Highest scored questions
159,035 questions
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$p=4x^2+27y^2$,with $p$ a prime [closed]
p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ ,
$p=1,9\pmod{20}$.
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votes
3
answers
3k
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Fuzzy Logic in Finance
Has fuzzy logic been commercially applied in finance fields and has it been successful ?
I have got knowledge that it has been applied in Algorithmic trading and operational risk, but I want to know ...
- 129
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votes
1
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540
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Measure Theories with a different convention to $\infty\cdot 0 =0$ [closed]
As we all know in a first course in measure theory we define a symbol $\infty$ to satisfy $\infty \cdot 0=0$, but there are more two possible choices for a convention as someone has shown to me; one ...
- 1,594
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1
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809
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Numbers representable as in the famous IMO question number 6 (1988)
The famous problem number 6 of the 1988 International Mathematical Olympiad is about showing that if $a,b$ are non-negative integers such that $\frac{a^2+b^2}{ab+1}$ is an integer, then it is a square ...
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1
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381
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Is it true that there are infinite palindromic primes that when squared give palindromic number? [closed]
Can you prove that there are infinite palindromic primes that when squared give a palindromic number?
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1
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290
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The logarith map as a contraction [closed]
Two Questions:
(1) Under what conditions(if any) can the logarithm map from a point on a Riemannian manifold, $q_1\in Q$, to the Tangent Space $T_{q_0}Q$, locally, be a contraction mapping?
Or more ...
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1
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375
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Opposite complex structure on Kaehler manifold
Let $(M,J)$ be a Kaehler manifold. How can one describe the opposite complex structure? What is the precise definition of the opposite complex structure? Can one describe the opposite complex ...
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3
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362
views
Are rotations generated by translations, scalar multiplications and inversions? [closed]
I read a paper saying the Mobius transformation from $\mathbb{R}^n\cup \infty \to \mathbb{R}^n\cup \infty$is generated by translations, scalar multiplications and inversions $x\to \frac{x}{|x|^2}$. So ...
- 488
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votes
2
answers
196
views
Which self homeomorphisms preserve measure on a torus, apart from affine? [closed]
Which self homeomorphisms preserve measure on a torus, apart from affines? Affine is the composition of rotation and automorphism. Measure is the Lebesgue measure.
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2
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160
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Non-vanishing of this ternary quadratic expression [closed]
I'm dealing with the expression $x^2+y^2+6z^2+8xy+4x+4y−6xz−6yz$. I want to show that this expression is always non-zero whenever $x,y$ and $z$ are positive integers. How does one do this? (Note that ...
- 1,109
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1
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271
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Is this Goldbach conjecture related quantity equal to the number of Goldbach decompositions up to a bounded quantity?
This question is a follow-up to About Goldbach's conjecture and as such deals with the notion of primality radius of a composite integer $n$, that is, a positive integer $r$ such that both $n-r$ ...
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1
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208
views
Conformal map from a 7-sided polyhedron to a square pyramid
I have a right-angled square pyramid, $A$, whose height and base-length are $l$. By 'right-angled', I mean that the apex of $A$ lies vertically above one of the vertices in its base. Now supposed I ...
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1
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363
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Can there be elementary embedding between a universe and a universe inside it?
[EDIT] the prior question (see the second section below) was trivially false, however the intention is to arrange a possible world of such universes, in other words the question is about if it is ...
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1
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267
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Number of ZFC proofs and are meta-proofs valid ZFC proofs [closed]
How many proofs in the language of ZFC are there? I would say countably infinite, since every proof is a finite sequence of symbols over a finite alphabet (e.g. ASCII).
Consider the proof in https://...
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1
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233
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Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$? [closed]
Let $E$ be an elliptic curve over $\mathbb{Q}.$
Is $\sharp E((\mathbb{F}_{p^{2}})/E(\mathbb{F}_{p}))=1$ for almost all primes $p$?
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3
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753
views
Riemann Mapping Theorem in Higher Dimensions for Continuous funcions [closed]
Is there any analogue for Riemann Mapping Theorem(!) in higher dimensions?
Or a much simpler question, is it true that every open subset of $\mathbb{R}^3$ with zero homology in dimensions 1 and 2 is ...
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1
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731
views
Do homeomorphic complements give homotopic knots?
Maybe this question is too trivial for a research site but there are so many notions of equivalence of knots that I am lost in literature. The question that interests me is the following.
A knot is a ...
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1
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677
views
Point me to an attempt to Proove Collatz Conjecture by Substitution and Factor analysis? [closed]
Summary of Question:
Where can I find a discussion of attempting to prove the Collatz Conjecture via substitution and abstract examination?
I've done a lot of reading on the problem, including ...
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1
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277
views
Why surreal numbers cannot be extended further in this way using measure approach?
