All Questions
Tagged with peano-axioms real-analysis
49
questions
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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 ...
- 283
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2
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37
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Confusion about the validity of the proof of Trichotomy of order for natural numbers in Tao's Analysis
It's well-known that in Tao's Analysis I P28, he provides a provement of Trichotomy of order for natural numbers as follows.
Denote the number of correct propositions among the three (i.e. $a<b,\ ...
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0
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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 ...
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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 ...
- 11
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266
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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 ...
- 1,123
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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 ...
- 587
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1
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202
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Fifth Peano axiom — Properties of the natural numbers
This question is kind of a follow-up question to this. I am also using Terence Tao's book and I still struggle to understand why the fifth Peano axiom is valid.
Tao defines the fifth axiom in the ...
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Landau Foundations of Analysis Axiom 4: Is it necessary?
Landau gives 5 axioms as the foundations for deriving the theorems in the first chapter:
Axiom 1: 1 is a natural number.
Axiom 2: If $x = y$ then $x' = y'$.
Axiom 3: 1 is not a successor to any ...
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1
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53
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Proof that the elements are distinct with Peano's axioms.
Consider the function successor function $s: \mathbb{N} \to \mathbb{N}$ and the Peano's axioms:
P1) $s: \mathbb{N} \to \mathbb{N}$ is injective.
P1) $\mathbb{N} \setminus s(\mathbb{N})$ has only one ...
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Prove the sum of two natural numbers is again a natural number, using the Peano Axioms.
I'm re-learning real analysis and decided to start from Tao's books (sorry Rudin) and Tao left a remark stating we can prove the sum of two natural numbers is again a natural number by the Peano ...
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Construction of the addition function
I am reading a book called Analysis I by Herbert Amann and Joachim Escher. I am currently stuck on page 33 where they construct the addition operator using functions. One property the addition ...
- 325
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58
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$s(n)=n+2$ and Peano axioms
Define $s:\mathbb{N}\rightarrow \mathbb{N} $ given by $s(n)=n+2$, with $n\in\mathbb{N}$. Prove that $\mathbb{N}$ and $s$ satisfy
every $n\in\mathbb{N}$ has only one sucessor and $s$ is one-to-one.
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- 511
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2
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284
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Addition of natural numbers in Edmund Landau's Foundation of Analysis
I am reading the proof of addition of numbers.
In the proof author first shows uniqueness of $x+y$ and then the existence of plus operation with the above listed properties.
The second proof is as ...
- 273
2
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2
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117
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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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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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