Questions tagged [peano-axioms]

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

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Model Theory in the Language of Peano Arithmetic

Most introductory textbooks on model theory establish the theory based on the ZF set theory (e.g. [1]). In particular, a structure is defined to be a 4-tuple of sets, and so on. In [2], I came to ...
Student's user avatar
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Why Löwenheim–Skolem theorem asserts the non-existence of such predicates in 1st order logic

Suppose there was a predicate, in the language of 1st order $ \mathsf {PA} $, such that it is only true for standard natural numbers i.e. it accepts ALL and ONLY standard natural number, and it ...
Alex Matyasaur's user avatar
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Confusion about $\mathsf{PA}$'s self-provable consistency sentences

Edit: This long question was basically answered by a quick comment! I'll accept an answer if someone posts one, but it's basically answered already. Background: In Peter Smith's Introduction to Gödel'...
WillG's user avatar
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Infinite statements from finite axioms

I want to know if a given finite subset of axioms of PA1 ( 1st order peano arithmetic ) can prove infinite sentences in PA1 such that those proofs need no other axioms except those in the given finite ...
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is arithmetic finitely consistent? [duplicate]

Let's take PA1( First order axioms of peano arithmetic ) for example. From godel's 2nd incompleteness theorem, PA1 can't prove its own consistency, more specifically it can't prove that the largest ...
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Modern reference on PA degrees?

I'm currently trying to work my way around some papers from Jockush et al, and PA degrees come up frequently. I'd be interested in a modern reference/survey summarizing the main results on the subject,...
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The Peano Axioms in Polish Notation

I am new to Polish Notation, and would like someone to translate the Peano axioms into PN for me. Either the first order or second order axioms would do, but if you can do both that would be much ...
Anthony Khodanian's user avatar
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Are there examples of statements not provable in PA that do not require fast growing (not prf) functions?

Goodstein's theorem is an example of a statement that is not provable in PA. The Goodstein function, $\mathcal {G}:\mathbb {N} \to \mathbb {N}$, defined such that $\mathcal {G}(n)$ is the length of ...
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Peano axioms - do we need a specific property to show that the principle of mathematical induction implies the "correct" set of natural numbers?

From Terence Tao's Analysis I, Axiom 2.5 for the natural numbers reads My intuition behind this axiom is that every natural number is an element of a "chain" of natural numbers that goes ...
jvf's user avatar
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Formally how do we view finite sets

This might be silly, but I have been thinking about how we would work with finite sets very formally. So, $\{1,2,3,\cdots,n\} = \{k \in \mathbb{Z}^+ \mid k \leq n\}$ gives a representee for which any ...
MathNerd23571113's user avatar
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Confusion about Löb's theorem [duplicate]

To quote wikipedia: Löb's theorem states that in any formal system that includes PA, for any formula P, if it is provable in PA that "if P is provable in PA then P is true", then P is ...
G. Bellaard's user avatar
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Is pointwise definability of a model of PA equivalent to it being the standard model? [duplicate]

The standard model of Peano Arithmetic is pointwise definable, because every finite natural number is parameter-free definable. What about the converse? That is, if a model $M$ of PA is pointwise ...
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What are the parameter-free definable elements of a model of Peano Arithemetic?

Let $M$ be a model of Peano Arithmetic. What are the parameter-free definable elements of $M$? I conjecture that they are precisely the standard natural numbers, meaning, no nonstandard infinite ...
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Non recursive provably total functions

It is provable that every primitive recursive function is total in 1st order PA. Some non primitive recursive functions are also provably total in PA. Can we show that a function is totally provable ...
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Meaning of Provable recursiveness

Is there any difference between provably total function and provable recursiveness of a function in first order PA ? From provably total I mean that the totality of the function itself is provable in ...
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Peano axioms-Analysis I by Terence Tao

Statement: Terence Tao in his book Analysis I states that the set N = {0,0.5,1,1.5,2,...} satisfies peano axioms 1 to 4. Axiom 2: if n is a natural number, n++ is also a natural number Definition 2.1....
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Is "standard $\mathbb{N}$" in fact not "fully formalizable"?

Note: "Update" at the end of this question hopefully summarizes/clarifies the original language (original text left in place for context). Philosophical Preface: For the purposes of this ...
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Proving the set of true expressions in a theory cannot be expressed in this theory

Suppose we have a first-order theory $T_C$ that includes a binary function $C$. $C$ is a bijection $\Sigma \to \mathbb{N}$ where $\Sigma$ is the alphabet of $T_C$. The function $C$ is defined as ...
ampersander's user avatar
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Is a set $T$ containing all natural numbers identical to $\mathbb{N}$?

