The necessity quantifier (aka Provability quantifier, or ~◊~, or Belief, or.... instead of the usual square I will be lazy and call it "N") is often allowed to be repeated as many (finite) times as one wishes, so NNNNNNψ is OK. Is it possible to somehow include into the axioms some restriction...
I know that the number systems we use are typically constructed from axiomatic set theory, and overall our choices along the way seam to have been largely informed by practical consideration (e.g. to resolve ambiguities, or do away with limitations).
Today I randomly started to think deeper...
Why can't we prove euclids fifth postulate
What's wrong in this proof:
why can't we prove that there is only one line which passes through a single point which is parallel to a line.
If we can prove that two lines are parallel by proving that the alternate angles of a transverse passing...
I will say that this question is coming from a lack of explanation in a classroom, however this particular proof is not homework and is just explanation over a proof that was discussed briefly in class, so I didn't think it belong in the homework section. I'm also not certain it belongs in the...
Hello everyone. I wanted to prove the following theorem, using the axioms of Peano.
Let ##a,b,c \in \mathbb{N}##. If ##ac = bc##, then ##a = b##.
I thought, this was a pretty straightforward proof, but I think I might be doing something wrong.
Proof:
Let ##G := \{c \in \mathbb{N}|## if ##a,b...
If the axiom of induction was extended to include imaginary numbers, what effect would this have?
The axiom of induction currently only applies to integers. If this axiom and/or the well ordering principle was extended to include imaginary numbers, would this cause any currently true statements...
I am not a mathematician but, as such, I think I have a pretty good background in mathematics. I have a good understanding and experience with calculus, differential equations, linear algebra, and probability theory. I also have interest in abstract algebra concepts, though I wouldn't say I am...
I am looking for a book that starts at the standard ZFC axioms and progresses to the point where some recognisable non-trivial mathematical statement is proved. By recognisable I mean something that you may encounter in school/early university level and is not purely set-theoretical (e.g...
Homework Statement
Let V be the set of all ordered pairs of real numbers. Suppose we define addition and scalar
multiplication of elements of V in an unusual way so that when
u=(x1, y1), v=(x2, y2) and k∈ℝ
u+v= (x1⋅x2, y1+y2) and
k⋅u=(x1/k, y1/k)
Show detailed calculations of one case...
I was looking for explanation why x^0=1.
thread
https://www.physicsforums.com/threads/my-simple-proof-of-x-0-1.172073/
is locked and i did't found solution in it from axioms. People using exp(x) and log(x) and xa-a=xax-a as given.
If you have xa-a=xax-a for a∈ℤ and x∈ℕ+ then there is no...