Intereting Posts

Elementary proof that $3$ is a primitive root of a Fermat prime?
Arrangements of MISSISSIPPI with all S's and P's separated
An attempt of catching the where-abouts of “ Mysterious group $Ш$ ”
Asymptotics of the lower approximation of a pair of natural numbers by a coprime pair
Prove that $C_n < 4n^2$ for all n greater than or equal to 1
Expanding problem solving skill
Prove using contour integration that $\int_0^\infty \frac{\log x}{x^3-1}\operatorname d\!x=\frac{4\pi^2}{27}$
Prove that series converge
If $x\in \left(0,\frac{\pi}{4}\right)$ then $\frac{\cos x}{(\sin^2 x)(\cos x-\sin x)}>8$
What is the intuition behind the name “Flat modules”?
How to calculate the expected value when betting on which pairings will be selected
How many 90 ball bingo cards are there?
Let A be a non-empty set, and p an equivalence relation on A . Let a , b be an element of A . Prove that = is equivalent to apb
An inequality about sequences in a $\sigma$-algebra
Does $a_n$ converges if and only if $a_{2n},a_{3n},a_{2n-1}$ converge?

A certain section of the chapter “The Axiom of Choice” on “Naive Set Theory” got me confused:

“[…] The assertion is that a set is infinite if and only if it is equivalent to a proper subset of itself. The “if” we already know; it says merely that a finite set cannot be equivalent to a proper subset. To prove the “only if,” suppose that $X$ is infinite, and let $v$ be a one-to-one correspondence from $\omega$ into $X$. If $x$ is in the range of $v$, say $x=v(n)$, write $h(x)=v(n^{+})$; if $x$ is not in the range of $v$, write $h(x)=x$. It is easy to verify that $h$ is a one-to-one correspondence from $X$ into itself. Since the range of $h$ is a proper subset of $X$ (it does not contain $v(0)$), the proof of the corollary is complete. The assertion of the corollary was used by Dedekind as the very definition of infinity.”

(The corollary would follow from “every infinite set has a subset equivalent to $\omega$”.)

- Why continuum function isn't strictly increasing?
- Sum of two countably infinite sets
- Question related with partial order - finite set - minimal element
- Show that $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$
- There is a well ordering of the class of all finite sequences of ordinals
- Proving with diagonal lemma

The question is: how can $h$ be a one-to-one correspondence from $X$ into itself if its range does not contain an element (namely, $v(0)$) that is in $X$?

- opposite of disjoint
- Does there exist a bijection between empty sets?
- What is the number of bijections between two multisets?
- Proving : Every infinite subset of countable set is countable
- Prove the principle of mathematical induction in $\sf ZFC $
- Namesake of Cantor's diagonal argument
- If $g \circ f$ is surjective, show that $f$ does not have to be surjective?
- Is the axiom of choice really all that important?
- Proving Cantor's theorem
- Why do we use both sets and predicates?

Note that the word “equivalent” requires context, literally it would just be interpreted as “satisfying some equivalence relation”, but what the nature of this relation is not automatically understood.

In the context of cardinality it means to have a bijection. Namely an infinite set has a bijection with a proper subset of itself. Of course this requires the axiom of choice (or rather a small fragment of it) to hold. This characterization is known as **Dedekind-infinite**. Without the axiom of choice it is consistent that there are sets which are infinite (in the sense that they are not with bijection with any finite ordinal), but they are not Dedekind-infinite.

For example note that $f(n)=n+1$ is a bijection from $\omega$ into $\omega\setminus\{0\}$. It shows that $\omega$ is a Dedekind-infinite set. For the more general case see my answer for Equivalent characterisations of Dedekind-finite proof.

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- Counting all possible legal board states in Quoridor
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- Quadratic Variation of Brownian Motion
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- If there is a branch of $\sqrt{z}$ on an open set $U$ with $0 \notin U,$ then there is also a branch of $arg$ $z.$
- Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$.
- Equality in Cauchy-Schwarz Inequality implies linear dependence.
- Why do we negate the imaginary part when conjugating?
- Number of monomials of certain degree
- Questions on symmetric matrices