# Questions tagged [peano-axioms]

For questions on Peano axioms, a set of axioms for the natural numbers.

82 questions
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### Is it possible to formalize all mathematics in terms of ordinals only?

Our experience shows that all finitary mathematical objects could be encoded using the natural numbers, and all operations on those objects could be expressed in terms of a few basic operations on ...
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### Axiomatizing a “bounded” companion to PA

There's nothing special about PA here, I'm just focusing on it since it's strong enough to ignore lots of minor technical issues around foundations. If switching to some other theory would yield a ...
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### Algebraically Constructing the Natural Numbers Using a Binary Operation Satisfying Some Properties

Here is the proposed theory: Definition: Let $M$ be a nonempty set with a binary operation $+$ satisfying the following properties: P-0: The operation $+: M \times M \to M$ is both associative and ...
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### Is there a weak set theory that can prove that the natural numbers is a model of PA?

$ZFC$ proves that $\mathbb N$ is a model of $PA$. Even $ZF$ does. How weak can go? In particular, is there some weak set theory that proves that $\mathbb N$ is a model of $PA$, but does not proof ...
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### Show that there are only $\aleph_0$ many countable models of the following theory.

Consider a language $L$ with $<0,1,S>$, where $S$ is the successor function. Show that there are only $\aleph_0$ many countable models of Th$(\mathbb{N})$, under $L$. This is one of the ...
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### Proving there is a unique binary operation we call multiplication

The following is a theorem from the book The Real Numbers and Real Analysis by Bloch which I am currently self-studying. I am pretty sure my proof for uniqueness is correct, but I am wondering is ...
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### Incompleteness theorem

Correct me if I am wrong at any point! Godel's incompleteness theorem allows us to express "PA is consistent" in the language of Peano arithmetic, and shows that this is not provable in PA. Let's ...
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### Mathematical Induction and Peano Arithmetic

Peano Arithmetic cannot employ Induction for any ε0 ordering. My question is too easy to be interesting and there is a reason obviously for why it has a negative answer. Can you please provide it for ...
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### True statements about natural numbers that are undecidable in Peano Arithmetic assuming consistency of PA only

I am looking for statements $P$ of Peano Arithmetic ($\textsf{PA}$) that have the following properties: They are as concrete and simple as possible. It is provable by finitistic means that neither $P$...