All Questions
9
questions
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Tarski axioms of real numbers
How does the Tarski axioms of real numbers imply that for each x,y,z ( x<y if and only if x+z < y+z ) ?
By using the 1st and 6th axioms it's easy to demonstrate that x+z<y+z implies x<y. ...
- 131
2
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2
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836
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How to properly define completeness of a set
A few weeks back I picked up a 1960 copy of General Theory of Functions and Integration by Taylor at a half price bookstore.
I started reading this and got up to the definition of completeness of an ...
- 3,105
3
votes
2
answers
913
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Confusion in least upper bound axiom
Least upper bound axiom: Every non-empty subset of $\mathbb R$ that has an upper bound must have a least upper bound.
This sounds too obvious as it works for both closed and open subsets of $\mathbb ...
- 1,141
1
vote
1
answer
149
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Why $x=x$? Why $x=y$ and $y=z$ imply $x=z$? (Assume, $x$, $y$ and $z$ are reals.)
I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body ...
2
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80
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Is an equivalence relation (= sign) needed for the real number system or is a consequence of the other axioms?
My math education is based on Calculus or Real Analysis didactical books intended for bachelor's degrees, mainly read in chunks, and never went any further.
Generally, the Real Number System is said ...
- 311
4
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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
4
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2
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1k
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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 ...
- 5,016
8
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1
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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
4
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1
answer
572
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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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