All Questions
Tagged with peano-axioms proof-writing
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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 ...
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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 \...
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Proof of $\forall y(Sx + y = S(x + y))$ in Peano Arithmetic.
I am following a proof of $\forall y(Sx + y = S(x + y))$. The base case is $Sx + 0 = S(x + 0)$, and for its proof the author says this:
"If $y = 0$, note that by PA2 ($\forall x x + 0 = x$) we ...
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Explicit Substitution in Peano Arithmetic proofs.
I am creating a proof of $S0 \times SS0 = SS0$ in Peano Arithmetic, and a big part of my proof is finding equivalences and then substituting in to relevant formulae. Now I know I can do that with the ...
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Proof in Peano Arithmetic
I am trying to prove $S0 \times SS0$ using the axioms of Peano Arithmetic. The axioms are:
$\forall x \hspace{0.05cm}0 \ne Sx$
$\forall x \forall y \hspace{0.05cm} (Sx = Sy \rightarrow x = y)$
$\...
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Building product in $\Bbb N$ using the function $s: n\mapsto n+1$
using the Peano's axioms we can give a description of the set of natural numbers.
Let's consider the functions
$s: \Bbb N\to \Bbb N$ defined by $s(n)=n+1$
and
$f^n=\begin{cases}
id, &\text{ if } n=...
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How do I prove that the addition of natural numbers, as defined by Peano, is unique?
In other words, for any two natural numbers, there exist no more than one natural number that equals the sum of the two numbers. Or rather, for any two natural numbers, their sum is unique.
In first ...
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Assuming the following definition of the addition of natural numbers, how do I prove that $\forall a:0+a=a$
Natural numbers and the succesor function S are defined according to the Peano axioms.
Addition is defined recursively (DIFFERENTLY from the traditional Peano definition, though I am trying to prove ...
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How to prove that $\forall a \forall b \exists x(a+x=b \lor b+x=a)$
The universe is the set of natural numbers including 0, defined by the Peano Axioms. I tried and failed to prove this by induction on b.
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How to prove the following statement regarding the successor function and addition of natural numbers?
Natural numbers (including 0) and the successor function are defined as per the Peano Axioms (you can check them on wikipedia). Addition is defined recursively as follows:
$a+0=a$
$a+S(b)=S(a)+b$
With ...
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If I define addition in the following way, how can I prove that it's commutative?
$a+b=a$, if $b=0$
$a+b=S(a)+S^{-1}(b)$, if $b\not=0$
Here a and b are natural numbers defined according to the Peano axioms, while S represents the successor function.
Basically, I am trying to prove ...
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How to prove that there are $n$ natural numbers that are less or equal than $n$ and what properties are allowed to use in induction.
Let $n \in \mathbf{N}$. I wondered how to prove that there are exactly $n$ natural numbers that are smaller or equal than $n$. This seems somewhat circular which confuses me. I guess the way to do ...
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For every $a$ in $\mathbb N^+$, prove that there exist a natural number $b$ such that $b++=a$.
The question is listed above. The question is that whether$\forall a\in \mathbb{N^+}, \exists b\in \mathbb{N} \,,s.t.a=b++$ is right.
My question is, in my proof, I’ve tried to use the mathematical ...
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peano arithmetic proof in fitch
I've been tasked with proving that any natural number times the successor of zero is equal with that natural number. I've been trying to solve this problem using induction in the Fitch proof system, ...
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Peano axioms proof
Given that $K$ is an ordered field satisfying the least upper bound property and $1$ as the multiplicative identity, the set of natural numbers $\mathbb{N}_K$ in $K$ is defined as: $1 \in \mathbb{N}_K$...
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