# Tagged Questions

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

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### An infinite set of axioms in ZF? What does that mean?

Before write this question, I lookeded around enough in this forum for a possible answer and although there are many similar questions, I couldn't find one answer which understand or satisfies me. I ...
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### This proof of peano's systems seemed so easy, perhaps i'm wrong?

Let $P=\{1,2,3,4\}$, where the succesor of a number is $S(n)=n+1$ and $S(4)=1.$ This last one means that 1 is the succesor of another number, hence this can't be a Peano's system. Do I have to ...
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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 ...
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### Decidability of quantifier-free formulae in Peano- and True Arithmetic

It is well-known that validity in Peano Arithmetic is undecidable. It is less well-known that validity is already undecidable in True Arithmetic (the theory of the standard model of Peano Arithmetic). ...
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### Prove that it is a theorem

I'm doing some excercises from the book "The Incompleteness Phenomenom" from Goldstern and Judah. I'm doing exercise about Peano Arithmetic and I have to show that: $\vdash p|x \wedge r·s=p \to r|x$...
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### Peano axioms with only sets and mapping

I've got Serge Lang's Undergraduate Algebra (2nd edition). In the Appendix is a treatment of the Peano Axioms, but, as he says: " The rules of the game from now on allow us to use only sets and ...
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### Could relational operators be used to construct formal theory of natural numbers which is “stronger” than Peano Axioms?

This is a beginner's question about foundational construction of (alternative?) number theory. The notion of mathematical equality is closely related to logico-philosophical notion of 'Law of Identity'...
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### Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
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### Understanding Demonstration There's no Bijection between a set and a proper set of it.

I'm in need of help to understand the demonstratrion of the theorem: Given $I_n=\{ p \in N / p\le n \}$ there's no bijection $f: A \to I_n$ between the set $I_n$ and the proper subset of it $A$. The ...
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### Defining Primes in Non-standard Models of Peano Arithmetic

I was recently reading a post basically discussing an "intuition" that Goldbach's Conjecture may be a statement which is undecidable (the post does not specify which axiomatic system the statement is ...
### How to prove $\mathbb{N}\subseteq\mathbb{R}$ given basic field axiomatisation.
I understand that $\mathbb{R}$ can be defined axiomatically in a number of different ways, including as an extension from more basic number systems (e.g., $\mathbb{N}$, $\mathbb{Z}$, or $\mathbb{Q}$) ...