Elementary Set Theory – How to call subsets of $ mathbf Z $?

Consider an integer $ n $ Where $ a leqslant n leqslant b, a, b in mathbf R, b gt a $, Then we can call it

$$ n in mathbf X, text {where} mathbf X: = {, k | k in mathbf Z land a leqslant n leqslant b , }. $$
But what if you don't want to define an additional quantity?

We know that $ (, a, b ,) $ is defined as

$$ (, a, b ,) = {x | x in mathbf R land a leqslant x leqslant b } forall a, b in mathbf R, b gt a. $$

Then we can designate $ n $ (defined above) as

$$ n in mathbf Z cap (, a, b ,) forall a, b in mathbf R, b gt a ? $$
Is this notation correct? I haven't seen it anywhere.

Elementary Number Theory – Prove that $ exists {c} in Bbb {N}: forall {k} in Bbb {Z} _ + $, $ mathrm {T} (2010, k) lt mathrm {T} (2, k + c) $ for the defined function $ mathrm {T} $.

given $ {v, k} in Bbb {Z} _ + $, we define $ mathrm {T} (v, 1): = v $ and $ mathrm {T} (v, k + 1): = v ^ { mathrm {T} (v, k)} $, Prove that $ exists {c} in Bbb {N}: forall {k} in Bbb {Z} _ + $. $ mathrm {T} (2010, k) lt mathrm {T} (2, k + c) $,

Couldn't figure out how to deal with this problem by induction, but here are my progress:
$ mathrm {P} (1) $: $ mathrm {T} (2010.1) = 2010 land mathrm {T} (2.1 + c) = 2 ^ { mathrm {T} (2, c)} $, but $ forall {n} geq {11} $, we have $ {2} ^ n gt {2010} also {c} geq {3} $; accept $ mathrm {P} (n) $ true; in the $ mathrm {P} (n + 1) $ we want to show that $ mathrm {T} (2010, n + 1) lt mathrm {T} (2, n + 1 + c) iff {2010} ^ { mathrm {T} (2010, n)} lt { 2} ^ { mathrm {T} (2, n + c)} $ where from the inductive hypothesis, $ mathrm {T} (2010, n) lt mathrm {T} (2, n + c) $but then I couldn't reach another bridge between $ mathrm {P} (1) $ and $ mathrm {P} (n + 1) $,

Set theory – elementary self-embedding conservative towards ZFC

Question: Is the following theory conservative towards ZFC? (If not, what is your strength?)
Language: $ ∈ $. $ j $ (unary function symbol)
axioms:
1. ZFC (without separation and replacement for formulas using $ j $).
2. (scheme) $ j $ is a non-trivial elementary embedding $ (V, ∈) → (V, ∈) $,
3. (scheme) $ S∩φ ≡ {s∈S: φ (s) } $ exists whenever $ φ $ is a formula with parameters and a free variable and quantity $ S $ can be defined from a quantity $ T $ With $ T = j (T) $,

Note that (3) implies the existence of $ S & # 39; ∩φ $ for all $ S & # 39; $ With $ | S & # 39; | ≤ | S | $ for some $ S $ as above. Thus, (3) simply confirms the complete separation scheme for quantities that are not too large, including all definable (allowable) $ j $) puts.

background

An important phenomenon in mathematical logic is that if we expand a theory with new symbols (and sometimes new types) and add meaningful new axioms, the new theory will sometimes be conservative of the original; and such correspondences often provide important structural insights into both theories. A relevant special case here is that the new theory is apparently stronger and covers part of the structure of a stronger theory, but a key ingredient is missing.

Taking various sensible formalizations (which imply substitutes for $ j $Formulas), existence of a non-trivial elementary embedding $ V → M $ ($ M $ transitive) is synonymous with the existence of a measurable cardinal, and $ V = M $ is not compatible with ZFC.

However, (1) + (2) is conservative towards ZFC: add Skolem functions for $ V $ and add $ ω $ Constant symbols for imperceptible atomic numbers, use the compactness to find a model, take the Skolem torso of the imperceptible elements and add a non-trivial order-preserving injection between the imperceptible elements, which then relates to the desired one elementary embedding stretches.

Such a model (see above) may be unfounded, but we can try to do well in small quantities. Any countable ZFC model $ M $ has a nontrivial elementary end extension $ N $; out of elementary, $ N $ is also a top extension that is for everyone $ α∈ mathrm {Ord} ^ M $. $ V_α ^ M $ and $ V_α ^ N $ have the same elements. And maybe a way to integrate $ j $ There will be a positive answer to the question.

