Questions tagged [peano-axioms]
For questions on Peano axioms, a set of axioms for the natural numbers.
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How do you understand $a\times1=a$ in Peano axioms?
How do you understand $a\times1=a$ in Peano axioms?
This should be understood as replacing $1$ with $a$? For example, when we multiply $3$ by $2$, that means: replace each $1$ in the number three with ...
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Provably total functions in $\mathsf{Q}$
I was interested in the relations between induction and recursion, and so a natural question (to my mind, anyway), was how much we can prove without appealing to induction, i.e. which functions are ...
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Show that $(P,S,1)$ is a Peano system if and only if it satisfies the Iteration Theorem
You can skip the first section(hints and useful things) if you want, I will keep editing this along I got more findings related to the question.
Hints and useful things: I find out that the book from ...
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Prove trichotomy law of addition in $\mathbb{N}$ (Peano Axioms).
I need help in my proof trichotomy law of addition in $\mathbb{N}$ (Peano Axioms). I have already proved that addition is associative and commutative. Also I proved the cancellation law and some ...
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How to show that a triple $(P, S, 1)$ constitutes a Peano System?
Mendelson (in Number Systems & the Foundations of Analysis) defines a Peano System as a triple $(P, S, 1)$ consisting of a set $P$, a distinguished element $1 \in P$, and a singulary operation $S :...
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$\Sigma_1$-soundness
In a theory of arithmetic like PA, "$\Sigma_1$-sound" means that it proves no false $\Sigma_1$-sentence. What is a false $\Sigma_1$-sentence in PA, does it involve TM ?
user175304
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The axiom 5 of Peano guarantees that 1 is not the successor of any natural number, however large this be?
The fifth axiom of peano is the axiom of induction, I would like to know what happens in both cases, when zero is considered natural and when it is not. If this statement is true then why consider ...
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$\forall x,y,z,u\in\mathbb{N}, (x\geq y \wedge z>u)\vee (x>y\wedge z\geq u)\Rightarrow xz>yu$ is true for all natural numbers? Even zero?
$\forall x,y,z,u\in\mathbb{N}, (x\geq y \wedge z>u)\vee (x>y\wedge z\geq u)\Rightarrow xz>yu$ is true for all natural numbers? Even zero?
I have come to the conclusion that this property is ...
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Prove that if a set (in a Peano System) is bounded above then it have a greatest element
The question is stated as: If $\emptyset \neq A \subseteq P$ and $A$ is bounded above(that is $(\exists w)(\forall u)(u \in A \Rightarrow u \leq w)$, then $A$ has a greatest element.
Here is assumed a ...
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What is wrong with this definition of a truth predicate?
Tarski's theorem, interpreted in Peano Arithmetic, says there is no predicate $T$ such that $PA\vdash T(\phi)\leftrightarrow \phi$. However, we know that there are partial truth predicates for each $k&...
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Prove the Principle of Complete Induction
The question is stated as:
Prove the Principle of Complete Induction
$$(\forall B)([B \subseteq P \land (\forall x)((\forall y)(y<x \Rightarrow y \in B)\Rightarrow x \in B)] \Rightarrow P=B)$$
(...
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Prove that the entire underlying set in a Peano System with the strict order relation($<$) forms a unique strictly ascending sequence
Original question: Prove that $1<2<3<4$,etc in a Peano System
That is the definition of Peano system by the used textbook.
Peano Systems: By a Peano System we mean a set $P$, a particular ...
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Kaye-Wong paper Theorem 6.5
In the paper On Interpretations of Arithmetic and Set Theory of Kaye and Wong they say that Theorem 6.5 (that PA can be interpreted in ZF-Inf*, that is, ZF plus every set is contained in a transitive ...
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Epsilon recursion and ZF-Inf+TC in the Inverse Ackermann Interpretation
In the paper On Interpretations of Arithmetic and Set Theory of Kaye and Wong they write:
Equipped with $\in$-induction, we obtain an inverse interpretation of
PA in ZF−Inf*. The plan is to define a ...
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Interpreting $\textbf{PA} + \neg\text{Con}(\textbf{PA})$ in $\textbf{PA}$
How does one interpret $\textbf{PA} + \neg\text{Con}(\textbf{PA})$ in $\textbf{PA}$, and what is the significance of such an aberrant interpretation?
I'm interested in the following: $\textbf{ZF} - \...
