2
votes
4answers
74 views

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 ...
2
votes
1answer
85 views

how to show associativity of multiplication for not just 3 operands but for n operands

ie Id like to show a(bc)=(ab)c but for any n operands eg abcdefg=gfdcabe etc I can see this is very intuitive that this should be true for all n operands, but as a logical exercise I would like to ...
1
vote
2answers
179 views

How can we know arithmetical axioms are consistent?

If we assume both distributivity and the opposite of the law of signs (ie, that $-1\times-1 = -1$) for the relative integers, then we can derive that two different numbers are actually equal. ...
14
votes
2answers
3k views

What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?

Ever since I learned the definition of a ring, I've wondered why the additive group is required to be abelian. What happens if we allow $\langle R, + \rangle$ to be nonabelian as well as $\langle R, ...
3
votes
1answer
623 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 ...
1
vote
2answers
95 views

Is using the - symbol with the Associative Law of multiplication invalid?

I was trying to prove that $-(x + y) = -x - y$ and as you can see in the image below, I took the liberty of using the $-$ symbol as a number and applying the associative law with it. Is it kosher in ...
5
votes
5answers
250 views

Axiomatization of $\mathbb{Z}$

Though I've seen several cool axiomizations of $\mathbb{R}$, I've never seen any at all for $\mathbb{Z}$. My initial guess was that $\mathbb{Z}$ would be a ordered ring which is "weakly" well-ordered ...
10
votes
4answers
448 views

Axiomatic approach to polynomials?

I only know the "constructive" definition of $\mathbb K [x]$, via the space of finite sequences in $\mathbb K$. It essentially tells a polynomial is its coefficients. Is there a way to define ...
2
votes
1answer
208 views

Which algebra satisfies this?

Do you know if the following set of rules has an algebra more general than the usual complex numbers? Maybe someone can also help me state the rules in some mathematically rigorous (fancy :) ) way so ...
8
votes
5answers
737 views

Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms?

Context: Rings. Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms? Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...