All Questions
Tagged with natural-numbers foundations
24
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Why does the principle of mathematical induction work for integers?
I took a course on the foundations of mathematics a while ago and we went through the construction of the natural numbers and then the integers. We did prove the principle of mathematical induction (...
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How do mathematicians know if they are extending the theory of natural numbers in the right direction?
According to Godel's incompleteness theorem, any formal system can never deduce all the truths about the set of natural numbers. Hence, to deduce more truths than we were able to before, we extend our ...
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How exactly is the Arabic numeral system defined?
I want to be able to do the following things:
Define the usual Arabic numeral system for natural numbers.
Show that every natural number has a corresponding numeral string.
Show that every natural ...
- 2,461
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Can One Discuss Induction without Sets?
The standard presentation of mathematical induction involves subsets having a certain property. Here is a typical formulation from Gallian's Contemporary Abstract Algebra, Ninth edition:
It seems to ...
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How do you define the natural numbers for different base systems?
When we define the natural numbers through set theory does it matter how the numbers are represented? For example, does the definition apply to a base 9 or base 16 system just like it would to a base ...
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Is ZFC arithmetically sound?
I apologize that this question is fairly philosophical and not purely mathematical. For the purposes of this question, I would like to take the point of view that that natural numbers are "real&...
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Why does my topology textbook (Munkres) define positive integers as the intersection of all inductive subsets of the reals?
This is how the topology textbook I'm reading (Munkres) defines integers:
A subset of the real numbers is "inductive" if it contains 1 and $1+x$ for all $x$ in the subset. The intersection ...
- 531
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What formula of ZFC defines the set of natural numbers?
Let $\mathsf{ZFC}'$ be the extension of $\mathsf{ZFC}$ containing the constant symbol $\Bbb N$, which we take to represent the natural numbers. In order to say that $\mathsf{ZFC}'$ is a definitional ...
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Peano axioms vs set construction of natural numbers
When I first looked at the construction of natural numbers the Peano axioms were shown as a way to do this, without the need of anything else, in the way described in this video and Analysis I by ...
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Existence of closures of sets
I am reading Stillwell's Elements of Algebra. And in Chapter 1, he introduces the real quadratic closure of $\mathbb Q$ as
the set of the numbers obtainable form $\mathbb Q$ by square roots of ...
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the (seeming) loop caused by defining Natural numbers through inductive sets.
I'm a first year student in mathematics, and we recently defined Natural numbers as following:
"For all numbers, a number is an element of the set Natural numbers, if and only if it is an element ...
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Constructing $(\Bbb N,+)$ via Peano function algebra duality.
In the next section we outline a $\text{ZF}$ construction of the natural numbers under addition using a 'duality' argument (the choice of the word duality is subjective and has no formal meaning).
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Why is $ℤ$ defined as $ℤ := ℕ×ℕ \text{ / ~} $, where $ \text{~} := \{ [ (a,b),(c,d) ] | a+d = b+c \} $
Why is $ℤ$ defined as $ℤ := ℕ×ℕ \text{ / ~} $, where $ \text{~} := \{ [ (a,b),(c,d) ] | a+d = b+c \} $
We already defined $ℕ$ in class as $ ℕ:=∩ \{ I | I \text { is an inductive set} \} $; we also ...
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Foundations of analysis - Natural number addition operator proof/definition
In Landau's "Grundlagen der Analysis" the author states the following proposition which at the same time is a definition.
$$\text{Proposition 4/at the same time Definition 1. There is precisely one ...
user672245
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How to construct natural numbers by set theory?
Definition 1: For any set $a$ , its successor $a^+=a\cup \{a\}$.
Informally , we want to construct natural numbers such that :
$0=\emptyset,1=\emptyset^+,2=\emptyset^{++},3=\emptyset^{+++}$,... ...
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