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Questions tagged [models-of-pa]

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8
votes
2answers
323 views

Models of arithmetic in a signature with exponentiation but not addition and multiplication

Let $\mathcal{L}_{\mathrm{exp}}$ be the language with signature $(0, ^\prime, <, \mathrm{exp})$ (with $0$ interpreted as zero, $^\prime$ as successor, and $\mathrm{exp}(x)$ as $2^x$) and let $\...
7
votes
1answer
417 views

Finding a PA cut in a nonstandard model of PA

For a certain project I am currently working on, I need to be able to find PA cuts in nonstandard models of PA, in desirable intervals. For example, I wonder if the following is true, where $\...
7
votes
2answers
506 views

On a theorem of Zhang Jinwen about models of arithmetic

In the paper ''A Nonstandard Model of Arithmetic Constructed by means of Forcing Method'', Zhang Jinwen states the following in his abstract: The first nonstandard model of arithmetic was given by ...
12
votes
1answer
785 views

Is there a nonstandard model of arithmetic having precisely one inductive truth predicate?

$\newcommand\Tr{\text{Tr}}$My question is whether there can be a nonstandard model of PA having a unique inductive truth predicate. Background. If $\mathcal{N}=\langle N,+,\cdot,0,1,<\rangle$ is ...
21
votes
6answers
1k views

What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?

I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic. Definitions. ...
7
votes
1answer
695 views

Nonstandard models of PA of large cardinal size

It is easy to overlook the fact that the existence of a given large cardinal provides us with a true arithmetical statement that would otherwise be false if the large cardinal notion were not ...
9
votes
1answer
1k views

Uncountable nonstandard models of PA

Standard techniques (no pun intended) can be used to show that countable nonstandard models of Peano Arithmetic are order isomorphic to $\mathbb{N} + \mathbb{Z} \cdot \mathbb{Q}$. Once we have used ...