# Tagged Questions

For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.

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### Defining the integers and rationals

What axiomatic system is most commonly used by modern mathematicians to describe the properties of the integers and the rationals? Properties like $a+0=a$, $a*1=a$, $a+b=b+a$, Also given these ...
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### Scotts axiom, representation theorems for Qualitative -Numerical Probability function relations

Scotts theorem/axiom and other representation theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, ...
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### Necessity of being rigorous

Disclamer I am no serious mathematician, just curious Context I recently discovered some set theories, ZFC and IZF in particular. It made me realize that I've studied math a whole year without ...
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### Definition: Mathematical way to define the “left” and the “right”

How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has ...
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### Axioms of Geometry?

I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it ...
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### Inner Product proof, axiom 1

I hate to argue that my text book is wrong. That being said, I am going to try and do just that. The text book says that this IS a valid inner product, I disagree. The vectors u and v are defined ...
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### Is the Bourbaki treatment of Set Theory outdated?

On page 5 of the following write up, the author asks why the Bourbaki did not notice that their system of Zermelo set theory with AC was inadequate for existing mathematics. Throughout the rest of the ...
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### Prove axiom of infinity from ZF$-$Infinity +Zermelo's version of infinity

Assume ZF-infinity and assume that there is a set $X$ such that $\varnothing\in X$ and for all $t\in X$, $\{t\}\in X$. How to deduce axiom of infinity from these new axioms?
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### Choice function without AC in special case [duplicate]

I read the Jech, Set theory, and saw following proposition. (☆) If S is a finite family of nonempty sets, existence of choice function of S can be proved without axiom of choice. I tried to prove ...
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### Proof using only field axioms

Prove if $x, y ∈ R$ and $xy > 0$ then either $x > 0$ and $y > 0$, or, $x < 0$ and $y < 0$ using only the field axioms. These include the Field axioms for addition, multiplication, ...