All Questions
Tagged with real-numbers axioms
76
questions
22
votes
1
answer
1k
views
Prove all 4 axioms of "less than" are necessary (for real numbers)
One way to define an ordered field is as a field $F$ with a relation $<$ that satisfies:
For all $x,y \in F$, exactly one of $x<y$, $x=y$, $y<x$ holds.
For all $x,y,z \in F$, if $x<y$ and ...
- 145k
10
votes
5
answers
5k
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Foundation of ordering of real numbers
This might be a silly question, but what is the mathematical foundation for the ordering of the real numbers? How do we know that $1<2$ or $300<1000$... Are the real numbers simply defined as ...
user116403
9
votes
2
answers
838
views
axioms of real numbers without multiplication
Consider the axioms of real numbers https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach
and suppose we remove the multiplication operation and its properties. Do we loose something?
I have ...
- 21.6k
8
votes
2
answers
237
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Is this part of axiom superfluous?
In "Analysis with an introduction to proof" (5th ed.) by Steven R. Lay, the existence of a set $\mathbb{R}$, and two binary operations $+$ and $\cdot$, satisfying 15 axioms is assumed.
The ...
- 2,681
8
votes
1
answer
397
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How can the axioms (and primitives) of Tarski's axiomatization of $\Bbb R$ be independent?
While reading through this Wikipedia page about Tarski's axiomatization of the reals, a particular bit of text jumped out at me:
Tarski proved these 8 axioms and 4 primitive notions independent.
...
- 104k
7
votes
4
answers
11k
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prove that if $a=b$ then $a+c=b+c$ where $a,b,c\in \mathbb R$
I was trying to prove if $l=m$ and $m=n$ then $l=n$ but when doing this I had to add $-m$ to both sides of both equations.i think it is not appropriate to proceed without proving "if $a=b$ then $a+c=b+...
- 657
7
votes
0
answers
701
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Geometric basis for the real numbers
I am aware of the standard method of summoning the real numbers into existence -- by considering limits of convergent sequences of quotients.
But I never actually think of real numbers in this way. I ...
- 2,146
6
votes
2
answers
1k
views
On the relation of Completeness Axiom of real numbers and Well Ordering Axiom
In my abstract algebra book one of the first facts stated is the Well Ordering Principle:
(*) Every non-empty set of positive integers has a smallest member.
In real analysis on the other hand one ...
- 4,395
5
votes
5
answers
590
views
Spivak's proof of $a\cdot 0 = 0$
I picked up Spivak's Calculus (3rd Edition) today and it seemed like a good idea to go through the section Basic Properties of Numbers. In this chapter, Spivak proves that
$$a \cdot 0 = 0$$
The ...
- 575
4
votes
2
answers
1k
views
Axiomatic definition of the real numbers and uncountability
There are several approaches in defining the real numbers axiomatically and suppose that we have some set of axioms $A$ which completely characterize rational numbers and which does not mention ...
- 5,016
4
votes
1
answer
322
views
The axiom of regularity and the real numbers
I'm having trouble understanding how there can be sufficiently many distinct elements for $\mathbb{R}$ to exist with its properties and yet still be a set.
(The ...
- 151
4
votes
1
answer
572
views
How do we decide which axioms are necessary?
I am studying the axioms for a complete ordered field. I have looked at different sources, some of which differ slightly in their listings.
Given some construction of the reals (e.g. Dedekind cuts or ...
- 9,666
4
votes
1
answer
105
views
Completeness Axiom of $\mathbb{R}$.
I use the following as the axiom of completeness of the reals $\mathbb{R}$:
$$\forall X,Y\in \mathcal{P}(\mathbb{R})\backslash\{\emptyset\}: (\forall x\in X\quad\forall y\in Y: x\leq y) \implies \...
- 77
4
votes
1
answer
204
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Can uncountability of reals be proved only from the axioms?
If we define real numbers, as is sometimes done, with field axioms, and order axioms, and completeness (or continuity) axiom, then, rational numbers fulfill field axioms and order axioms, but they do ...
- 1,802
4
votes
1
answer
623
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The axiomatic method to real number system VS the constructive method(genetic method)
According to book Georg Cantor: His Mathematics and Philosophy of the Infinite -
Joseph Warren Dauben , David Hilbert claimed that the axiomatic method to real number system is more secure than the ...
- 2,267