Also under (1) – (3), $ {s: s = j (s) } $ (under & # 39; ∈ & # 39;) is an actual elementary substructure of $ (V, ∈) $, but it doesn't exist as a set unless its cardinality is $ n $– great for everyone $ n $,

That being said, there is a rich hierarchy theory based on how & # 39; close & # 39; $ j $ is to $ V $, For example (see this question) if (3) is replaced by the axiom of the critical point (i.e. the smallest atomic number that has been shifted) $ j $ exists), the resulting theory is conservative towards ZFC + {there is $ n $-inffable cardinal}$ _ {n∈ℕ} $,

Partitioning – Elementary operating system hard drive not recognized

I have a single boot option for the basic operating system. I have both a hard drive and an SSD. I have booted this operating system in SSD. I am now in the operating system but cannot find my hard drive. There is no hard drive option in the> Application> Files> Devices tab. It only shows 128 GB SSD.Cannot Find hard disk

What's the problem? Lead me.

elementary set theory – use of the implicit symbol in evidence

Is it valid to use the implicit symbol ($ implies $) or the if and only if symbol ($$ iff $$) to replace assumptions in mathematical evidence in writing? For example, if I want to prove

If A subseteq B, then A union B = B
Can I write my proof as follows:
Let A be a subset of B.
$$
x in A union B
iff x in A or x in B
implies x in B or x in B ( since x in A x in B implies)
iff x in B
also A union B subseteq B
x in B
implies x in B or x in A
iff x in A union B
also B subseteq A union B
therefore A union B = B
$$

Elementary theorem – About a certain proof of the countability of $ mathbb {Q} $

I came across this proof of the countability of $ mathbb {Q} $ in a textbook:

Any rational can be expressed as a fraction $ a / b $, Where $ a $ and $ b $ are whole numbers. We know the set $ {(a, b): a, b in mathbb {Z} } $ is countable $ mathbb {Q} $ is countable.

My question is: Clearly, every rational can be clearly expressed in lowest terms as a fraction $ a / b $, However, each fraction can be expressed in a variety of different ways that are included in the set $ S = {(a, b): a, b in mathbb {Z} } $specially $ (na) / (nb) $, Where $ n in mathbb {Z} $, So we only know from that $ | mathbb {Q} | leq | S | $and not that $ | mathbb {Q} | = | S | $, For this reason, does this mean that we start from CH?

* As a side note: I am not wondering whether the rations can be counted or not, I am absolutely certain that they are. I'm just asking if this evidence accepts CH or not.

Thank you in advance for your help.

elementary number theory – possible values ​​of expression

Real numbers $ x, y, z $ are selected so that $ frac {1} {| x ^ 2 + 2yz |} $. $ frac {1} {| y ^ 2 + 2xz |} $. $ frac {1} {| z ^ 2 + 2xy |} $ are sides of a triangle (they fulfill the triangle inequality). Determine the possible values ​​of the expression $ xy + xz + yz $,

It is easy to prove that all positive and negative real numbers can be expressed. How about $ 0 $? Any help is appreciated.

Elementary topology of surfaces – MathOverflow

To let $ S $ be a compactly connected, orientable, bordered surface of the genus $ g $ With $ n $ Holes (a hole
is a component of the boundary that is homeomorphic to a circle). Consider
a cell disassembly (the closure of each cell is a closed disc of the same dimension as the cell) with $ f $ faces, $ e_i $ Inner edges, $ e_b $ Boundary edges $ v_i $ inner key points and $ v_b $ Boundary vertices. Is there a function $ F $ so that $ g = F (f, e_i, e_b, v_i, v_b) $?

The Euler property gives me $ 2g + n $and I want to relax $ g $ and $ n $
separate from $ f, e_i, e_b, v_i, v_b $, A counterexample would be two non-homeomorphic surfaces for which the five numbers apply $ f, e_i, e_b, v_i, v_b $ are the same.

elementary number theory – using the pigeonhole principle as proof

Suppose I have a set of numbers {a1 .. an + 1} that contain n + 1 numbers, so that
1 <= a1 <= a2 … an + 1 <= 2n

How do I use the pigeonhole principle to prove that there are at least two numbers where one divides the other?

I can see that when two numbers are equal, they obviously split. If they are all different, I would need n + 1 numbers.

I tried to think of each number as ai = 2 ^ ki * bi, where bi is an odd number.
I notice that for every set {1 … 2n} n there are odd numbers. I'm not sure where to go from here or how to translate it into a mathematical proof.

Any explanation would be great!

Elementary Theory – How do I refer to the only element in a singleton sentence?

You can rotate every element $ n $ into a singleton phrase by adding curly braces, $ {n } $, Is there an inverse to this operation, so if I know that the set is a singleton set, I can easily reference its element?

Suppose I have the set $ A = {5,6,7,8 } $and then a process that iteratively removes all elements except for one element of $ A $, and then I want to see what 10 plus is the resulting element, how could I write that?

The only way I can think of is to reference the first element by its index, e.g. $ 10 + A_1 $, Is there another way to do this? In my case the elements of $ A $ If there are already indices, I would have to index eg an index $ 10 + x_ {A_1} $, Which is not the worst thing in the world, I just wondered if there is a better way.