All Questions
Tagged with real-numbers axioms
76
questions
6
votes
2
answers
1k
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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
-1
votes
1
answer
76
views
If the product of two numbers is nonnegative than either both are nonnegative or both are nonpositive
In trying to prove the following inequality: $0\leq ab\Longrightarrow (0\leq a\wedge 0\leq b)\vee(a\leq 0\wedge b\leq 0)$ the following proof by contradiction was tried
Proof:
Let $0\leq ab$ and let,$...
- 131
0
votes
4
answers
109
views
How to show $a+b=b+a$ correctly?
I have to show that $a+b = b+a$ without the use of the first axiom, which states exactly this. I may use commutativity of multiplication, associativity of addition and multiplication, existence of the ...
0
votes
2
answers
870
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Are the logarithm rules and exponentiation rules (e.g. $a^{x+y}=a^xa^y$) axioms when talking about real numbers?
Are the logarithm rules and exponentiation rules (e.g. $a^{x+y}=a^xa^y$) axioms when talking about real numbers?
I know many of them can be proved via induction for integers, but no professor has ...
- 13.5k
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
8
votes
1
answer
397
views
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
0
votes
0
answers
73
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Order Axioms: '<' or '$\leq$'?
In the axioms given for the real numbers, I see that the order axioms are sometimes given for the '<' relation and sometimes for '$\leq$'. Which is more commonly used these days?
- 14.8k
1
vote
4
answers
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Prove $(-x)y=-(xy)$ using axioms of real numbers
Working on proof writing, and I need to prove
$$(-x)y=-(xy)$$
using the axioms of the real numbers. I know that this is equivalent to saying that the additive inverse of $xy$ is $(-x)y$ but I am ...
user124862
1
vote
1
answer
52
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Contrapositive of: $(x,y \in P) \implies xy \in P$, where $P$ is the set of real numbers
One of the axioms of order for real numbers read:
$$(x,y \in P) \implies xy \in P$$
where P is the set of positive real numbers.
Then, the contrapositive of this statement is:
$$\sim ((x,y \in P) \...
- 623
2
votes
1
answer
331
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a problem in Stein's book 'Real analysis', relate to continuum hypothesis.
The question is from chapter 2, problem 5 in Stein's book 'Real analysis':
There is an ordering $≺$ of $\mathbb R$ with the property that for each $y\in\mathbb R$ the set $\{x\in\mathbb R : x ≺ ...
- 2,802
0
votes
2
answers
232
views
Equality of Real Numbers
Is the following statement provable from the axioms of $\mathbb{R}$?
If $\forall \epsilon>0$, $|r-s|\leq \epsilon$, then $r=s$.
- 9,666
4
votes
1
answer
572
views
How do we decide which axioms are necessary?
I am studying the axioms for a complete ordered field. I have looked at different sources, some of which differ slightly in their listings.
Given some construction of the reals (e.g. Dedekind cuts or ...
- 9,666
3
votes
3
answers
467
views
Formal proof of: $x>y$ and $b>0$ implies $bx>by$?
Property: If $x,y,b \in \mathbb{R}$ and $x>y$ and $b>0$, then $bx>by$.
What is a formal (low-level) proof of this result? Or is this property taken as axiomatic?
The motivation for this ...
2
votes
2
answers
898
views
Proving order in real number set
Can we prove such a statement, or is it axiomatic ?
$$
\forall (x,y,z) \in \mathbb{R}^3 ∶(x \leq y)∧(y \leq z)⇒(x \leq z)
$$
- 147
1
vote
4
answers
8k
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The product of any number and zero is zero
I don't understand why this proof is valid. The proof just swaps the left and right hand side of the equality. Is this a valid method of proof for equalities? If so, what's the logical intuition ...
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