## lo.logic – Reference on a corollary for applying Deduction Theorem multiple times in FOL

This question is from “Introduction to Mathematical Logic” by Elliot Mendelson , forth edition , page 75.

In page 75 of the book , there is a corollary in a paragraph that can be used to apply the Deduction Theorem multiple times in a row.

The new proof of $$Gamma vdash mathscr B to mathscr C$$ (in Proposition 2.4 $$Gamma vdash mathscr C$$ )
involves an application of Gen to a wf depending upon a wf $$mathscr E$$ of $$Gamma$$ only if
there is an application of Gen in the given proof of $$Gamma, mathscr B vdash mathscr C$$ that involves
the same quantified variable and is applied to a wf that depends upon $$mathscr E$$.

But there seems to be no justification/proof about this corollary.I searched online but non of them seem to mention this corollary.Are there any references about this I can read or is it just too trivial?

Here is an failed attempt of mine for justifying this corollary.

1.Technique for applying deduction multiple times in Mendelson Logic.

## linear algebra – How to prove this deduction?

When I was working with mathematics, I encountered with a problem.

For a given $$A,Bin mathcal{R}$$ and $$Ninmathcal{N}$$, I have to solve the equations below (where $$U_iinleft{0,1right}$$):

$$A = frac{U_1}{1}+frac{U_2}{2}+frac{U_3}{3}+cdots+frac{U_N}{N}$$

$$B = frac{U_1}{N}+frac{U_2}{N-1}+frac{U_3}{N-2}+cdots+frac{U_N}{1}$$

From my point of view, I thought it has a unique solution for each $$U_i$$. However, I do not know how to prove it.

## Formal Logic – Natural deduction: Problem with assumptions about exists-negation

I’m stuck on how to progress with this proof, despite I have tried, I cannot see the next move.

Given this proof without predicate:

So far what I’ve accomplished:

My idea is, as I can’t see any other option using `(-(Sv(P->Q))` as the first assumption in order to introduce a conditional, so the assumption must end in `P ^ -Q ^ -S`. As you can see I have obatined `-Q and -S` but, how do I proof `P`?

## Natural deduction problem that has no premise

I have the following problem: ⊢A→((A→B)→B)

I can’t seem to be able to start! I am very confused and any tips in the right direction will help! Thanks a lot

## Deduction theorem for the modal mu-calculus

Does the modal mu-calculus have a deduction theorem?

If yes, how is it stated? Does it have the ‘classical’ form (i.e. as in classical propositional logic) or is it more involved?

## general topology – The deduction of two points is topologically indistinguishable by a sub-base

I have the following argument that I would like to check:

Let $$X$$ to be a topological space with a sub-base $${S_ alpha } _ alpha$$ such as $$cup S_ alpha neq X$$. Then each $$x, y in X setminus cup S_ alpha$$ are topologically indistinguishable.

And when I say topologically indistinguishable, I mean that for any open set $$U$$, $$x, y in U$$ or $$x, y notin U$$. My argument is as follows:

Conquer the collection $$mathcal {S}: = {S_ alpha } _ alpha cup {X }$$. Then it generates a base $$mathcal {B}$$ whose elements are finite intersections of sets in $$mathcal {S}$$. For all $$x, y in X setminus cup S_ alpha$$ be separate points, the only open set that contains $$x$$ or $$y$$ must be $$X$$. Let $$B in mathcal {B}$$, that is to say:

$$B = cap_ {i = 1} ^ n S _ { alpha_i}$$

Yes $$x in B$$, then $$x in S _ { alpha_i}$$ for everyone $$i in [n]$$, and so $$S _ { alpha_i} = X$$ for everyone $$i in [n]$$, and $$B = X$$. Since $$mathcal {B}$$ is a base, any open set containing $$x$$ must be all space $$X$$. And the same for $$y$$.

This would imply that if a topological space has a sub-base that is not a cover, then it cannot be a $$T_0$$ space. Is this reasoning valid?

## nt.number theory – A deduction on the asymptotic of \$ | {1 leq n leq x | gcd (n, S (n)) = 1 } | \$, with \$ S (n) \$ the sum of leftovers, and get a sense of another mixing problem

Let $$n geq 1$$ to be an integer, in this post we denote the sum of the remaining functions as $$S (n) = sum_ {k = 1} ^ n n text {mod} k,$$ for example $$S (1) = S (2) = 0 + 0$$ and $$S (5) = 0 + 1 + 2 + 1 + 0 = 4$$. In the literature, there are problems that have been studied related to the condition $$gcd (n, f (n)) = 1$$, for a given arithmetic function $$f (n)$$.

Question. A) Is it possible to provide roughly a cheap terminal for cardinality $$# {1 leq n leq x | gcd (n, S (n)) = 1 }$$
as $$x$$ grows for $$infty$$? B) The sequence of prime numbers $$p$$ those who meet the condition $$pgcd (p, S (p))> 1$$ starts like $$2.11.17.2161, ldots$$. Can you tell us if this sequence has a finite number of terms? Thank you so much.

Just to emphasize, since I ask two questions, it suffices to provide a cheap terminal for A) and appropriate reasoning / heuristics for B), to get an idea of ​​these problems.

Computer evidence and documentation for question B. We have the following script in Pari / GP showing the first terms

`for(n=1, 10000, if(gcd(n,sum(k=1,n,n%k))>1&&isprime(n)==1,print(n)))`

that you can rate on the site Sage cell server choose language GP. Here is the chain `sum(k=1,n,n%k)` is our sum of leftovers $$S (n)$$ with `n%k` coding $$n text {mod} k$$ for each integer $$1 leq k leq n$$.

## first order logic – Duration of a deduction

Suppose we have a sentence whose number of symbols is less than $$n$$. Suppose this sentence is provable in Peano Arithmetic (with the first order induction scheme).

Suppose also that we want to code the deduction by a number less than $$g (n)$$. Are there theorems on the length of this coding?