All Questions
Tagged with peano-axioms natural-numbers
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Axiomatic reason why $a=4 \implies a>1$ for $a \in \mathbb{N}$
This is a trivial task:
Given $a \in \mathbb{N}$ and $$a=4$$
Show $$a > 1$$
Part of the challenge for newcomers like me is that "easy" tasks actually make it harder to think about the ...
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Proof that each natural number has a unique successor
I've proven that every positive natural number has a unique predecessor using Peano's axioms. But now, I was wondering how I could prove that every natural number has a unique successor using the same ...
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Can we modify the Peano axioms like this? [closed]
I am wondering if the following modifications of the Peano axioms result in a set of axioms equivalent to the Peano axioms, in the sense that any set of numbers satisfies these modified axioms if and ...
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Proving that the set of non-negative half-integers satisfies Peano's axioms
I postulate that the following set $\{0,0.5,1,1.5,...\}$ represents the natural numbers. Of course, intuitively, this isn't true. But let me try to show this using Peano's axioms. I'll first define ...
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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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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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Peano axioms-Analysis I by Terence Tao
Statement: Terence Tao in his book Analysis I states that the set N = {0,0.5,1,1.5,2,...} satisfies peano axioms 1 to 4.
Axiom 2: if n is a natural number, n++ is also a natural number
Definition 2.1....
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Is "standard $\mathbb{N}$" in fact not "fully formalizable"?
Note: "Update" at the end of this question hopefully summarizes/clarifies the original language (original text left in place for context).
Philosophical Preface: For the purposes of this ...
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On the uniqueness of the addition operation on $\mathbb{N}$
My textbook (Amann and Escher, Analysis I) gives a theorem which says that the operations of addition and multiplication (and a partial order $\leq$) exist and are uniquely defined by a whole host of ...
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Is my understanding of Peano Axioms as mentioned below correct? I’ll also be grateful if questions below are answered definitively [closed]
I have concluded the reading of second chapter of Prof. Tao’s Analysis books in which he covers natural numbers and defines addition and multiplication operation on them,
He states the following ...
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What is 'increment' in Peano Axioms?
I am reading Tao's book on Analysis in which the first two axioms apropos natural numbers are,
0 is a natural number.
If n is a natural number, then n++ is also a natural number.
As a motivation ...
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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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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 ...
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How do Peano's axioms make it clear what the successor is equal to?
Most likely this subject has been covered many times here, still I fail to grasp this.
I can't understand how do we know that the successor of $1$ is $2$ based on Peano's axioms, given that we start ...
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What is the mathematical definition of "standard arithmetic/standard natural numbers"?
As a consequence of Godel's incompleteness theorem, no axiomatic system can be the definition of standard natural numbers because any axiomatic system of arithmetic will always be satisfied by non-...
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