Intereting Posts

Understanding branch cuts by manually choosing the branch cuts of a function
What's special about $C^\infty$ functions?
$X \to Y$ flat $\Rightarrow$ the image of a closed point is also a closed point?
Asymptotics of $1^n + 2^{n-1} + 3^{n-2} +\cdots + (n-1)^2 + n^1$
What is the the $n$ times iteration of $f(x)=\frac{x}{\sqrt{1+x^2}}$?
$U_q$ Quantum group and the four variables: $E, F, K, K^{-1}$
Coefficient of $n$th cyclotomic polynomial equals $-\mu(n)$
Ellipses given focus and two points
Proving that if $f>0$ and $\int_E f =0$, then $E$ has measure $0$
Counterexamples to “Naive Induction”
Higher dimensional analogues of the argument principle?
Non-unital rings: a few examples
Prove that the number of jump discontinuities is countable for any function
Measure of a set in $$
Understanding the Musical Isomorphisms in Vector Spaces

Why is (1) a copy of $\mathbb{N}$ “followed by” a copy of $\mathbb{Z}$ **not** a (non-standard) model of arithmetic, neither (2) a copy of $\mathbb{N}$ followed by an infinite sequence of copies of $\mathbb{Z}$, but (3) a copy of $\mathbb{N}$ followed by infinitely many densely ordered copies of $\mathbb{Z}$ **is**? (see the Wikipedia entry on non-standard models)

Can this intuitively be seen, or be explained in laymans terms?

The axioms concerning the successor function hold in all of these (pseudo-)models, don’t they?

- nth roots of negative numbers
- The last digit of $n^5-n$
- Sum of the sum of the sum of the first $n$ natural numbers
- Continuum between addition, multiplication and exponentiation?
- Given dividend and divisor, can we know the length of nonrepeating part and repeating part?
- How do I find two integers - $x$ and $y$ - whose values satisfy the expression $x^2 + y^2 = z$, where $z$ is a perfect square?

But the induction axiom really puzzles me! Naively, it can be interpreted as describing essentially an infinite row of dominoes: knocking over the first will let fall all of them. How can this be understood in the non-standard model (3), where there is no immediate “contact” between the building blocks? What’s the “true” interpretation of induction then? And why does it work in (3) but not in (2) or (1)?

- How much faster is the Trachtenberg system?
- Does elementary embedding exist between two elementary equivalent structures?
- Missing dollar problem
- Will Division by Zero be Defined Eventually?
- A quick question about categoricity in model theory
- Converting decimal(base 10) numbers to binary by repeatedly dividing by 2
- Combination of quadratic and arithmetic series
- Convergents of square root of 2
- Do the axioms of set theory actually define the notion of a set?
- Topological spaces as model-theoretic structures — definitions?

${\mathbb N}$ is more than an ordered set with successor. You have addition (and multiplication) as well. Let $M$ be a nonstandard model; we can identify its beginning with ${\mathbb N}$. Given $n\in M$ let $[n]$ be the collection of all $m$ that are at finite distance from $n$. Note $[n]={\mathbb N}$ if $n$ is finite, and otherwise $[n]$ has the same order type as ${\mathbb Z}$, because every number has a successor, and every number other than zero has a predecessor.

If $n$ is an infinite nonstandard number, then $n+n$ is also infinite but, moreover, it is infinitely away from $n$, so just from $M$ being nonstandard we deduce that there is no largest copy of ${\mathbb Z}$ in the order of $M$.

Also, either $n$ or $n+1$ is even, so there is an $m$ such that $m+m=n$ or $m+m=n+1$. This $m$ is infinite, so the copy $[m]$ of ${\mathbb Z}$ is before the copy $[n]$. This shows that there is no first copy of ${\mathbb Z}$ in $M$.

Finally, given $n<k$ and infinitely apart, $n+k$ or $n+k+1$ is even, and if $l$ is its half, then $n<l<k$, and $l$ is infinitely apart from both. This shows that between any two copies of ${\mathbb Z}$ we have another.

We have shown that, removing the original ${\mathbb N}$, and taking the quotient that identifies all elements in a class $[n]$ as a single point, we are left with a linear order that is dense in itself and has no end points. If $M$ is countable, this order is ${\mathbb Q}$. Otherwise, it is even more complicated.

- Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$
- An example of a (necessarily non-Noetherian) ring $R$ such that $\dim R>\dim R+1$
- Global sections of $\mathcal{O}(-1)$ and $\mathcal{O}(1)$, understanding structure sheaves and twisting.
- Continuous function preserving rational difference.
- How is the general solution for algebraic equations of degree five formulated?
- Show that the equation $y^2 = x^3 + 7$ has no integral solutions.
- Prove that $C(r, r) + C(r+1, r) +\dotsb+C(n, r) = C(n+1, r+1)$ using combinatoric arguments.
- All $11$ Other Forms for the Chudnovsky Algorithm
- Given a rational number $x$ and $x^2 < 2$, is there a general way to find another rational number $y$ that such that $x^2<y^2<2$?
- How is this proof correct in regard to a $k$-subalgebra (Eisenbud)?
- Show that a function from a Riemann Surface $g:Y\to\mathbb{C}$ is holomorphic iff its composition with a proper holomorphic map is holomorphic.
- Show that unit circle is compact?
- Summing over General Functions of Primes and an Application to Prime $\zeta$ Function
- Why does a meromorphic function in the (extended) complex plane have finitely many poles?
- Series which are not Fourier Series