All Questions
Tagged with peano-axioms solution-verification
68
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Using Peano's axioms to disprove the existence of self-looping tendencies in natural numbers
Let me clarify by what I mean by "self-looping". So, we know that Peano's axioms use primitive terms like zero, natural number and the successor operation. Now, I want to prove that the ...
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Is Gödels second incompleteness theorem provable within peano arithmetic?
All following notation and assumptions follow Gödel's Theorems and Zermelo's Axioms by Halbeisen and Krapf.
Exercise 11.4 c) states "Conclude that the Second Incompleteness Theorem is provable ...
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Help me check my proof of the cancellation law for natural numbers (without trichotomy)
can you guys help me check the fleshed out logic of 'my' proof of the cancellation law for the natural numbers? It's in Peano's system of the natural numbers with the recursive definitions of addition ...
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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 ...
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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 ...
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Does this in complete detail prove $\forall x, y \in N, x + (y + 0) = (x + y) + 0$?
My goal is to prove math theorems without skipping any steps. Is this proof correct?
I googled Peano exercises
I found: http://www.public.coe.edu/~jwhite/s11/fndho5.s11.pdf
Question 1 says to prove ...
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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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Peano' s Systems
Lets consider $(A,g,a_{0})$ , where $A$ is a set, $g$ is a function and $a_{0}$ is the first element of $A$. If $(A,g,a_{0})$ follows all the Peano's axioms, than we can find an isomorphism f between $...
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Does this proof of the binomial expansion (a+b)^2 work?
I was rereading Terence Tao's Analysis 1 and found this question in the section:
Exercise $2.3.4.$ Prove the identity $(a + b)^2 = a^2 + 2ab + b^2$ for all natural
numbers a, b.
Prior to this we ...
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Is this proof of the cancellation law for natural numbers alright? [duplicate]
For starters:
$\mathbb{N_{0}}$ means $0\in \mathbb{N}$
We define the addition between natural numbers (the suggested reading for my class does) as follows: Let $m$ and $n$ be natural numbers.
\begin{...
user923938
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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 attempt
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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$\pi$ is isomorphism from one Peano system $(N, S, e)$ to another $(N', S', e')$, then $\pi^{-1}$ is isomorphism from $(N', S', e')$ to $(N, S, e)$
This is an exercise from Cunningham's book "Set Theory: A First Course".
Theorem: Let $(N, S, e)$ and $(N', S', e')$ be Peano systems. Let $\pi$ be an isomorphism from $(N, S, e)$ onto $(N', ...
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Proof critique of least number principle, please!
I am independently working through Elliot Mendelson's "Number Systems and the Foundations of Analysis," which I find very well-written and rigorous, along with providing a healthy set of ...
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