In this comment, I see that Mauro Allegranza and I were saying the same thing as shown in the table below: Allegranza Zeynel $0 \in T$ $0 \in T$ $n \in T$ $n \in T$ $S(n) \in T$ $S(n) \in T$ $ T =...
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How to find what a definition defines?

Are these definitions of the "+" and "mod" operators? $m+0=m$........(1) $0 \;\text{mod} \;2 = 0$.....(2) To me, (1) defines the identity property of zero and (2) defines zero as ...
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There are no loops in a Peano system. I've attempted a proof. Is my proof correct?

Definition (Peano systems). Suppose $P$ is a set, $1 \in P$, and $S: P \to P$ is a function. The triple $(P, S, 1)$ is a Peano system if the following conditions hold. (P1) $\forall x (1 \neq S(x))$, ...
Mostafizur Rahman's user avatar
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How to prove commutative from Peano axioms WITHOUT proving associativity first?

I am reading this https://en.wikipedia.org/wiki/Peano_axioms#:~:text=The%20Peano%20axioms%20define%20the,0%20is%20a%20natural%20number. They use S(x) for successor ...
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Is Russell's proof of addition with Peano's 5. Axiom valid?

This is a follow up question to my previous question: Why define addition with successor? In this one I'd like to ask about Russell's use of Peano's 5. Axiom to prove his definition of addition: ...
zeynel's user avatar
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On the consistency of satisfiable first order theories

Considering this question, we know that a first order theory that admits a model has to be consistent. A model for a theory $T$ in a language $\mathcal L$ is an interpretation of $\mathcal L$ in which ...
Spasoje Durovic's user avatar
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3 answers
566 views

Why define addition with successor?

I'm reading Russell's Introduction to Mathematical Philosophy Russell defines the sum of two numbers in terms of successors. I don't understand why: Suppose we wish to define the sum of two numbers. ...
zeynel's user avatar
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Peano’s Fifth axiom as stated by Russell

I’m translating Russell’s Introduction to Mathematical Philosophy. I’m having difficulty understanding his formulation of the fifth axiom: (5) Any property which belongs to 0, and also to the ...
zeynel's user avatar
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On the uniqueness of the addition operation on $\mathbb{N}$

My textbook (Amann and Escher, Analysis I) gives a theorem which says that the operations of addition and multiplication (and a partial order $\leq$) exist and are uniquely defined by a whole host of ...
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Why universal closure?

In logic, why do we talk of universal closure of a formula, and don't consider its "existential closure" (as far as I know)? I guess that one of the reasons may be that interesting systems ...
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Is my understanding of Peano Axioms as mentioned below correct? I’ll also be grateful if questions below are answered definitively [closed]

I have concluded the reading of second chapter of Prof. Tao’s Analysis books in which he covers natural numbers and defines addition and multiplication operation on them, He states the following ...
Quorthon's user avatar
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1 answer
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What is 'increment' in Peano Axioms?

I am reading Tao's book on Analysis in which the first two axioms apropos natural numbers are, 0 is a natural number. If n is a natural number, then n++ is also a natural number. As a motivation ...
Quorthon's user avatar
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Irregular Induction Theorem for $\mathbb{N}\times\mathbb{N}$

I am trying to prove this irregular induction theorem that would help prove a recursion theorem I am working on. Can you help? Here is the theorem: $\forall X (\forall x \in \mathbb{N} (\langle x,0 \...
Isaac Sechslingloff's user avatar
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Are the axioms of analysis a combination of Peano axioms and set theory axioms?

Observing the use of mathematical induction in proving the finite version of the axiom of choice, I began to ponder. Why is mathematical induction from Peano axioms employed to prove facts about sets? ...
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How to prove natural number addition using induction? [duplicate]

I am a self learner so excuse me if I am asking a seemingly easy question , But I ve been stuck at this point for couple of days , I think I understand mathematical induction and what the author is ...
skipping tutorial's user avatar
3 votes
1 answer
88 views

Is there a concept of finiteness independent of the successor function?

Why is there no infinite natural number, and why does finiteness need to be closed under the successor function? I think can understand why something like $…S(S(0))…$ is not a natural number because ...
Teddy Astor's user avatar
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Gödels incompleteness theorem false for natural numbers

Do I understand it correctly that in an axiomatic system that includes the natual numbers defined by using Peano's axiom, where the 9th axiom (induction) is formulated using 2nd-order logic, then ...
Axel Bregnsbo's user avatar
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1 answer
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Does the property $T\vdash Pvbl_T(\ulcorner \sigma \urcorner) \implies T\vdash \sigma$ apply to set theories?

