Linked Questions
10 questions linked to/from isomorphism of Dedekind complete ordered fields
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What's the proof that the only Dedekind-complete field is the reals? [duplicate]
I know that the field of the rational numbers is ordered but not Dedekind-complete. What's the proof that the only Dedekind-complete field is the reals?
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Is the real number structure unique?
For a frame of reference, I'm an undergraduate in mathematics who has taken the introductory analysis series and the graduate level algebra sequence at my university.
In my analysis class, our book ...
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Are different constructions of an algebraic structure always isomorphic?
Any two complete ordered fields are isomorphic (as proved, e.g., in Spivak's Calculus; see also this question).
While I understand this proof, I cannot yet appreciate why it is necessary. Given any ...
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What does isomorphism mean in "$\mathbb{R}$ is the Dedekind-complete ordered field up to isomorphism"?
This is an embarrassing question, because I learned about this theorem in basic analysis, but haven't realized that I don't really understand its statement until now.
Anyway, it's a famous result ...
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Do there exist another $\mathbb{R}$?
Can we find a set other the $\mathbb{R}$ satisfying all the field axioms, order properties and completeness axiom?
By another set I mean, it differs from $\mathbb{R}$ may be in terms of topology, ...
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Real numbers axiomatization without natural numbers
I think to remember that there is way to uniquely characterize the real numbers $\mathbb{R}$ via an axiom set. I wonder if this is possibly without introducing some notion of the natural numbers $\...
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2
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Uniqueness of the real line
A few days ago, I came across this question in a review queue. I tried my luck at it. Here is what I did:
If I want a homomorphism (isomorphism, but even just homomorphism) $f:\mathbb{R}\to F$, then ...
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How can it be true both that Complete Ordered Fields are unique up to isomorphism and that anything that can prove Peano Arithmetic is incomplete?
The real numbers include the natural numbers which presumably satisfy the Peano axioms, I don't know how you could be a strong enough theory to prove the existence of a set that satisfies the Peano ...
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Can the so-called completeness of real numbers be understood as closure under limits in the real number system?
Source of background information：《The Real Analysis Lifesaver》ISBN:9780691172934
P37: “the axiom of completeness”—here, completeness is just another word for the least upper bound/greatest lower ...
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Characterizations of the reals
I know that one characterization of the reals is that it is the only Dedekind-complete ordered field. Are there any other characterizations of the reals as a field?
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