# Questions tagged [peano-axioms]

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

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### Godel's incompleteness theorem: Question about effective axiomatization

I am studying Godel incompleteness theorems and I am struggling with the definition of effective axiomatization. From Wikipedia: A formal system is said to be effectively axiomatized (also called ...
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### Understanding Peano Arithmatic and Axioms

I am new to analysis and started reading a PDF I found on Reddit, the link is here. I stumbled on a few question about basic Peano axioms and the definitions that the PDF derived from it. In case ...
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### Proving the existence of hyperoperations in a Peano system

In Mendelson's "Number Systems" he defines and subsequently proves the existence and uniqueness of binary operators $+$ and $\times$, as well as exponentiation, in $P$ using the so-called ...
1 vote
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### Why is the material conditional treated like logical entailment in second order quantification? [closed]

According to this Wikipedia article the second order axiom of induction is: $$\forall P(P(0)\land \forall k(P(k) \to P(k+1)) \to \forall x(N(x) \to P(x))$$ Where N(x) means x is a natural number. That ...
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### Finite axiomatization of EFA

According to the paper Fragments of Peano's Arithmetic and the MRDP theorem (Section 6), elementary function arithmetic (EFA) is finitely axiomatizable. Is there a known explicite finite ...
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### Naive Set Theory - proof of commutativity of products

I am working through Halmos's Naive Set Theory on my own and trying to do all the exercises, including what are merely suggestions in the text. I am right now in section 13, which shows a derivation ...
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### Ability of Peano axiom with integer set?

Axioms: Peano Axioms (defines natural number, introducing 0 and ') For each predicate φ, there exist exactly one set X, s.t. forall x, φ(x) <=> x∈X. So, it's possible to define less-than in ...
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### Axiomatic reason why $a=4 \implies a>1$ for $a \in \mathbb{N}$

This is a trivial task: Given $a \in \mathbb{N}$ and $$a=4$$ Show $$a > 1$$ Part of the challenge for newcomers like me is that "easy" tasks actually make it harder to think about the ...
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### not precisely understanding what is asked by ex 3.5.13 Tao Analysis I (only one natural number system)

I am failing to understand the intent of the question posed by exercise 3.5.13 of Tao's Analysis I 4th ed. The purpose of this exercise is to show that there is essentially only one version of the ...
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### Proof that each natural number has a unique successor

I've proven that every positive natural number has a unique predecessor using Peano's axioms. But now, I was wondering how I could prove that every natural number has a unique successor using the same ...
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### How to justify why succession and addition cannot be circularly defined like this?

I am reading Tao's Analysis I, in which he states: One may be tempted to [define the successor of $n$ as] $n + 1$. . . but this would introduce a circularity in our foundations, since the notion of ...
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### Why is addition not completely defined here?

Say for the natural numbers, we define addition this way: $0 + 0 = 0$, and if $n+m = x$, then $S(n) + m = n+S(m) = S(x)$ Say we have the regular Peano axioms, except we delete the axiom of ...
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### Question about the Peano axioms + linear order axioms

The signature consists of $S$, $0$ & $<$ and the axioms are: I - $\forall x (S(x) \not= 0)$ II - $\forall x \forall y (S(x) = S(y) \to x = y)$ III - First-order Induction schema IV - $<$ is ...
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### Is it circular to include reachability from $0$ like this as a Peano axiom?

I am wondering whether it makes logical and semantic sense to include, as an axiom to define the natural numbers, that "every natural number is either $0$ or the result of potentially repeated ...
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### Specific example of a property $P$ that Peano arithmetic proves holds true for every specific number, but not for all numbers.

Can someone give a specific example, if there is any, of a predicate $P(x)$ expressible in the language of Peano arithmetic, such that the first-order theory of Peano Arithmetic proves $P(0)$, $P(1)$, ...
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### Can we modify the Peano axioms like this? [closed]

I am wondering if the following modifications of the Peano axioms result in a set of axioms equivalent to the Peano axioms, in the sense that any set of numbers satisfies these modified axioms if and ...
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1 vote
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### Order type of cuts satisfying $\mathsf I\Sigma_n$

When $M$ is a model of Peano arithmetic, a cut of $M$ is an initial segment $I$ of $M$ such that $I$ is closed under successor. There is some work on cuts that satisfy $\mathsf I\Sigma_n$, Peano ...
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### Are the models of PA recursively enumerable?

Is there a Turing machine (TM from now on) which lists every model of PA (without the induction axiom schema, so just addition and multiplication)? More specifically, can such a TM list all infinite ...
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### Proving that the set of non-negative half-integers satisfies Peano's axioms

I postulate that the following set $\{0,0.5,1,1.5,...\}$ represents the natural numbers. Of course, intuitively, this isn't true. But let me try to show this using Peano's axioms. I'll first define ...
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1 vote
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### Using Peano's axioms to disprove the existence of self-looping tendencies in natural numbers

Let me clarify by what I mean by "self-looping". So, we know that Peano's axioms use primitive terms like zero, natural number and the successor operation. Now, I want to prove that the ...
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### Peano Arithmetic can prove any finite subset of its axioms is consistent

Timothy Chow writes in a MathOverflow answer [...] here is a classical fact: for any finite subset of the axioms of PA (remember that PA contains an axiom schema and hence has infinitely many axioms),...
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