Intereting Posts

Derive $\frac{d}{dx} \left = \frac{1}{\sqrt{1-x^2}}$
Conics definition from Lens formula
If $A$ is a Dedekind domain and $I \subset A$ a non-zero ideal, then every ideal of $A/I$ is principal.
Bases are looping using simplex method
Is it mathematically valid to separate variables in a differential equation?
Sufficiency to prove the convergence of a sequence using even and odd terms
Multivariable calc “second course” that does differential forms
Constructing a local nested base at a point in a first-countable space
Is the union of finitely many open sets in an omega-cover contained within some member of the cover?
Find $a, b, c, d \in \mathbb{Z}$ such that $2^a=3^b5^c+7^d$
What would happen if ZFC were found to be inconsistent?
Calculate half life of esters
How can I find a curve based on its tangent lines?
Calculate Camera Pitch & Yaw To Face Point
How to solve $n < 2^{n/8}$ for $n$?

Given a family of sets $(A_i : i \in I)$, we define the disjoint union:

$$\sum_{i \in I} A_i = \bigcup_{i \in I} (\{i\} \times A_i).$$

There is a surjection $\sum_{i \in I}A_i \to \bigcup_{i \in I} A_i$ given by $(i,a) \mapsto a$, so by the Axiom of Choice there is an injection $\bigcup_{i \in I} A_i \to \sum_{i \in I}A_i$.

My question is whether AC (or some fragment of it) is required to prove that there is an injection $\bigcup_{i \in I} A_i \to \sum_{i \in I}A_i$.

- Prove that functions map countable sets to countable sets
- $F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$
- Explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$
- Can one construct an algebraic closure of fields like $\mathbb{F}_p(T)$ without Zorn's lemma?
- About a paper of Zermelo
- Does linear ordering need the Axiom of Choice?

(If I remember correctly, it is not known whether “every surjective image of any set $X$ injects into $X$” implies AC.)

- (ZF)subsequence convergent to a limit point of a sequence
- Without the Axiom of Choice, does every infinite field contain a countably infinite subfield?
- What goes wrong when you try to reflect infinitely many formulas?
- Is there a constructive way to exhibit a basis for $\mathbb{R}^\mathbb{N}$?
- Cardinality of a vector space versus the cardinality of its basis
- The comprehension axioms follows from the replacement schema.
- Does the assertion that every two cardinalities are comparable imply the axiom of choice?
- Do the proofes in set theory rely on the semantics of the formulas used in the axioms?
- Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?
- Linearly ordered sets “somewhat similar” to $\mathbb{Q}$

The assertion “If $f\colon X\to A$ is surjective then there is $g\colon A\to X$ injective” is known as **The Partition Principle**.

Indeed it is still open whether or not the partition principle implies the axiom of choice. (It is one of my goals for the foreseeable future, to solve this problem, although it’s a goal I am fully aware I am unlikely to achieve.)

To see that the formulation $\bigcup A_i\leq\sum A_i$ is equivalent to the Partition Principle, one direction is trivial (clearly PP implies this thing), and in the other direction suppose $f\colon A\to B$ is surjective, take $I=A$ and $X_a=\{f(a)\}$. We have that $\bigcup X_a=B$ while $\sum X_a=f$ which is naturally in bijection with $A$. Requiring, if so, an injection from the union into the sum implies an injection from $B$ back into $A$, as wanted.

The assertion “every surjection splits” indeed implies the axiom of choice, that is to say the map $(i,a)\mapsto i$ is a surjection then there is an injection which splits it, is exactly asserting a choice function.

But requiring only the existence of at least one injection whenever there is a surjection is not enough to ensure every surjection splits, at least not as we know it. It goes even further, I don’t think that we know that many counterexamples too.

The only sets I am aware of that have the property that whenever $f\colon X\to Y$ is surjective then there is $g\colon Y\to X$ injective are sets whose surjections split, and those are strong $\kappa$-amorphous sets, that is sets that every partition into two sets implies one is smaller than $\kappa$; and that every partition into no less than $\kappa$ parts is almost all singletons (almost all: meaning all but $<\kappa$ many parts).

For example, if $A$ is strongly amorphous (which means that it cannot be split into two infinite sets, and every infinite partition is all but finitely many singletons) then every surjection from $A$ splits.

I am not aware of any sets other than those which have the partition principle in ZF, or even consistently having partition properties.

- euclidean geometry books…
- How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt{2+\sqrt3}}{\sqrt2}$
- Examples of fields of characteristic 0?
- Why does specifying an interval for a function make the function odd or even?
- Is the Lie Algebra of a connected abelian group abelian?
- Principal ideal of an integrally closed domain
- relation between $W^{1,\infty}$ and $C^{0,1}$
- Solution of nonlinear ODE: $x= yy'-(y')^2$
- Slope of the tangent line, Calculus
- Let $f:\mathbb{R} \to \mathbb{R}$ be a function, so that $f(x)=\lfloor{2\lceil{\frac x2}\rceil}+\frac 12 \rfloor$ for every $x \in \mathbb{R}$.
- Riesel and Gohl's Approximation of the Modified Prime Counting Function, $\pi_{0}$
- Prove that $\prod_{k=1}^{\infty} \big\{(1+\frac1{k})^{k+\frac1{2}}\big/e\big\} = \dfrac{e}{\sqrt{2\pi}}$
- Mathematical literature to lose yourself in
- Effect the zero vector has on the dimension of affine hulls and linear hulls
- Identifying the series $\sum\limits_{k=-\infty}^{\infty} 2^k x^{2^k}$