:

.

• (1) M has a self-embedding onto a proper initial segmentof itself such that Fix(j) = K 1(M).

.

• (2) N is a strong cut of M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Some Recent Results (4)

• Theorem. (A.E., to appear) The following conditions areequivalent for a countable nonstandard model of PA:

.

• (1) M has a self-embedding onto a proper initial segmentof itself such that Fix(j) = K 1(M).

.

• (2) N is a strong cut of M.

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Envoi

• Thank you, and Congratulations Harvey! .

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Envoi

• Thank you, and Congratulations Harvey! .

Variations ona Theme by

Friedman

Ali Enayat,GöteborgsUniversitet

Envoi

• Thank you, and Congratulations Harvey! .