Nobel laureate Hans Bethe was a friend of mathematician-physicist John von Neumann, and he once said:
"I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man"
and
"[von Neumann's] brain indicated a new species, an evolution beyond man"...
Summary: What did Omnès mean with this?
I found an old article by Roland Omnès which analyzes the EPR paradox and offers a solution to it (https://www.sciencedirect.com/science/article/abs/pii/0375960189900182).
At some point, the article says:
"Some macroscopic systems do not satisfy the...
Hi
I am currently trying to learn about smooth manifolds (Whitneys embedding theorem and Stokes theorem are core in the course I am taking). However, progress for me is slow. I remember that integration theory and probability became a lot easier for me after I learned some measure theory. This...
First, I want to be pedantic here and underline the distinction between a set (in the model, or interpretation) and a sentence (in the theory) which is fulfilled by that set, and also constant symbols (in the theory) versus constants (in the universe of the model)
Given that, I would like to...
I am fascinated by Einstein’s quote that the most unbelievable aspect of the universe was that it was intelligible. So my question is does anyone know whether it is so unlikely as to be absurd to suppose that random unguided processes could produce a rational brain in man in as little as 3...
Homework Statement
Let ##n## be odd and a composite number, prove that all of its prime is at most ##\frac{n}{3} ##
Homework Equations
Some theorems might help?
Any ##n>1## must have a prime factor
if n is composite then there is a prime ##p<√n## such that ##p|n##
The Attempt at a Solution...
This thread is a shoot-off from this thread.
Assuming some relation between human language and logical reasoning, how would this relate, let's say, to the arrival and evolution of logic in human and animal psychology?
I would presume that some logic, for example classical logic, can be more or...
I have received (unasked) a digital edition of "Laws of Form" (1969) by G. Spencer-Brown; I have glanced at it, and also at the Wikipedia article https://en.wikipedia.org/wiki/Laws_of_Form. OK, another logical system; logical journals (e.g. by ASL) are full of them, and I am not sure whether...
So I am very, very new to logic based questions, and in the past have solved some with relative ease but whilest scrolling through the next to find some example stuff I came across a web site that gives a question and hints to the question if stuck, so I thought this would be good practice. But...
I can't remember where this subject came forward in my topics, so I created a new topic.
Suppose that:
If X happens, we observe A, and:
If Y happens, we observe B.
Could we then say:
If we observe A, Y did not happen, and:
If we observe B, X did not happen,
if we apply this to...
It seems like we keep chasing "reality", and by "reality" I think Physicists would mean the apparent rules of quantum physics which we hope would (if applied) lead to all the apparent known rules of macro-physics.
However ... It seems like we have had to create a few things to do that:
1. Ideas...
I just saw this proof.... And, I don't understand why this is true. How am I supposed to think about problems like this one?
Edit:
Here's another one:
The only steps here I understand are 1 to 5. I don't know why 6 and 7 are true.
Homework Statement
I refer to part G of this little problem:
I don't see how to arrive at any conclusion, especially when I can't even see how ##z## comes into play. Assistance in interpreting the problem is appreciated!
Homework Equations
The Attempt at a Solution
I know that the...
Homework Statement
Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
I found an article written by physicist George Ellis that confused me a little.
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.498.4569&rep=rep1&type=pdf
At some part, he says:
3.2 Non-uniqueness:
Possibilities There is non-uniqueness at both steps. Stating “all that is possible...
It seems to me that, despite several systems developed for higher-order logics, almost all the attention in Logic is devoted to first-order. I understand that higher-order logics have some drawbacks, such as the compactness theorem and the Löwenheim-Skolem theorems and other such not holding in...
Gödel's incompleteness theorem only applies to logical languages with countable alphabets. So it does not rule out the possibility that one might be able to prove 'everything' in a language with an uncountable infinite alphabet.
Is that a loophole in Godel's Incompleteness Theorem?
Doesn't...
Hello all, I have only seen this paper brought up here once before based on the search function 2 years ago, and the thread devolved into something off topic within the first page.
I am asking in reference to this paper:
https://arxiv.org/pdf/1604.07422.pdf
Which claims to show that single...
I learned something new today: the “Axiom of Dependent Choice”:
The axiom can be stated as follows: For every nonempty set ##X## and every entire binary relation ##R## on ##X##, there exists a sequence ##(x_n)_{ n \in \mathbb{N} }## in ##X## such that ##x_nRx_{n+1}## for all ##n \in...
Homework Statement
Need to demonstrate this proposition: (P→Q)↔[(P ∨ Q)↔Q] . My textbook use truth tables, but I'd like to do without it. It asks me if it's always truth
The Attempt at a Solution
Im unable to demonstrate the Tautology and obtain (¬Q) as solution.
I start by facing the right...
I want to do tricky physics and maths questions that require only A-level knowledge of maths and physics to solve. Questions that don't have a straightforward answer and will help me to develop my logic and problem-solving skills. I'm looking books, online resources and etc.
So , what I was wondering about was a slight difference in notation, for which I am not certain if correct (mine, in particular.).
The induction axiom says: If M is a subset of ℕ, and if holds that:
a)1∈M
b)(∨n∈ℕ)(n∈M→s(n)∈M)
then M=ℕ.
Now my question is: why do we write (∨n∈ℕ)(n∈M→s(n)∈M)...
1. Homework Statement
"Use the truth table method in Boole to determine whether the conclusion is a tautological consequence of the premises."
(Tet(a) ^ Small(a)) v Small(b)
------
Small(a) v Small(b)
2. Homework Equations
Taller(claire,max) v Taller(max,claire)
Taller(claire,max)...
So, here's my question. I read somewhere that all universal truths on empty domains are vacuously true, whereas all existential are false. However, if all statements of the form (∀x∈A)(P(x)) , where A is an empty set, are vacuously true, then the statement (∃x∈A)(P(x)) should also be true...
The sentence is : "For all real numbers there exists a natural number that is smaller". That is (∀x∈R)(∃n∈N)n>x. This is what I thought of: we can write this sentence as:"If x is a real number, then there exists a natural number n that satisfies n>x." So how would I make a converse statement...
https://en.wikipedia.org/wiki/Fitch's_paradox_of_knowability It begins by assuming that 'All truths are knowable' and then logically proves that that assumption means 'All truths are already known.' The proof is like this:
Suppose p is a sentence that is an unknown truth; that is, the sentence...
Let me start by saying that this is from a 30 question assessment on Big Oh, Big Theta, and Big Omega. I understood every other question, however, even after being given the correct answer, I do not understand why my answer was wrong for this one. If you could point me in the direction of any...
Good day.
I'm currently in the soldering phase of an RF circuit designed to operate an L293D motor chip. On the breadboard I managed to get it to work, but after soldering (and checking for shorts) I am not getting proper high/low signals sent through the encoder/decoder from 2 2p2t switches to...
I'm reading Velleman's book titled "How to Prove it" and I'm very confused when I'm reading about conditional statements. I understand that there exists some issue with the conditional connective and I accept that because that's the cost of espousing a truth-functional view. I came here to ask...
1. Suppose P(x) and Q(x) are propositional functions and D is their domain.
Let A = {x ∈ D: P(x) is true}, B = {x ∈ D: Q(x) is true}
(a) Give an example for a domain D and functions P(x) and Q(x) such that A∩B = {}
(b) Give an example for a domain D and functions P(x) and Q(x) such that A ⊆ B...