All Questions
Tagged with real-numbers axioms
10
questions
4
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2
answers
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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 ...
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9
votes
2
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838
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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
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
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2
answers
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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
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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 ...
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4
votes
1
answer
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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 ...
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3
votes
1
answer
206
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Abstracting Magnitude Measurement Systems (i.e. subsets of ${\mathbb R}^{\ge 0}$) via Archimedean Semirings.
I did some googling but could not find any easily accessible theory so I am going to lay out my ideas and ask if they hold water.
Definition: A PM-Semiring $M$ satisfies the following six axioms:
(1)...
- 11.5k
1
vote
1
answer
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How to derive the axiom no. 15 from the Cantor's and Archimedean axiom?
How could one substitute the (15th) axiom of completeness with Archimedean and Cantor's axiom?
We discussed Cantor's axiom as well as Archimedean in analysis lectures and were told this question might ...
- 4,670
0
votes
0
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Proving the well ordering principle starting from the axiom of completeness. Is this topological proof valid?
While reading this SE thread, I saw in the comments someone say "the proof [that the completeness axiom implies the well ordering principle] will take some work".
However, this other thread ...
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0
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2
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Is it possible to create the smallest real positive number by axiome?
I know that with standard math there is no "smallest positive real number". But, the same way we created Aleph Null by axiome, can we create the axiome below?
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