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Non-isolated types of Presburger Arithmetic
I want to show that there are $2^{\aleph_0}$ countable models of Presburger Arithmetic. Now, there is a (more or less) easy argument for this using the fact that every subset of $\mathbb{N}$ is coded ...
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Concrete examples of statements not provable in PRA + $\epsilon_0$-induction that are provable in PA?
It is well-known that $\mathbf{PRA}$ plus $\epsilon_0$-induction on bounded formulas cannot prove all $\mathbf{PA}$ theorems (essentially because $I\Sigma_1$ plus $\epsilon_0$-induction on bounded ...
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Determine whether or not the following structure $(P,S,1)$ is a Peano System
First this is how the book define as a Peano System.
By a Peano System we mean a set $P$, a particular element $1$ in $P$, and a singulary operation $S$ on $P$ such that the following axioms are ...
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What is the intersection of inductive definable subsets of a real closed field?
Let $X$ be a real closed field. Let us call a subset of $X$ definable if it is definable using a first-order formula in the language of ordered fields without parameters from $X$. And let us call a ...
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Nonstandard models of PA
Reading The Incompleteness Phenomenon, by Goldstern and Judah, they show there are nonstandard models of PA by adding a constant greater than any natural number. They then show that any countable ...
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How to add new axioms to classical Peano Arithmetic to obtain a non-standard theory.
What is a simple (the simplest?) axiom which can be added to the usual PA axioms so that the new "non-standard PA theory" no longer has the Standard Model as one of its models? Assuming of ...
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Proving that $\mathbb{Z_{-}} \cap \mathbb{N}=\emptyset$
In assumption that $\mathbb{N}$ and successor function ($\overline{x}$) over $\mathbb{N}$ is defined by 5 Peano axioms:
$1\in\mathbb{N}$
$n\in\mathbb{N} \Rightarrow \overline{n}\in\mathbb{N}$
$\...
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Function by recursion on a set $X$ satisfy Peano's axioms
I've been stuck on this theorem for like two days and I still don't really get it.
I'm reading the construction of natural numbers using "classic set theory for guided independent study", ...
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What is a set with a function $f$ that don't satisfy peano's axioms?
I'm reading "classic set theory for guided independent study", and i'm studying the construction of natural numbers using peano's axioms.
The three axioms they give me are:
A Peano system is ...
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Peter Smith's Hilbertian argument
Peter Smith in "An Introduction to Gödel's Theorems" presents a broadly Hilbertian argument (in the sense of Hilbert's program) on page 276 (2nd edition):
Theorem 37.2 If $I$ is consistent ...
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Kripke's proof of the incompleteness of PA
In Hillary Putnam's writeup on Kripke's beautiful incompleteness proof of PA, which I learned about from Noah Schweber's answer here, I can convince myself that (see the top of p.56) "$s$ ...
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Prove that for there is no $n \in \mathbb{N} \setminus \{1\}$ for which there is another $m \in \mathbb{N}$ so that $nm = 1$.
Question is in the title, here's my proof attempt:
Let $n \in \mathbb{N} \setminus \{1\}$ be arbitrary but fixed. Define the following set:
$$S = \{m \in \mathbb{N}: nm \neq 1\}$$
Let $m = 1$. Since $...
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Peano arithmetics formulae examples [closed]
Help me please to come up with an example of two arithmetic formulae $\varphi$ and $\psi$ such that $PA\vdash\varphi\vee\psi$, but neither $\varphi$ nor $\psi$ is derivable in $PA$ ($PA$ is Peano ...
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Prove that $+:\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is a binary operation
The question is in the title. The definition of $+:\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is given as follows:
$$\forall m,n \in \mathbb{N}: [n+1 := n'] \land [n+m' := (n+m)']$$
where $n'$ ...
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How do I prove this? (Set Theory, Mathematical Logic)
I'm studying for an exam of mathematical logic.
This question envolves the Peano axioms, I think.
Prove that, for all $ n \in \omega$, $ n \notin n$.
It's kind of obvious that it's true but I don't ...
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Is the axiom of induction required for proving the first Gödel's incompleteness theorem?
I'm reading a book about the mathematical logic. In the 6.3 chapter of that book, a theory $Q$ is introduced that contains precisely these axioms:
$Q1: \forall x. (S(x) \not= 0)$
$Q2: \forall x,y. (...
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Every provably recursive function in PA is bounded by a Hardy function
The following lemma and proof are from Takeuti's Proof Theory (2nd edition, pp. 126-127), I've highlighted the problematic part in blue:
How does Takeuti get this inequality? If the proof $x$ is not ...
