All Questions
Tagged with peano-axioms analysis
22
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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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2
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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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1
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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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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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A corollary of two lemmas regarding the definition of addition of real numbers
In Terence Tao's Analysis, he mentioned that two lemmas contribute to a corollary, which I can not fully understand.
To start with, Tao defined two axioms of addition:
0 + m := m
(n++) + m := (n+m)++...
- 862
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104
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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 ...
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Which is the axiom: well ordering principle, principle of induction, both, or none?
From analysis 1 by Terence Tao, I learn that the principle of induction is a peano axiom. In many other analysis books, like analysis by Bartle and Sherbert, the well ordering principle is used to ...
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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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377
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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", ...
- 686
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347
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Proof of Edmund Landau's Foundation of Analysis [duplicate]
The theorem is as following:
The proof was split into two parts, namely: Uniqueness and Existence, I have hard time understanding the existence part:
Namely: in the 10-12th line, what did Edmund ...
- 428
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Prove if $x \neq 1$ then there exists exactly one $u$ such that $x=u'$
While I'm reading E. Landau's Grundlagen der Analysis (tr. Foundations of Analysis, 1966), I couldn't understand the proof of Theorem 3 at the segment of Natural Numbers which I've quoted below.
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How can I prove this proposition from Peano Axioms: (Cancellation law). Let a, b, c be natural numbers such that a + b = a + c. Then we have b = c.
Peano Axioms.
Axiom 2.1
$0$ is a natural number.
Axiom 2.2
If $n$ is a natural number then $n++$ is also a natural number. (Here $n++$
denotes the successor of $n$ and previously ...
- 13
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Proving distributive law of natural numbers
Is my proof correct?
If we define multiplication for natural numbers as
$a \times S(b) = (a \times b) + a$
$a \times 0 = 0$
And addition as
$a + 0 = a$
$a + S(b) = S(a+b)$
Where $S(n)$ is the ...
- 5,651
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Are my proofs correct for basic addition properties for natural numbers?
Are my proofs correct?
Additive Identity: $a + 0 = a$
Definition of Addition: $a + S(b) = S(a + b)$
where $S(a)$ is the successor of $a$.
Claim: $0 + a = a$.
Base Case: When $a=0$, we have $0 + ...
- 5,651
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2
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Peano axioms-Mathematical Induction
This is from the book we're using in my Analysis class:
The Peano Axioms of the set $\Bbb N$ are:
$1.$ Every natural number has a successor, i.e. $\forall n\in\Bbb N, \exists!s(n)\in\Bbb N$ ...
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