Basically, a lebesgue measure of dimension $n$ of a set of the same dimension $n$ is $n$-volume, $\lambda_n(S)$.
If the dimension of a set is greater than the dimension of the measure, the measure is ...
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1
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154
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Proving that $P($$\{\text{$a$ and $b$ are co-prime}$ }$)=0$ for $a,b$ following the Uniform distribution over $[n, 2n]$ as $n \rightarrow \infty$
I have been working on a problem concerning the "line of sight" from a fixed integer co-ordinate — let's say $(0,0)$ — to a variable co-ordinate $(a,b)$. Having a line of sight means that ...
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2
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496
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Not triangulirazable over $\Bbb R$ implies diagonalizable over $\Bbb C$ [closed]
Let $A$ be a real matrix. Suppose $A$ is not triangulirazable over $\Bbb R$ then $A$ is diagonalizable over $\Bbb C$.
My proof: Since $A$ is not triangularizable over $\Bbb R$ it has a complex ...
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1
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232
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Function satisfying $f(x)^{f^{-1}(x)}=x^2$ with $ f^{-1}$ is a compositional inverse of $f$ and $f:\mathbb{R+}\to \mathbb{R+}$?
Let $f$ be a function such that :$f:\mathbb{R+}\to \mathbb{R+}$ and $f^{-1}$ is a compositional inverse of $f$ , I have tried to find solution of the following functional $f(x)^{f^{-1}(x)}=x^2$, I ...
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votes
1
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479
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PCP theorem to check hard proofs [closed]
Is it technically possible to check formidable proofs like Mochizuki's using PCP theorem before mathematicians spend time in understanding the mechanics of the proof? If so why have mathematicians not ...
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1
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418
views
Holomorphic line bundles with torsion Chern class [closed]
Suppose you have a holomorphic line bundle $L$ such that $L^{n}$ is a trivial holomorphic line bundle and the base complex manifold $M$ has no torsion cohomology classes in second degree (i.e. $H^{2}...
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2
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349
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Concepts of criticality in graph theory
A graph $G=(V,E)$ is said to be vertex-critical if removing a vertex $v\in V$ reduces the chromatic number $\chi(\cdot)$. Edge-criticality is defined in a similar manner. Moreover, $G$ is called ...
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3
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338
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Can we decide whenever a function is the derivate of another function in this Language?
Our EXP functions are made in the following way:
Any constant $ \in \Bbb R$ is a EXP
$X \in \Bbb R$ is a EXP
$sin( g(x))$, $cos( g(x))$ are in EXP if $g(x)$ is a EXP
$tan( g(x))$ is a EXP if $g(x)$...
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2
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1k
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Dirichlet's Divisor Function
We know that by Dirichlet's formula for the Divisor function $ \displaystyle \sum\limits_{n \leq x} d(n) = x \log{x} + (2C-1)x + \mathcal{O}(\sqrt{x})$.
What is the best approximation available till ...
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1
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691
views
Why is a partition function of a Topological Conformal Field Theory related to Deligne-Mumford space
I find when I read a paper, Costello" The Gromov-Witten potential associated to TCFT"
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1
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104
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when does $h$ exist?
Let $\zeta(s)$ denote the Reimann zeta function in the critical strip. It is easy to see that $$ \zeta(s) = 0 \Longleftrightarrow \Re(\zeta(s))+\Im(\zeta(s)) = 0 ~~~~ \text{and} ~~~~~~ \Re(\zeta(s)) \...
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2
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318
views
When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]
When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...
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1
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349
views
Weyl tensor of a Riemannian metric $g$
Does Weyl tensor of a Riemannian metric $g$ give information about the conformally-flatness of $g$?
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1
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319
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What is a good formalization of this classic math puzzle? [closed]
Here is a classic math olympiad problem (but this is NOT my question!): Each of the girls A and B tells the teacher a positive integer but neither of them knows the other's number. The teacher writes ...
-3
votes
1
answer
272
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Show that a polynomial of degree 4 is birational equivalent to a polynomial of degree 3 [duplicate]
Suppose that $f_{4}(x)$ is a polynomial of degree 4 with no multiple roots, $C$ is the curve defined by $y^{2}=f_{4}(x)$, I want to show that there is a polynomial $f_{3}(x)$ of degree 3 with no ...
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1
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953
views
combinatoric proof $\sum_{i=0}^{n}(-1)^i\binom{n}{i}\binom{n-i+k-1}{k}=\binom{k-1}{k-n}$ [closed]
I would like help with combinatorial proof ,
not algebraic proof . Thank you for your time
$\sum_{i=0}^{n}(-1)^i\binom{n}{i}\binom{n-i+k-1}{k}=\binom{k-1}{k-n}$
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1
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200
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Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$
The Euler equations are given as $$ \pmb{u}_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$
Where $$u = [u_1,u_2,\ldots u_n]^T$$
Now I want to rewrite these same equations but with a new ...
