# Questions tagged [peano-axioms]

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

565 questions
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### Why does induction only allow numbers connected to $0$ to be natural?

When learning Peano axioms normally we use induction to suggest that if a property $P(0)$ is true, and if $P(k)$ being true implies $P(k + 1)$ being true (for natural number $k$), then if $P$ is the ...
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### Is it true that $0\in 1$?

From Zermelo–Fraenkel set theory and Peano axioms, we have $0=\varnothing$ and $1=\varnothing\cup{\{\varnothing\}}\implies0\in 1$. Many thanks for your help!
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### How powerful is PA+Con(PA)+Con(PA+Con(PA))… etc?

From what i remember from Godel encoding there was alot of freedom in how one chooses to expresses the statement Con(PA), my question is if one can classify all statements, or some subclass of all ...
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### Is there an error in this proof the the “strong induction” theorem? Is this Escherian logic?

By set predicates, I mean the "tests for inclusion" used in defining sets. That may be more of a computer scientific than mathematical use of the the term predicate. I included predicate logic in ...
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### Is this a valid proof that $\forall_{x,y}\left[x\ne y\implies x<y\lor x>y\right]$ in $\mathbb{N}_1$?

I do not consider the following to prove all of the so-called trichotomy of order, since I take that to be a statement that exactly one of $<.=,>$ holds for any given pair of numbers. The ...
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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 ...
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### Peano axioms: S(1)=1?

I was reading through the Peano axioms here, and a question came up: Can we define $S(0)=1$, and $S(1)=1$? It seems to me (at least as it is stated) that it would satisfy all of the axioms listed. ...
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### Terminology: arithmetic vs. expressible vs. represented

A function $f:\mathbb{N}^k\rightarrow\mathbb{N}$ is arithmetic iff its graph is arithmetic, i.e., there is a formula $\psi(\vec{x},y)$ in the language of Peano arithmetic such that for all $\vec{a}$ ...
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### Proof of cancellation law for multiplication of natural numbers.

The cancellation law for the multiplication of natural numbers is: $$\forall m, n\in\mathbb N, \forall p\in\mathbb N-\{0\}, m\cdot p=n\cdot p\Rightarrow m=n.$$ Is it possible to show this using ...
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### Must heredity be explicilty stated in addtion to Peano's axioms when defining natural numbers?

My question is stated in bold text. This question pertains to the following formulation of Peano's axioms used to formalize an introduction of the natural numbers (beginning with 1) consisting of ...
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### How would I show that PA proves the godel sentence for PA implies Con(PA)

Started off with definitions: godel sentence for PA is the sentence in PA that cannot be proved nor disproved Con(PA) formalizes PA is consistent We know godels first incompleteness theorem is ...
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### Do Gödel's incompleteness theorems apply to finitist axiomatic systems that reject the axiom of infinity?

I'm curious which axiomatic frameworks Gödel's incompleteness theorems apply. Do the theorems apply to any axiomatic system that adopts potential infinity? Or just to those that adopt completed ...
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### Prove cancellation law using peano axioms.

Using Peano axioms, prove $∀x∀y∀z(x+y=x+z→y=z)$. I have been stuck on it for some time, could someone please give a proof? Thanks!
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### Peano axioms and commutativity of addition

I'm trying to prove the commutativity of addition in Peano arithmetic in something like a natural deduction system, but am stuck halfway (I say "something like a natural deduction system" because I ...
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### Peano axioms: prove that there is no natural number between n and sucessor of n

Let $m \in \mathbb{N}$. Show that $\nexists n \in \mathbb{N}$ such that $m < n < s(m)$, where $s$ is the successor function. Here's my proof using only the Peano Axioms I was introduced. I'd ...
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### What is Complete Induction without Hypothesis?: Ordering of $\mathbb{N}$ from Peano's Axioms

The following is from Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra; Edited by H. Behnke, F. Bachmann, K. Fladt, W. Suess and H. Kunle I've used ...
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### Nonstandard proof codes for Con(PA)

Note: This question is inspired by a paper being discussed in the FoM mailing list. The formal statement Con(PA) is a statement about the existence of a proof code for the derivation of a ...
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### How to prove that first-order PA proves the consistency of each of its finite sub-theories?

The locus classicus of this theorem (the ''reflexivity'' of PA) is Mostowski's 1952 On models of axiomatic systems. I freely admit that I can't read the rather archaic formalism of this paper. Is ...
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### Is the 'copy' of the naturals in ZF a unique way to represent the second-order arithmetic in first-order logic?

The following line of thought came into me while studying introductory logic and set theory. I feel like there is an error in it somewhere, but I can't point it. It might even be correct. We have the ...
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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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### Peano axiom of induction- why cannot it be replaced by something simpler?

The Peano axiom of induction for natural numbers says that For any property $P(n)$ , if $P(0)$ holds, and that whenever $P(n)$ holds, $P(n++)$ holds, then $P(n)$ holds for all natural numbers. Can ...