Tagged Questions
1
vote
1answer
50 views
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 ...
20
votes
6answers
566 views
Axiomatic characterization of the rational numbers
We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this).
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3
votes
1answer
324 views
Proving $1 > 0$ using only the field axioms and order axioms
How do I prove $1 > 0$ using only field axioms and order axioms? I have tried using the cancellation law, with the identities in a field and I cannot get anywhere. Does anybody have any ...
5
votes
4answers
150 views
Why would the author ask if I used the Associative Law to prove + is not equiv. to *?
I just started reading An Introduction to Mathematical Analysis by H.S. Bear and problem 1 goes as follows:
Problem 1: Show that + and * are necessarily different operations. That is, for any ...
2
votes
3answers
108 views
Show $xy\neq0$ is the same as $x\neq0 \wedge y \neq0$
I have to show:
$$xy\neq0 \Leftrightarrow x\neq0 \wedge y \neq0 $$
I think I can "simplify" it to this:
$$xy=0 \Leftrightarrow x=0 \vee y=0 $$
Since $a\cdot0=0$ is an proven theorem, I can show:
...
0
votes
6answers
465 views
Is it possible to have a field without an additive identity?
If I drop the axiom that Zero is the identity of an addition what consequences does this entail? What do I need to change to my axiomatization?
By definition it is not possible, but are there ...
