## 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.