Intereting Posts

What is the free abelian group on $M \times N$ where $M,N$ are modules.
When $\min \max = \max \min$?
Intuition behind logarithm inequality: $1 – \frac1x \leq \log x \leq x-1$
A high-powered explanation for $\exp U(n)=2\iff n\mid24$?
How much math education was typical in the 18th & 19th century?
How to compute the following integral?
The Join of Two Copies of $S^1$
If $b_n$ is a bounded sequence and $\lim a_n = 0$, show that $\lim(a_nb_n) = 0$
Better Proofs Than Rudin's For The Inverse And Implicit Function Theorems
Unique expression of a polynomial under quotient mapping?
Is is possible to obtain exactly 16 black cells?
Golbach's partitions: is there always one common prime in $G(n)$ and $G(n+6)$ , $n \ge 8$ (or a counterexample)?
how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $ (x,y)$
There is a number of three digits, 1 is never immediate right of 2 is?
Proof that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$

Natural numbers can be represented as

$0=\emptyset$

$1=\{\emptyset\}$

- Axiom schema and the definition of natural numbers
- Given real numbers: define integers?
- How many combinations of $3$ natural numbers are there that add up to $30$?
- Determine the divisibility of a given number without performing full division
- The sum of three consecutive cubes numbers produces 9 multiple
- What is the meaning of set-theoretic notation {}=0 and {{}}=1?

$2=\{\{\emptyset\}\}$

$…$

or as

$0=\emptyset$

$1=\{0\}=0\cup\{0\}$

$2=\{0,1\}=1\cup\{1\}$

$…$

What are the names of these representations?

Aren’t they identical?

What are advantages of second representation?

- Unions and the axiom of choice.
- What are large cardinals for?
- Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals
- axioms of equality
- Set of All Groups
- Principle of Transfinite Induction
- What goes wrong when you try to reflect infinitely many formulas?
- Why do we need to learn Set Theory?
- What is needed to make Euclidean spaces isomorphic as groups?
- Cardinality of Irrational Numbers

These are respectively Zermelo’s and von Neumann’s representations/implementations of the naturals in set theory.

If you just want to reconstruct arithmetic and then classical analysis inside set theory, Zermelo’s representation works just fine and some standard textbooks do things that way.

But once we go beyond, and want to deal with *infinite* ordinals as well as finite ones, then you need a von Neumann-style representation. So many authors use it from the start, even for finite numbers.

- Proving that for infinite $\kappa$, $|^\lambda|=\kappa^\lambda$
- If every compact set is closed, then is the space Hausdorff?
- Defining the determinant of linear transformations as multilinear alternating form
- Ring homomorphisms from $\Bbb Q$ into a ring
- Prove that the complex expression is real
- Isomorphism involving tensor products of homomorphism groups
- expectation of $\int_0^t W_s^2 dW_s $ (integral of square of brownian wrt to brownian)
- Isomorphic Group with $G=(\mathbb Z_{2^\infty}\oplus \frac{\mathbb Q}{\mathbb Z}\oplus \mathbb Q)\otimes_{\mathbb Z}\mathbb Q $
- Maximum area of a square in a triangle
- System with infinite number of axioms
- Any nonabelian group of order $6$ is isomorphic to $S_3$?
- For bounded real valued function $f$ show that $\omega_f$ is upper continuous
- Proving $\binom{m}{n} + \binom{m}{n-1} = \binom{m+1}{n}$ algebraically
- Can $\frac{SO(5)}{N}= SU(2) \times SO(2)$ be true?
- Exponent of a number is a square root?