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:

Proof

So far what I’ve accomplished:

My approach

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?

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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?

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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?