Questions tagged [peano-axioms]

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

In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of consistency and completeness of number theory.

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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 Peano arithmetic (PA), yet true in certain extended…
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How is exponentiation defined in Peano arithmetic?

How would exponentiation be defined in Peano arithmetic? Unless $n$ is fixed natural number, $x^n$ seems to be hard to define. Edit 2: So, what would be the way to define $x^n+y^n = z^n$ using $\Sigma_1^0$ formula? Edit: OK, I say Peano arithmetic…
UQT
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A question on Terence Tao's representation of Peano Axioms

While reading Terence Tao's book on Analysis I had some questions regarding the implication of the Peano Axioms. After writing the following four axioms (which I will write without changing their original numbering), Axiom 2.1 $0$ is a natural…
user170039
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Can Peano arithmetic prove the consistency of "baby arithmetic"?

I am reading Peter Smith's An Introduction to Gödel's Theorems. In chapter 10, he defines "baby arithmetic" $\mathsf{BA}$ to be the zeroth-order version of Peano arithmetic ($\mathsf{PA}$) without induction. That is, $\mathsf{BA}$ is the…
WillG
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Why does induction have to be an axiom?

I noticed that there is an axiom that says that if $S(n)\implies S(n+1)$, and $S(1)$ is true, then $\forall n \in \Bbb N, S(n).$ My question is why is this an axiom? why can't we derive this from the other axioms?
yes
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Is there a natural model of Peano Arithmetic where Goodstein's theorem fails?

Goodstein's Theorem is the statement that every Goodstein sequence eventually hits 0. It is known to be independent of Peano Arithemtic (PA), and in fact, was the first such purely number theoretic result. It is provable in ZFC. One way of…
Jason DeVito
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How does Peano Postulates construct Natural numbers only?

I am beginning real analysis and got stuck on the first page (Peano Postulates). It reads as follows, at least in my textbook. Axiom 1.2.1 (Peano Postulates). There exists a set $\Bbb N$ with an element $1 \in \Bbb N$ and a function $s:\Bbb N \to…
Solomon Tessema
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What is an example of a non standard model of Peano Arithmetic?

According to here, there is the "standard" model of Peano Arithmetic. This is defined as $0,1,2,...$ in the usual sense. What would be an example of a nonstandard model of Peano Arithmetic? What would a nonstandard amount of time be like?
user223391
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Statement provable for all parameters, but unprovable when quantified

I've been reading a book on Gödel's incompleteness theorems and it makes the following claim regarding provability of statements in Peano arithmetic (paraphrased): There exists a formula $A(x)$ such that the statements $A(0), A(1), A(2), \dots$ are…
Matěj G.
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Do we have to prove how parentheses work in the Peano axioms?

One thing that has bothered me so far while learning about the Peano axioms is that the use of parentheses just comes out of nowhere and we automatically assume they are true in how they work. For example the proof that $(a+b)+c = a+(b+c)$. Given…
user537069
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Why is Peano arithmetic undecidable?

I read that Presburger arithmetic is decidable while Peano arithmetic is undecidable. Peano arithmetic extends Presburger arithmetic just with the addition of the multiplication operator. Can someone please give me the 'intuitive' idea behind…
chinu
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Purpose of the Peano Axioms

Wikipedia says the Peano Axioms are a set of axioms for the natural numbers. Is the purpose of the axioms to create a base on which we can build the rest of mathematicas formally? If this is true were they chosen because they are agreed to be basic…
integrator
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Is there a 'nice' axiomatization in the language of arithmetic of the statements ZF proves about the natural numbers?

It's well known that ZF (equivalently ZFC by this question) proves more about the natural numbers than PA. The set of such statements is recursively enumerable so it is recursively axiomatizable. Is it difficult to explicitly axiomatize these…
James Hanson
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A model-theoretic question re: Nelson and exponentiation

EDIT: I am not asking about the validity of exponentiation, or PA. My question is about a specific technical claim which Nelson makes in this article (pp. 9-12): that a certain theory does not prove a certain sentence, and more generally that that…
Noah Schweber
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Leaving out one of the Peano Axioms

What happens if you leave N4 (from Ross' book) out of the Peano axioms which states that if $n$ and $m$ in $\mathbb{N}$ have the same successor, then $n = m$?
michael straws
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