It seems to me that any formula in the language of first-order arithmetic which has only bounded quantifiers can be written as a formula without any quantifiers. For instance, “There exists an n < 1000 such that P(n)” can be written as “P(1) or P(2) or … or P(999)”, and “for all n < 100, […]

Statement of L’Hospital’s Rule Let $F$ be an ordered field. L’Hospital’s Rule. Let $f$ and $g$ be $F$-valued functions defined on an open interval $I$ in $F$. Let $c$ be an endpoint of $I$. Note $c$ may be a finite number or one of the symbols $-\infty$,$+\infty$. Suppose $f'(x)$ and $g'(x)$ are defined everywhere on […]

Is most of mathematics independent of set theory? Reading this quote by Noah Schweber: most of the time in the mathematical literature, we’re not even dealing with sets! it seems that the answer to my question is “yes”. But why? When I read in the mathematical literature, sets appear everywhere – we need them in […]

As a mathematician interested in novel applications I am trying to gain a deeper understanding of (the non-constructiveness of) Gödel’s Completeness Theorem and have recently studying two texts: Mathematical Logic for Mathematicians (by Y.Manin) and the book on Reverse Mathematics by Simpson [2009]. Having read Mendelsohn and Crossley (and other works) I am actually surprised […]

Intereting Posts

If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?
If $n = m^3 – m$ for some integer $m$, then $n$ is a multiple of $6$
Is it possible to generate truly random numbers using a computer?
Use mathematical induction to prove that any integer $n\ge2$ is either a prime or a product of primes.
Problem with my floor…
Why is the expected number coin tosses to get $HTH$ is $10$?
Are there any interesting semigroups that aren't monoids?
proving that this ideal is radical or the generator is irreducible
Finding a limit using change of variable- how come it works?
Prove $p^2=p$ and $qp=0$
Proving $\pi^3 \gt 31$
Upper bound number of distinct prime factors
Find the number of permutation in $S_6$ which are conjugate to $\sigma$
Why don't we have an isomorphism between $R$ and $ R]$?
Complement of a set and inverse image.