Questions tagged [peano-axioms]

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

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How Can the Peano Postulates Be Categorical If They Have NonStandard Models?

Having just read Noah Schweber's excellent answer to this question, I am reminded of something that has always mystified me. I was taught that the Peano Postulates are categorical (that is, any two ...
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Difference between provability and truth of Goodstein's theorem

I have been thinking about the difference between provability and truth and think this example can illustrate what I have been wondering about: We know that Goodstein's theorem (G) is unprovable in ...
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Is every true statement about the natural numbers provable in ZFC?

Related questions: Difference between undecidable statements in set-theory and number theory? Is the arithmetic most mathematicans use a modelled within first or a second order logic? Peano's axioms ...
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For any two models of PA, is one model a cut of the other?

For two models of arithmetic $M_1=(S_1, 0_1, 1_1, +_1, \times_1, <_1)$, $M_2(S_2, 0_2, 1_2, +_2, \times_2, <_2)$, we say that the $M_1$ is a cut of the $M_2$ if there is a $S \subseteq S_2$ such ...
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Proving that no number is the successor of itself

I'm doing my research project on Peano Arithmetic, and need to show the PA can prove that no number is the successor of itself. I've seen an answer here: Peano's Axiom: Is it implied that ...
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Logic proof for $(x\leq y\to 0\leq y-x), x+y-y=x$, and the property that the product two consecutive numbers is even

I need to find the proof for $(x\leq y\to 0\leq y-x)$ $x+y-y=x$ $(\exists u ((u+u)=s(x)\times x))$, where s is the successor function. Or, the product of two consecutive natural numbers is even. I ...
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What's a good source on how to construct all sets of numbers?

Lately I've been interested in building, formally, all sets of numbers, starting from ℕ, then ℤ, then ℚ, then ℙ, then ℝ, then iℝ, then ℂ. The only book I have come up, so far, is "Foundations of ...
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Who first proved Peano Arithmetic is not finitely axiomatizable?

By Peano Arithmetic I mean first order Peano Arithmetic. The earliest proof that it is not finitely axiomatizable that I know of is R. Montague, Semantical Closure and Non-Finite Axiomatizability I. ...
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Why the following initial segment is NOT a model of PA

Let $M$ be a non-standard model of PA, fix an $a \in M$ \ $\mathbb{N}$ Consider the collection: $$I:=\{b\in M\ |\ b<a^n,\ \text{for some }n \in \mathbb{N}\}$$ It is easy to see that this is ...
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Formulation of the Successor Function as an Endofunction in First-Order Logic

The Peano axioms are often listed (among other ways) as a set of 5 axioms in an informal language or 3 axioms in a formal language. For example: Informal (see, e.g., http://mathworld.wolfram.com/...
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Precise reason why a non standard model of arithmetic is not isomorphic to the natural numbers

So I'm looking for a (more) precise reason why a non standard model of arithmetic is not isomorphic to the natural numbers. Under the usual language of arithmetics, +, <, S, $0$, $\times$ I.E) Let ...
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Why do we need second-order logic?

I often read that without second-order logic we can't have a theory of natural numbers because we wouldn't be able to express the well-ordering principle. OK, then there must be a flaw in the ...
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Upper bound of the “number” of countable models of Th$\mathbb{N}$ up to isomorphism

So basically I want to find out what is the upper bound of the "number" of countable models of Th($\mathbb{N})$ up to isomorphism. In the book by Richard Kaye, on Peano arithmetics, he simply said ...
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How to prove $n\leq x \rightarrow x = n \vee Sn \leq x$ using Robinson Arithmetic

Given the definition $n \leq x \Leftrightarrow \exists y \ni y+n=x$, how can one prove $n\leq x \rightarrow x = n \vee Sn \leq x$ in Robinson Arithmetic? I think this should be a proof by induction, ...
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Decomposition into primes in Peano arithmetic.

The language of first-order Peano arithmetic seems to me rather limited. As far as I am aware, you have only the symbols $S, 0, +, \times ,=$. Now the theorem of unique factorization into primes, ...
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Questions about Peano axioms and second-order logic

I have what I hope are some simple questions regarding the Peano axioms. I am reading Terence Tao's book "Analysis 1" where he constructs the natural numbers using the Peano axioms, but the axioms are ...
Prove that it is impossible to define $0$ and “multiplication” on $\Bbb N^2$ such that it becomes a model of **PA** [closed]
Suppose we define addition on $\Bbb N^2$ as $(x_1,x_2)+(y_1,y_2)=(x_1+y_1,x_2+y_2)$. I need to prove that there is no way to define $0$ andd multiplication so that becomes a model of the Peano Axioms ...