I know from other posts that $PA\vdash Pvbl_{PA}(\ulcorner \sigma \urcorner ) \implies PA\vdash \sigma$ and this applies to other extensions/restrictions of PA as well. Does it also apply to set ...
Ari's user avatar
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Analysis I, can Tao's construction of the integers be further simplified?

In chapter 4 of Analysis I by Terence Tao, we have the following note about the set theoretic construction of the integers: In the language of set theory, what we are doing here is starting with the ...
HJE's user avatar
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1 answer
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Circular reasoning when explaining positional notation

I am trying to explain from scratch the foundation of mathematics and couldn't start anywhere but from Peano's axioms. After that I introduced the set of digits = {1, 2, 3, 4, 5, 6, 7, 8, 9} where ...
Alessandro's user avatar
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1 answer
144 views

On the Axiomatic Foundation of Elementary Number Theory

I am under the impression that there is a set of axioms on which elementary number theory unfolds. If this is true, what are the axioms? Are they the five Peano axioms (at least, thought there were ...
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Showing that adding by $1$ is the same as the successor function

In the answer to this question I asked previously, it was stated that I can identify the successor function $a \to a^+$ with $a+1$. I finished reading the section in Jacobson's Basic Algebra I and ...
Richard K Yu's user avatar
2 votes
1 answer
260 views

Is my proof of $1+1=2$ correct?

Here is the proof: Note: I will denote the successor of a natural number $n$ by $(n++)$ If one assumes the Peano axioms then they may define addition as follows: $0+m:=m$ $(n++)+m=(n+m)(++)$ $\forall ...
Person's user avatar
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1 vote
1 answer
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Are the natural numbers definable in ZFC-Inf

While reading up on the incompleteness theorems on this page, I came upon the statement that the incompletess theorems hold in ZFC-Inf. According to the page ZFC-Inf is 'proof-theoretically equivalent'...
leon.fuchsler's user avatar
3 votes
3 answers
174 views

If Goldbach's conjecture G is undecidable in PA, then can we prove $\mathbb N\models G$?

Suppose Goldbach's conjecture $G$ is undecidable in first-order Peano arithmetic, $\sf{PA}$. That would mean there are models in which $G$ is true and other in which it is false. But intuitively, this ...
WillG's user avatar
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4 votes
1 answer
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Representability of Goodstein function in PA

I have a doubt concerning the representability of Goodstein's function in Peano Arithmetic ($PA$). Specifically, the function $G(n) =$ “The length of the Goodstein sequence starting from $n$” is ...
Keplerto's user avatar
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Peano axioms and meaning of successor map in Jacobson's Basic Algebra I

From Jacobson's Basic Algebra I on P. 16, the Peano axioms are stated as: $0 \neq a^+$ for any $a$ (that is, $0$ is not in the image of $\mathbb{N}$ under $a \to a^+$). $a\to a^{+}$ is injective. (...
Richard K Yu's user avatar
3 votes
1 answer
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Role of Induction among Peano Axioms

I am working with Peano Axioms as follows: $0 \in \mathbb{N}$. $n \in \mathbb{N} \implies S(n) \in \mathbb{N}$. $(\forall n \in \mathbb{N})(S(n) \neq 0)$. $n \neq m \implies S(n) \neq S(m)$. Let $A$ ...
Promethèus's user avatar
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1 answer
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Can we replace these two Peano axioms with this single axiom?

Peano Axioms from Mathworld: Zero is a number. If $a$ is a number, the successor of $a$ is a number. zero is not the successor of a number. Two numbers of which the successors are equal are ...
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1 answer
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Proof of Recursive definition, Analysis 1 by Terence Tao.

I got Proof of a proposition regarding recursive definitions (from Terence Tao's Analysis I) Here i understood that what tao done in the proof. But still i have some confusion. Question: Why ...
Afzal's user avatar
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2 answers
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Does every well-ordered set obeying the non-induction Peano axioms have a well-ordering compatible with the successor operation?

Let $N$ be a well-ordered set together with a unary operation $s$ that obeys the following axioms (they are just the Peano axioms without induction): $0 \in N$ for each $n \in N$ we have $s(n) \in N$ ...
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Why do we require that $1 \in A$ in defining n-admissible functions?

In the proof of the Iteration Theorem in Eliot Mendelson's book Number Systems and the Foundations of Analysis (1973), page 57, the author defines a function $f: A \to W$ to be n-admissible iff: $A \...
Mostafizur Rahman's user avatar

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