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Peano's Parlour

From Peano's Parlour. This site is dedicated to open problems in nonstandard models of Peano Arithmetic and related theories. As in most areas of mathematics, we do not suffer from a shortage of open problems. The problems are arranged by topics. Cuts in models of PA and independence results. Recursively saturated, resplendent models, and saturated models. Standard systems and the Scott set problem. Lattices of elementary substructures. Uncountable models with interesting second-order properties.

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Peano's Parlour | modelsofpa.info Reviews

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From Peano's Parlour. This site is dedicated to open problems in nonstandard models of Peano Arithmetic and related theories. As in most areas of mathematics, we do not suffer from a shortage of open problems. The problems are arranged by topics. Cuts in models of PA and independence results. Recursively saturated, resplendent models, and saturated models. Standard systems and the Scott set problem. Lattices of elementary substructures. Uncountable models with interesting second-order properties.

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1 peano's parlour

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4 automoprhism groups

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Peano's Parlour | modelsofpa.info Reviews

https://modelsofpa.info

From Peano's Parlour. This site is dedicated to open problems in nonstandard models of Peano Arithmetic and related theories. As in most areas of mathematics, we do not suffer from a shortage of open problems. The problems are arranged by topics. Cuts in models of PA and independence results. Recursively saturated, resplendent models, and saturated models. Standard systems and the Scott set problem. Lattices of elementary substructures. Uncountable models with interesting second-order properties.

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modelsofpa.info

1

Nonstandard satisfaction classes - Peano's Parlour

http://modelsofpa.info/Nonstandard_satisfaction_classes

From Peano's Parlour. A converse to Tarski? A satisfaction class for a model $M$ is $e$-full, if $S$ decides all If for every $ Sigma e$ sentence (in the sense of $M$) $ varphi$ with parameters either $ varphi in S$ or $ lnot varphi in S$. Fullness of $M$, ${ rm Full}(M) }$ is the set of those $e in M$ M for which $M$ has an $e$-full inductive partial satisfaction class. Kaufmann and Schmerl [ 1. Problem: Suppose ${ rm Full}(M)=M$, does $M$ have a full inductive satisfaction class? A converse to Tarski?

2

Lattices of elementary substructures - Peano's Parlour

http://modelsofpa.info/Lattices_of_elementary_substructures

Lattices of elementary substructures. From Peano's Parlour. For $M models PA$ let ${ rm{Lt}(M)}=( {K: K prec M }, prec)$ and for $M prec N$, let ${ rm Lt}(N/M)=( {K: M prec K prec M }, prec)$. In general, the lattice problem is: Which lattices can be represented as ${ rm Lt}(N/M)=( {K: M prec K prec M }, prec)$, for some $M prec N$? There is a vast literature on the problem and many special cases remain open. Here are basic references: [ 1. Chapter 4 of [ 7. Is devoted to the lattice problem. What finite...

3

Cuts in models of PA and independence results - Peano's Parlour

http://modelsofpa.info/Cuts_in_models_of_PA_and_independence_results

Cuts in models of PA and independence results. From Peano's Parlour. Diversity in elementary cuts. Diversity in elementary cuts. Let $M models PA$ be countable and recursively saturated. Smorynski proved that here are $2 { aleph 0}$ theories of pairs $(M,K)$, such that $K$ is not semiregular in $M$. Kossak proved that there are $2 { aleph 0}$ theories of pairs $(M,K)$, such that $K$ is strong in $M$. The problem was stated in [ 1. Related results can be found in [ 2. Question 6.6 of [ 6. Roman Kossak and...

4

Automoprhism groups - Peano's Parlour

http://modelsofpa.info/Automoprhism_groups

From Peano's Parlour. Automorphism groups in general. Automorphisms groups of countable recursively saturated models. Retrieved from " http:/ modelsofpa.info/index.php? This page was last modified on 6 February 2013, at 15:34. This page has been accessed 3,023 times. About Peano's Parlour.

5

Complexity and classification of countable models - Peano's Parlour

http://modelsofpa.info/Complexity_and_classification_of_countable_models

Complexity and classification of countable models. From Peano's Parlour. This question is about the complexity of the classification problem for finitely generated models of a given completion $T$ of $PA$. For an introduction and survey of this subject of study. It was shown in [ 2. What is the precise complexity of $ cong {fg} T$? There is a wide gap between $E 0$ and $E infty$. Is $ cong {fg} T$ bireducible with $E infty$? Is $ cong {fg} T$ bireducible with $E 0$? It should be remarked that the classif...

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cantorsattic.info

The parlour - Cantor's Attic

http://cantorsattic.info/Parlour

From Cantor's Attic. PT Loma Lighthouse photo by wagbogwest. Welcome to the parlour, where finite numbers dream. And into the attic. Over to Peano's Parlour. The site for open problems in nonstandard models of Peano Arithmetic and related theories. The smallest non-describable number. The smallest uninteresting number. And the googol plex bang stack hierarchy. Retrieved from " http:/ cantorsattic.info/index.php? This page was last modified on 30 July 2013, at 13:15. About Cantor's Attic.

boolesrings.org

Victoria Gitman | Mathematical Explorations

http://boolesrings.org/victoriagitman

Home at Booles’ Rings. 8220;If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” — John von Neumann. Welcome to my blog! I am a visiting scholar at the CUNY Graduate Center. Starting Fall 2014, I will be co-organizing the CUNY Set Theory Seminar. With Thomas Johnstone. My most recent CV is here. Visit Cantor’s Attic. Here is my WordPress profile. Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom. V Gitman ...

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modelsofpa.info

Peano's Parlour

From Peano's Parlour. This site is dedicated to open problems in nonstandard models of Peano Arithmetic and related theories. As in most areas of mathematics, we do not suffer from a shortage of open problems. The problems are arranged by topics. Cuts in models of PA and independence results. Recursively saturated, resplendent models, and saturated models. Standard systems and the Scott set problem. Lattices of elementary substructures. Uncountable models with interesting second-order properties.

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