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votes
2
answers
225
views
Zeroes of linear combination of sines [closed]
Let us consider $$f(z):=\sum\limits_{j=1}^{j=n}a_j\sin(\lambda_jz) $$
where all $a_j$ and $\lambda_j$ (of course, $\lambda_j$ are distinct) are real numbers and $n \in \mathbb{Z},\, n \ge 3$. The ...
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1
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178
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Asymptotic behavior of $\sum_{k=1}^{n}\frac{p_{k+1}}{p_{k+1}-p_k}$
I refer to my previous question Asymptotic behavior of a certain sum of ratios of consecutives primes.
We can split the sum
$$\sum_{k=1}^{n}\frac{p_{k+1}+p_k}{p_{k+1}-p_k}$$
where $p_k$ stands for the ...
- 1,008
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votes
1
answer
245
views
An interesting phenomenon of the analytic continuation of Riemann zeta function [closed]
It is known that
$$\Gamma (s) \zeta (s)=\int_0^{\infty} \frac{x^{s-1}}{e^x-1}dx$$
this function is valid only for $\Re{s}>1$.
However, if we ignore this restriction, and integrate by using
$$\frac{...
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1
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245
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Can this weakening of Polignac's conjecture be proven?
Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the ...
- 6,982
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votes
2
answers
646
views
Maximal degree and chromatic number [closed]
Given a positive integer $n\in \mathbb{N}$, is there a positive integer $k\in{\mathbb N}$ such that
for every finite, simple, undirected graph $G$ with $\Delta(G) = n$ we have $\chi(G) \leq k$
?
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1
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651
views
A question about Yitang Zhang's paper "On the zeros of ζ’(s) near the critical line"
We conclude that, in the case $\sigma = 1/2$ and $\zeta’\left(s\right) \neq 0$,
$$\mathrm{Re}\frac{\eta’}{\eta}\left(s\right) = \sum_{\beta’ \gt 0}{\mathrm{Re}\frac{1}{s - \rho’}} + O(1)$$
It is easy ...
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votes
1
answer
792
views
How many sequences of length n satisfy these constraints? [closed]
I want to count the number of unique sequences of length n with the following constraints.
Each element of the sequence is an integer in $\lbrace 1,2,\dots,n\rbrace$.
Each two adjacent elements of ...
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votes
1
answer
274
views
Isomorphic quotient of a Module over Noetherian commutative algebra [closed]
I have a nice solution to the following problem and I thought of writing a paper about it but beforehand, I wanted to ask the problem here to see if this is an easy problem and if you people can solve ...
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votes
1
answer
446
views
Category of sheaves on the topological space X [closed]
For a topological space $X$, the category of sheaves on $X$ with its values in $Ab$ will form an Abelian Category.
Q1: Is it difficult to prove this?
Next, for the short exact sequence $0 \to F ...
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votes
2
answers
529
views
Uncountable Sets [closed]
I've read that the set of real numbers R is uncountable. It was proved by contradiction. A number x that is not in the f(n) side was constructed.
Ultimately it was said that "x is not f(n) for any n ...
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votes
2
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260
views
On \ell_3 norm in R^2
Let $v,w\in\mathbb{R}^{2}$ and $v\perp w$. Is it true that $\left\Vert v\right\Vert _{3}\leq\left\Vert v+w\right\Vert _{3}$,
in which $\left\Vert \left(x,y\right)\right\Vert _{3}:=\sqrt[3]{\left|x\...
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2
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2k
views
What are the fractal parameters?
There are so many fractal which are not uniquely characterize by some fractal parameters like Fractal dimension, Succolarity, Lacunarity, Morphological entropy. Can you suggest some fractal parameters ...
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1
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74
views
Non-Ramsey function $f:[\omega]^{<\omega}\to\{0,1\}$ [closed]
Let $\newcommand{\o}{\omega}\o$ be the set of non-negative integers, and for any set $X$, let $\newcommand{\oo}{[\o]^{<\o}}X^{<\o}$ denote the collection of all finite subsets of $X$.
What is an ...
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votes
1
answer
201
views
Formula for gaps between primes [closed]
The twin prime conjecture refers to:
$$
\liminf_{n\to \infty}\; p_{n+1} - p_{n} = 2.
$$
By reasoning I arrive at the following simple formula for gaps between primes:
\begin{align}
p_{...
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votes
1
answer
117
views
Can stratification be used to internalize functions on models of $\sf Z$?
Suppose $M$ is a model of $\sf Z +\neg AC$ that is externally bijective to an element $k \in M$. Obviously if $j$ denotes such an external bijection, then it cannot be used in Separation within $M$, ...
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