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Characterization of provably recursive functions in PA
This concerns Takeuti's Proof Theory: the book contains a lot of wonderful material, but the presentation is sometimes lacking (and so many typos!). At least that has been my experience so far, ...
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While using the method of proof by contradiction, are we "assuming" consistency?
I am aware of threads here and here which asks something similar. However, I had something very specific to ask under the same context.
I have a very elementary question about the connection between ...
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What's the strength of weakening induction in PA by relativizing it to naturals?
If we weaken induction in PA to the following scheme:
$\forall m [\phi(0) \land \forall n < m (\phi(n) \to \phi(n+1)) \to \forall n \leq m (\phi(n))]$
Where $\phi$ is any formula in the language ...
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Proving addition preserves order in natural numbers
I have the following question:
Prove $a≥b$ if and only if $a+c≥b+c$ $\forall$ natural numbers. (using Peano Axioms)
The solution I checked is different from what I used and same applies to the ones I ...
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When can a statement in Second Order Logic be converted into a statement in First Order Logic
I was reading on first and second order logic (The first has quantifiers over individuals of the domain and the second can have quantifiers over predicates as well - see this question: Differentiating ...
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Are mathematical operations axioms?
Are mathematical operations axioms?
I will give an example of multiplication, but this also applies to division, subtraction and addition. Idea of multiplication was invented by people to increase/...
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Exercise 2.2.1 in Analysis by Terence Tao Induction Proof of Addition Associative Law Natural Numbers
The proposition 2.2.5 and also exercise 2.2.1 of Analysis by Terence Tao is as below:
Show for any natural numbers $a, b, c$, we have $(a+b)+c = a+(b+c)$ the associative rule
The proof should use ...
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Defining specific Exponentiation in PA
Given the dictionary {0,1,+,*,<} can I write a simple formula with only x occurs free, that states "x is an exponent of 6"?
When I say simple, I mean, without using complicated techniques like ...
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Show that the decidability of a statement is undecidable.
In this system of axioms, it can be shown that A = 5 is undecidable
Axioms: A is a prime number, A is greater than 4, A is less than 8
This is because both A = 5 and A = 7 are consistent with these ...
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Proving Natural Predecessor existence using Well Order
I know that the Principle of Induction is equivalent to Well Order at least in Natural Numbers, but I have seen that the demonstration of Induction using Well Order uses the existence of Predecessor:
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Prove trichotomy of order WITHOUT induction?
Claim. If $x,y \in \mathbb N$, then at least one of these is true: (a) $x>y$; (b) $x=y$; or (c) $x<y$.
It seems that the usual proof of this claim uses induction. Is it possible to prove this ...
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How do I derive fraction multiplications from Peano axioms
and sorry for the noob questions :)
Trying to teach my 10 year old daughter some math and came across the Peano axioms.
On the following resource there are two sets of axioms, one is based on symbols ...
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Is there a place which has the functional (programming) definition of Numbers in various forms?
I am familiar with the Peano Natural Number definition which is what you see in intro to functional programming courses and examples. But I have also much later encountered a practical implementation ...
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Peano arithmetic - Why is adding n to m the same as incrementing m n times?
Addition(+) is defined using the successor function(++) in Peano arithmetic as:
0 + m = m
(n++) + m = (n + m)++
While these are intuitive axioms that are consistent with my previous, elementary, ...
user487586
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Is there a natural intermediate version of PA?
Below, "logic" means "regular logic containing first-order logic" in the sense of Ebbinghaus/Flum/Thomas.
Setup
For a logic $\mathcal{L}$, let $\mathcal{PA}(\mathcal{L})$ be the ...
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How to prove $x=5$ using the Peano Axioms?
This question relates to Proposition V of Gödel's 1931 Incompleteness theorem (and another one posted on math.stackexchange here ) which says:
For every recursive relation $ R(x_{1},...,x_{n})$ there ...
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A question about Dedekind-infinite sets and Peano natural integers.
I've doubt about Dedekind-infinite sets, sets which are in bijection with a proper part, in the ZF axiomatic framework, without Axiom of Choice.
Assume a Dedekind-infinite set X exists.
Then it can ...
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A fundamental question about relations between axioms
In mathematics, there are several sets of axioms. For example, we have ZFC axioms, Peano axioms, Hilbert's axioms of Euclidean geometry(https://en.wikipedia.org/wiki/Hilbert%27s_axioms), and so on.
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